Citation: Qichun Zhang. Performance enhanced Kalman filter design for non-Gaussian stochastic systems with data-based minimum entropy optimisation[J]. AIMS Electronics and Electrical Engineering, 2019, 3(4): 382-396. doi: 10.3934/ElectrEng.2019.4.382
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In the last years the regularity theory for two phase problems governed by uniformly elliptic equations with distributed sources has reached a considerable level of completeness (see for instance the survey paper [10]) extending the results in the seminal papers [2,4] (for the Laplace operator) and in [17,18] (for concave fully non linear operators) to the inhomogeneous case, through a different approach first introduced in [7].
In particular the papers [15] and [8] provides optimal Lipschitz regularity for viscosity solutions and their free boundary for a large class of fully nonlinear equations.
Existence of a continuous viscosity solution through a Perron method has been established for linear operators in divergence form in [3] (homogeneous case) and in [9] (inhomogeneous case), and for a class of concave operators in [19]. The main aim of this paper is to adapt the Perron method to extend the results of [19] to the inhomogeneous case. Although we are largely inspired by the papers [3] and [9], the presence of a right hand side and the nonlinearity of the governing equation presents several delicate points, significantly in Section 6, which require new arguments.
We now introduce our class of free boundary problems and their weak (or viscosity) solutions.
Let Symn denote the space of n×n symmetric matrices and let F:Symn→R denote a positively homogeneous map of degree one, smooth except at the origin, concave and uniformly elliptic, i.e. such that there exist constants 0<λ≤Λ with
λ‖N‖≤F(M+N)−F(M)≤Λ‖N‖ for every M,N∈Symn with N≥0, |
where ‖M‖=max denotes the (L^2, L^2)-norm of the matrix M.
Let \Omega\subset{\mathbb{R}}^n be a bounded Lipschitz domain and f_1, f_2 \in C(\Omega)\cap L^\infty(\Omega). We consider the following two-phase inhomogeneous free boundary problem (f.b.p. in the sequel).
\begin{cases} F(D^2u^+) = f_1& \text{in }\Omega^+(u): = \{u \gt 0\} \\ F(D^2u^-) = f_2\chi_{\{u \lt 0\}} & \text{in }\Omega^-(u) = \{u\leq 0\}^o\\ u_\nu^+(x) = G(u^-_\nu, x, \nu) & \text{along }{\mathcal{F}}(u): = \partial \{u \gt 0\}\cap \Omega. \end{cases} | (1.1) |
Here \nu = \nu(x) denotes the unit normal to the free boundary {\mathcal{F}} = {\mathcal{F}}(u) at the point x, pointing toward \Omega^+, while the function G(\beta, x, \nu) is Lipschitz continuous, strictly increasing in \beta, and
\inf\limits_{x\in\Omega, |\nu| = 1} G(0, x, \nu) \gt 0. | (1.2) |
Moreover, u_\nu^+ and u^-_\nu denote the normal derivatives in the inward direction to \Omega^+(u) and \Omega^-(u) respectively.
As we said, the main aim of this paper is to adapt Perron's method in order to prove the existence of a weak (viscosity) solution of the above f.b.p., with assigned Dirichlet boundary conditions
For any u continuous in \Omega we say that a point x_0\in{\mathcal{F}}(u) is regular from the right (resp. left) if there exists a ball B\subset\Omega^+(u) (resp. B\subset\Omega^-(u)) such that \overline{B}\cap{\mathcal{F}}(u) = x_0. In both cases, we denote with \nu = \nu(x_0) the unit normal to \partial B at x_0, pointing toward \Omega^+(u).
Definition 1.1. A weak (or viscosity) solution of the free boundary problem (1.1) is a continuous function u which satisfies the first two equality of (1.1) in viscosity sense (see Appendix A), and such that the free boundary condition is satisfied in the following viscosity sense:
(ⅰ) (supersolution condition) if x_0\in \mathcal F is regular from the right with touching ball B, then, near x_0,
u^+(x)\geq \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|) \qquad\text{in }B, \text{ with } \alpha\ge0 |
and
u^-(x)\leq \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|) \qquad\text{in }B^c, \text{ with }\beta\ge0, |
with equality along every non-tangential direction, and
\alpha\leq G(\beta, x_0, \nu(x_0)); |
(ⅱ) (subsolution condition) if x_0\in \mathcal F is regular from the left with touching ball B, then, near x_0,
u^+(x)\leq \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|) \qquad\text{in }B^c, \text{ with }\alpha\ge0 |
and
u^-(x)\geq \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|) \qquad\text{in }B, \text{ with }\beta\ge0, |
with equality along every non-tangential direction, and
\alpha\geq G(\beta, x_0, \nu(x_0)); |
We will construct our solution via Perron's method, by taking the infimum over the following class of admissible supersolutions {\rm{class}}.
Definition 1.2. A locally Lipschitz continuous function w\in C(\overline{\Omega}) is in the class {\rm{class}} if
(a) w is a solution in viscosity sense to
\begin{cases} F(D^2w^+)\le f_1& \text{in }\Omega^+(w) \\ F(D^2w^-)\ge f_2\chi_{\{u \lt 0\}} & \text{in }\Omega^-(w); \end{cases} |
(b) if x_0\in {\mathcal{F}}(w) is regular from the left, with touching ball B, then
w^+(x)\leq \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|) \qquad\text{in }B^c, \text{ with }\alpha\ge0 |
and
w^-(x)\geq \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|) \qquad\text{in }B, \text{ with }\beta\ge0, |
with
\alpha \le G(\beta, x_0, \nu(x_0)); |
(c) if x_0\in {\mathcal{F}}(w) is not regular from the left then
w(x) = o(|x-x_0|). |
The last ingredient we need is that of minorant subsolution.
Definition 1.3. A locally Lipschitz continuous function \underline{u}\in C(\overline{\Omega}) is a strict minorant if
(a) \underline{u} is a solution in viscosity sense to
\begin{cases} F(D^2\underline{u}^+)\ge f_1& \text{in }\Omega^+(\underline{u}) \\ F(D^2\underline{u}^-)\le f_2\chi_{\{\underline{u} \lt 0\}} & \text{in }\Omega^-(\underline{u}); \end{cases} |
(b) every x_0\in {\mathcal{F}}(\underline{u}) is regular from the right, with touching ball B, and near x_0
\underline{u}^+(x)\geq \alpha \left\langle x-x_0, \nu \right\rangle^+ + \omega(|x-x_0|)|x-x_0| \qquad\text{in }B, \text{ with }\alpha \gt 0, |
where \omega(r)\to0 as r\to0^+, and
\underline{u}^-(x)\leq \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|) \qquad\text{in }B^c, \text{ with }\beta\ge0, |
with
\alpha \gt G(\beta, x_0, \nu(x_0)). |
Our main result is the following.
Theorem 1.4. Let g be a continuous function on \partial\Omega. If
(a) there exists a strict minorant \underline{u} with \underline{u} = g on \partial \Omega and
(b) the set \{w\in {\rm{class}} : w\geq \underline{u}, \ w = g\text{ on }\partial\Omega\} is not empty, then
u = \inf \{w : w\in {\rm{class}} , w\geq \underline{u}\} |
is a weak solution of (1.1) such that u = g on \partial \Omega.
Once existence of a solution is established, we turn to the analysis of the regularity of the free boundary.
Theorem 1.5. The free boundary {\mathcal{F}}(u) has finite (n-1)-dimensional Hausdorff measure. More precisely, there exists a universal constant r_{0}>0 such that for every r < r_{0}, for every x_{0}\in {\mathcal{F}}(u),
\mathcal{H}^{n-1}({\mathcal{F}}(u)\cap B_{r}(x_{0}))\leq r^{n-1}. |
Moreover, the reduced boundary {\mathcal{F}}^{\ast }(u) of \Omega ^{+}(u) has positive density in \mathcal{H}^{n-1} measure at any point of F(u), i.e. for r < r_{0}, r_{0} universal
\mathcal{H}^{n-1}({\mathcal{F}}^{\ast }(u)\cap B_{r}(x))\geq cr^{n-1}, |
for every x\in {\mathcal{F}}(u). In particular
\mathcal{H}^{n-1}({\mathcal{F}}(u)\setminus {\mathcal{F}}^{\ast }(u)) = 0. |
Using the results in [8] we deduce the following regularity result.
Corollary 1.6. {\mathcal{F}}(u) is a C^{1, \gamma } surface in a neighborhood of {\mathcal{H}}^{n-1} a.e. point x_{0}\in {\mathcal{F}}(u).
Notation. Constants c, C and so on will be termed "universal" if they only depend on \lambda, \Lambda, n, \Omega, \|f_i\|_\infty and g.
In this section we show that positive solutions of F(D^2u) = f (with f continuous up to the boundary) have asymptotically linear behavior at any boundary point which admits a touching ball, either from inside or from outside the domain. We need the following preliminary result.
Lemma 2.1. Let r>0, \delta>0, \sigma>0, B_1^+: = B_1\cap\{x_1>0\} and let E\subset \partial B_1^+\cap\{x_1>0\} be any subset such that there exists \bar x\in E with
E \supset \partial B_1^+\cap\{x_1 \gt 0\}\cap B_\sigma(\bar x). |
Let u be the solution to
\begin{cases} F(D^2u) = r & in \;B_1^+\\ u = \delta g_{E} & on \;\partial B_1^+, \end{cases} | (2.1) |
where g_E is a cut-off function, g_E = 1 on E. If r is sufficiently small then there exists a positive constant C = C(\delta, \sigma) such that
u(x)\geq C x_1\qquad in \;B_{1/2}^+. |
Proof. We write
F(D^2 u) = \sum\limits_{i, j = 1}^n a_{ij}(x)u_{x_ix_j}\equiv L_u u, |
with (F = F(M))
a_{ij} = \int_0^1\frac{\partial F}{\partial M_{ij}}(tD^2 u)dt. |
We have
\lambda |\xi|^2\leq a_{ij}(x)\xi_i\xi_j\leq \Lambda|\xi|^2. |
Denote u = v+w with L_u v = 0, v = \delta\chi_E on \partial B_1^+ and L_u w = r, w = 0 on \partial B_1^+. By [11] we have that v(e_1/2)\geq C\delta, for some constant C = C(n, \lambda, \Lambda, \sigma), and by the Boundary Harnack principle applied to v and u_1(x) = x_1 we get that, in B^+_{1/2}, for some positive constants c_0 and c_1,
c_0\delta x_1\leq v \leq c_1\delta x_1. |
Put z(x) = \frac{1}{2 \min a_{11}}(x_1-x_1^2)r. The function z is positive in B^+_1 and
L_u z = -\frac{a_{11}}{\min a_{11}}r\leq-r. |
Therefore L_u(w+z)\leq 0 in B_1^+ and w+z\geq 0 on \partial B_1^+. By the maximum principle w\geq -c_2r x_1 in B_1^+, where c_2 = \frac{a_{11}}{\min a_{11}}>0.
Summing up we get, in B_{1/2}^+,
u = v+w\geq(c_0\delta-c_2r)x_1\geq c_3x_1 |
for r small enough, having c_3>0.
Lemma 2.2. Let \Omega_1 be a bounded domain with 0\in \partial \Omega_1 and
B_1^+: = B_1\cap\{x_1 \gt 0\}\subset\Omega_1. |
Let u be non-negative and Lipschitz in \overline{\Omega}_1\cap B_2, such that F(D^2u) = f in {\Omega}_1\cap B_2 and that u = 0 in \partial{\Omega}_1\cap B_2 . Then there exists \alpha\ge0 such that
u(x) = \alpha x_1^+ + o(|x|)\qquad as \;x\to0, \ x_1 \gt 0. |
Proof. Let \alpha_k = \sup\{\beta:u(x)\geq\beta x_1\text{ in }B^{+}_{1/k}\} for k\ge 1. Then the sequence \{\alpha_k\}_k is increasing and \alpha_k\leq L for any k, where L is the Lipschitz constant of u. Let \alpha = \lim_k \alpha_k. By definition, u(x)\geq \alpha x_1+o(|x|) in B_1^+, where x = (x_1, x_2, ..., x_n).
Suppose by contradiction that u(x)\neq \alpha x_1+o(|x|) in B_1^+. Then there exist a constant \delta_0>0 and a sequence \{x_k\}_k = \{(x_{1, k}, x_{2, k}, ..., x_{n, k})\}_k\subset B_1^+, with |x_k| = r_k\rightarrow 0, such that
u(x_k)\ge\alpha x_{1, k}+\delta_0r_k. |
Since u is Lipschitz, a simple computation implies that
u(x)\ge\alpha x_1 + \frac{\delta_0}{2}r_k\ge \alpha_k x_1 + \frac{\delta_0}{2}r_k\qquad \text{in }\left\{x:|x| = r_k, |x-x_k|\leq \frac{\delta_0 r_k}{4L}\right\}. |
Let
u_k(x) = \frac{u(r_k x)}{r_k}-\alpha_k x_1. |
The functions u_k are defined in B_1^+ and, by assumption of homogeneity on F, we have
F(D^2u_k(x)) = F(r_k D^2u(r_kx)) = r_k F(D^2u(r_k x)) = r_k f(r_k x)\le r_k\|f\|_\infty. |
Moreover u_k(x)\geq 0 on \partial B_1^+ and u_k\geq \delta_0/2 in E_k = \{x : x \in \partial B_1^+, x_1>0, |x-x_k|\leq\frac{\delta_0}{4L}\}. We deduce that u_k is a supersolution of (2.1). By comparison and Lemma 2.1, there exists C>0, not depending on k, such that
u_k(x) = \frac{1}{r_k}u(r_k x)-\alpha_k x_1 \geq C x_1\qquad\text{in }B^+_{1/2}. |
Writing z = r_k x we obtain u(z)\geq (\alpha_k +C)z_1 in B^+_{r_k/2}. Choosing k, k' in such a way that \alpha_k + C > \alpha and k'>2/r_k we obtain
\alpha_{k'} \gt \alpha, |
a contradiction.
Lemma 2.3. Let \Omega_1 be a bounded domain such that, writing B_1^-: = B_1\cap\{x_1 < 0\},
\overline{B_1^-}\cap\overline{\Omega}_1 = \{0\}. |
Let u be non-negative and Lipschitz in \overline{\Omega}_1\cap B_2(0), such that F(D^2u) = f in {\Omega}_1\cap B_2(0) and that u = 0 in \partial{\Omega}_1\cap B_2(0) . Then there exists \alpha\ge0 such that
u(x) = \alpha x_1^+ + o(|x|)\qquad as \;x\to0, \ x\in\Omega_1. |
Proof. By assumption, we have that
\Omega_1\cap B_1 \subset B_1^+. |
Then we can extend u as the zero function on B_1^+\setminus \Omega_1 so that it is a Lipschitz, non-negative solution to
F(D^2u)\ge -\|f\|_\infty\qquad\text{in }B_1^+. |
Reasoning in a similar way as in Lemma 2.2, we define \alpha_k = \inf\{\beta:u(x)\leq\beta x_1\text{ in }B_{1/k}\}, k\ge1. Then 0\le \alpha_k < +\infty (u is Lipschitz), and \alpha_k\searrow \alpha\ge 0, with u(x)\leq \alpha x_1+o(|x|) in B_1^+. Again, let us suppose by contradiction that
u(x_k)\le\alpha x_{1, k}-\delta_0r_k. |
where \delta_0>0 and \{x_k\}_k = \{(x_{1, k}, x_{2, k}, ..., x_{n, k})\}_k\subset B_1^+, is such that |x_k| = r_k\rightarrow 0. As before, such inequality propagates by Lipschitz continuity:
u(x)\le\alpha x_1 - \frac{\delta_0}{2}r_k\le \alpha_k x_1 - \frac{\delta_0}{2}r_k\qquad \text{in }\left\{x:|x| = r_k, |x-x_k|\leq \frac{\delta_0 r_k}{4L}\right\}. |
Defining the elliptic, homogeneous operator F^*(M) = - F(-M), we have that the functions
u_k(x) = \alpha_k x_1 - \frac{u(r_k x)}{r_k} |
solve
F^*(D^2u_k(x)) \le r_k\|f\|_\infty\qquad\text{in }B_1^+, |
with u_k(x)\geq 0 on \partial B_1^+ and u_k\geq \delta_0/2 in E_k = \{x : x \in \partial B_1^+, x_1>0, |x-x_k|\leq\frac{\delta_0}{4L}\}. As a consequence, a contradiction can be obtained by reasoning as in Lemma 2.2.
Lemma 2.4. Let \Omega_1 be bounded domain with 0\in \partial \Omega_1 and
either \;B_1(e_1)\subset\Omega_1\qquad or \;\overline{B}_1(-e_1)\cap\overline{\Omega}_1 = \{0\}. |
Let u be non-negative and Lipschitz in \overline{\Omega}_1\cap B_2(0), such that F(D^2u) = f in {\Omega}_1\cap B_2(0) and that u = 0 in \partial{\Omega}_1\cap B_2(0). Then there exists \alpha\ge0 such that
u(x) = \alpha x_1+o(|x|) |
as x\to0 and either x\in B_1(e_1) or x\in\Omega.
Proof. In both cases, we use the smooth change of variable
\begin{cases} y_1 = x_1 - \psi(x')\\ y' = x', \end{cases} |
where \psi(x') is smooth, with \psi(x') = 1-\sqrt{1-|x'|^2} for |x'| small. Then, by direct calculations, the function \tilde u(y) = u(y_1 + \psi(y'), y') satisfies
\tilde F(D^2\tilde u, \nabla\tilde u, y') = F(D^2u), |
where \tilde F is still a uniformly elliptic operator. As a consequence the lemma follows by arguing as in the proofs of Lemmas 2.2, 2.3, with minor changes.
We conclude this section by providing a uniform estimate from below of the development coefficient \alpha, in case the touching ball is inside the domain.
Lemma 2.5. Let u\in C(\overline{B_r(re_1)}), r\le1, be such that
\begin{cases} F(D^2u) = f & in \;B_r(re_1), \\ u\ge0, \\ u(0) = 0. \end{cases} |
Moreover, assume that u(re_1)\ge Cr, for some C>0. Then
u(x)\ge\alpha x_1+o(|x|), \qquad where \;\alpha\ge c_1 \frac{u(re_1)}{r} - c_2 r \|f\|_\infty, |
as x\to0, for r\le \bar r, where c_1, c_2 and \bar r only depend on \lambda, \Lambda, n.
Proof. Let
u_r(x) = \frac{u(r(e_1+x))}{r}, \qquad x\in B_1(0). |
Then
\begin{cases} F(D^2u_r) = rf &\text{in }B_1, \\ u_r\ge0, \\ u_r(-e_1) = 0. \end{cases} |
By Harnack's inequality [5,Theorem 4.3] we have that
\inf\limits_{\partial B_{1/2}} u_r \ge c(u_r(0) - r\|f\|_\infty) = :a, |
where c only depends on \lambda, \Lambda, n. We are in a position to apply Lemma A.2, which provides
u_r(x) \ge \alpha(x_1 + 1) + o (|x+e_1|), \quad\text{with }\alpha\ge c_1 a - c_2r\|f\|_\infty = c_1' u_r(0) - c_2'r\|f\|_\infty, |
as x\to -e_1, and the lemma follows.
Remark 2.6. Notice that the above results can be applied both to F(D^2 u^+) = f_1 in \Omega^+(u) and to F(D^2u^-) = f_2\chi_{\{u < 0\}} in \Omega^-(w).
In this section we adapt the strategy developed in [3], in order to show that u^+ is locally Lipschitz. To this aim we need to use the following almost-monotonicity formula, provided in [6,14].
Proposition 3.1. Let u_i, i = 1, 2 be continuous, non-negative functions in B_1, satisfying \Delta u_i \ge -1, u_1\cdot u_2 = 0 in B_1. Then there exist universal constants C_0 and r_0, such that the functional
\Phi(r) : = \frac{1}{r^4} \int_{B_r}\frac{|\nabla u_1|^2}{r^{n-2}} \int_{B_r}\frac{|\nabla u_2|^2}{r^{n-2}} |
satisfies, for 0 < r\le r_0,
\Phi(r) \le C_0\left(1 + \|u_1\|_{L^2(B_1)}^2 + \|u_2\|_{L^2(B_1)}^2\right)^2. |
Lemma 3.2. Let w\in {\rm{class}} . There exists \tilde w\in {\rm{class}} such that
1. F(D^2\tilde w) = f_1 in \Omega^+(\tilde w),
2. \tilde w\leq w, \tilde w^- = w^-, and
3. \tilde w \geq \underline{u} in \Omega.
Proof. Let w\in{\rm{class}} and \Omega^+ = \Omega^+(w). We define
{\mathcal{V}}: = \{v\in C(\overline{\Omega^+}) : F(D^2v)\ge f_1\chi_{\{v \gt 0\}}\text{ in }\Omega^+, \ v\ge0\text{ in }\Omega^+, \ v = w\text{ on }\partial\Omega^+\} |
and
\tilde w(x): = \begin{cases} \sup\{v(x):v\in{\mathcal{V}}\} & x\in\Omega^+\\ w(x) & \text{elsewhere}. \end{cases} |
Since \underline{u}^+\in{\mathcal{V}} we obtain that {\mathcal{V}} is not empty and that \underline{u} \le \tilde w \le w. Moreover \tilde w is a solution of the obstacle problem (see [13])
\begin{cases} F(D^2\tilde w) = f_1 & \text{in }\{\tilde w \gt 0\}\\ \tilde w \ge 0 & \text{in }\Omega^+\\ \tilde w = w & \text{on }\partial\Omega^+. \end{cases} |
In particular, regularity results for the obstacle problem for fully nonlinear equations imply that \tilde w is C^{1, 1} in \Omega^+ (see [13]). To conclude that \tilde w\in{\rm{class}}, we need to show that the free boundary conditions in Definition 1.2 hold true. Let x_0\in{\mathcal{F}}(\tilde w): if x_0\in{\mathcal{F}}(w) too, then the free boundary condition follows from the fact that \tilde w\le w; otherwise, x_0 \in \Omega^+ is an interior zero of \tilde w, and the free boundary condition follows by the C^{1, 1} regularity of \tilde w.
Lemma 3.3. Let w\in {\rm{class}} with F(D^2 w) = f_1 in \Omega^+(w), and let x_0\in{\mathcal{F}}(w) be regular from the right. Then u admits developments
\begin{split} w^+(x)& = \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|), \\ w^-(x)&\geq \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|), \end{split} |
with 0\le \alpha \le G(\beta, x_0, \nu(x_0)), and
\alpha\beta \le C_0\left(1 + \|u_1\|_2^2 + \|u_2\|_2^2\right). |
Proof. If x_0 is not regular from the left, then by definition of {\rm{class}} the asymptotic developments hold with \alpha = \beta = 0 and there is nothing to prove. On the other hand, if x_0 is also regular from the left, then the asymptotic developments and the free boundary condition hold true by definition of {\rm{class}} and by Lemma 2.4. Also in this case, if \alpha = 0 then there is nothing else to prove, thus we are left to deal with the case \alpha>0.
Reasoning as in [3,Lemma 3], see also [19,Lemma 4.3], one can show that
\Phi(r) \ge C(n)(\alpha+o(1))^2(\beta+ o(1))^2 | (3.1) |
(recall that \Phi(r) is defined in Proposition 3.1). On the other hand, since F is concave,
\Delta w^\pm \ge - c\|f\|_\infty. |
The conclusion follows by combining Proposition 3.1 with (3.1).
Proposition 3.4. For every D\subset\subset\Omega there exists a constant L_D, depending only on D, G, \underline u and {\rm{class}} , such that
\frac{|w^+(x)-w^+(y)|}{|x-y|} \leq L_D |
for every x, y\in D, x\neq y, and for every w\in {\rm{class}} with F(D^2 w) = f_1 in \Omega^+(w).
Proof. Let x_0\in \Omega^+(w) \cap D such that
r : = {\rm{dist}}(x_0, {\mathcal{F}}(w)) \lt \frac12 {\rm{dist}}(\overline{D}, \partial \Omega). |
We will show that there exists M>0, not depending on w, such that
\frac{w(x_0)}{r} \le M, |
and the lemma will follow by Schauder estimates and Harnack inequality. By contradiction, let M large to be fixed and let as assume that
\frac{w(x_0)}{r} \gt M. |
Then Lemma 2.5 applies and we obtain
w(x)\ge \alpha_M \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|), |
where \alpha_M = c_1 M - c_2 r \|f\|_\infty > 0 for M sufficiently large. Then x_0 is regular from the right and Lemma 3.3 applies, with \alpha_M \le \alpha \le G(\beta, x_0, \nu(x_0)), providing
\alpha_M G^{-1}(\alpha_M) \le C_0\left(1 + \|u_1\|_2^2 + \|u_2\|_2^2\right), |
where G^{-1}(\alpha): = \inf_{x, \nu}G^{-1}(\alpha, x, \nu). This provides a contradiction for M sufficiently large.
Corollary 3.5. u^+ is locally Lipschitz and satisfies F(D^2 u) = f_1 in \Omega^+(u).
Now we turn to the Lipschitz continuity of u.
Lemma 4.1. If w_1, w_2\in {\rm{class}} then \min\{w_1, w_2\}\in {\rm{class}}.
Proof. This follows by standard arguments, see e.g. [9,Lemma 4.1].
To prove that u is Lipschitz continuous we use the double replacement technique introduced in [9]. Let w\in {\rm{class}} with w(x_0) < 0 and
B: = B_R(x_0), \qquad \Omega_1: = \Omega^+(w)\setminus\overline{B}. |
Working on \Omega_1, we define
{\mathcal{V}}_1: = \{v : F(D^2v)\ge f_1\chi_{\{v \gt 0\}}\text{ in }\Omega_1, \ v\ge0\text{ in }\Omega_1, \ v = w\text{ on }\partial\Omega_1\setminus\partial B, \ v = 0\text{ on }\partial B \} |
(which is non empty, for R sufficiently small, as \underline{u}^+\in{\mathcal{V}}_1). Then
w_1: = \sup {\mathcal{V}}_1 |
solves the obstacle problem (see [13])
F(D^2w_1)\ge f_1\text{ in }\{w_1 \gt 0\}, \quad w_1\ge0\text{ in }\Omega_1. | (4.1) |
On the other hand, working on B, let
{\mathcal{V}}_2: = \{v : F(D^2v)\ge f_2\chi_{\{v \gt 0\}}\text{ in }B, \ v\ge0\text{ in }B, \ v = w^-\text{ on }\partial B \} |
(which is non empty, as w^-\in{\mathcal{V}}_2). Again,
w_2: = \sup {\mathcal{V}}_2 |
solves the obstacle problem
F(D^2w_2)\ge f_1\text{ in }\{w_2 \gt 0\}, \quad w_2\ge0\text{ in }B. | (4.2) |
Under the above notation, the double replacement \tilde w of w, relative to B, is defined as
\tilde w: = \begin{cases} w_1 & \text{in }\Omega_1\\ -w_2 & \text{in }B\\ w & \text{otherwise.} \end{cases} |
Lemma 4.2. Let w\in {\rm{class}} with w(x_0) = -h < 0. There exists \varepsilon >0 (depending on {\rm{dist}}(x_0, \partial\Omega) and \underline{u}) such that:
1. the double replacement \tilde w of w, relative to B_{\varepsilon h}(x_0), satisfies \underline{u}\le \tilde w \leq w in \Omega;
2. \tilde w < 0 and F(D^2\tilde w) = f_2 in B_{\varepsilon h}(x_0), with
|\nabla \tilde w|\le \frac{C}{\varepsilon} + \varepsilon C \|f_2\|_\infty\quad in \;B_{\varepsilon h/2}(x_0); |
3. \tilde w \in {\rm{class}}.
Proof. The inequality w_1 \leq w in \Omega_1 follows by the maximum principle, while w_2 \ge -w in B because w^-\in{\mathcal{V}}_2. On the other hand, provided \varepsilon is sufficiently small (depending on the Lipschitz constant of \underline{u}), we have that \underline u < 0 in B: = B_{\varepsilon h}(x_0) and u^*\in{\mathcal{V}}_1, so that w_1\ge \underline{u}; finally, by the maximum principle in \{w_2>0\}, also -w_2\ge \underline u, and part 1. follows.
Turning to part 2., assume by contradiction that \partial\{w_2>0\}\cap B_{\varepsilon h}(x_0)\neq\emptyset. Then, by the regularity properties of the obstacle problem (4.2) (see [13]), we obtain that
w_2(x_0) \le C (\varepsilon h)^2. |
Since -w_2(x_0) \le w(x_0) = -h, we obtain a contradiction for \varepsilon sufficiently small. Then w_2 > 0 in B_{\varepsilon h}(x_0), w_2 solves the equation by (4.2), and the remaining part of 2. follows by standard Schauder estimates and Harnack inequality.
Coming to part 3., the fact that \tilde w satisfies (a) in Definition 1.2 follows by equations (4.1), (4.2) and by part 2. above, and we are left to check the free boundary conditions. For \bar x\in{\mathcal{F}}(\tilde w), three possibilities may occur. If \bar x\in {\mathcal{F}}(w) then, since \tilde w \le w, then \tilde w has the correct asymptotic behavior both when \bar x is regular and when it is not (recall that G(0, \cdot, \cdot)>0. If \bar x\in\partial \{w_1>0\} \cap \Omega_1, then we can use again the regularity of the obstacle problem]4.1 to obtain the correct asymptotic behavior. We are left to the final case, when \bar x \in \partial B \cap \Omega^+(w). By Proposition 3.4, let us denote with L the Lipschitz constant of w in B_{{\rm{dist}}(x_0, \partial\Omega)/2}(x_0). Then
\tilde w \le w^+ \le L\varepsilon h\qquad \text{in }B_{2\varepsilon h}(x_0). |
Defining
\tilde w_\varepsilon(x): = \frac{\tilde w(x_0 + \varepsilon h x)}{\varepsilon h}, |
we have that
\begin{cases} F(D^2\tilde w_\varepsilon^+) = \varepsilon h f_1 & \text{in }(B_2\setminus \overline{B}_{1}) \cap \Omega^+(w_\varepsilon)\\ \tilde w_\varepsilon^+\le L & \text{on }\partial B_2\\ \tilde w_\varepsilon^+\le 0 & \text{on }\partial B_{1}\cap\partial\Omega^+(w_\varepsilon). \end{cases} |
Then Lemma A.1 applies, yielding
\tilde w_\varepsilon^+ \le \alpha\langle x-\bar x_\varepsilon, \nu(\bar x_\varepsilon) \rangle^+ + o (| x-\bar x_\varepsilon|), |
where
\alpha\le c_1 L + c_2 \varepsilon h \|f_1\|_\infty, \qquad \bar x_\varepsilon : = \frac{\bar x -x_0}{\varepsilon h}, |
for universal c_1, c_2. Going back to \tilde w we obtain
\tilde w^+ \le \alpha\langle x-\bar x, \nu\rangle^+ + o (| x-\bar x|), \qquad \alpha \leq \bar L | (4.3) |
where \nu = (\bar x-x_0)/|\bar x-x_0|.
On the other hand, we can apply Lemma 2.5 to (-w_2)_\varepsilon, obtaining
\tilde w^-_\varepsilon = - (w_2)_\varepsilon\ge \beta\langle x-\bar x_\varepsilon, \nu(\bar x_\varepsilon) \rangle^+ + o (| x-\bar x_\varepsilon|), |
where
\beta \ge \frac{c'_1}{\varepsilon} - c'_2 \varepsilon h \|f_1\|_\infty, |
for universal c'_1, c'_2, and thus
\tilde w^- \ge \beta\langle x-\bar x, \nu\rangle^- + o (| x-\bar x|), \qquad \beta \geq \ \frac{\bar c}{\varepsilon}. | (4.4) |
Comparing (4.3) and (4.4) we have that, choosing \varepsilon small so that
\bar L \lt \inf\limits_{x, \nu}G(\bar c/\varepsilon, x, \nu), |
the free boundary condition holds true.
Corollary 4.3. Let u(x_0) = -h < 0. There exist an non-increasing sequence \{\tilde w_k\}\subset {\rm{class}}, \tilde w_k\ge \underline{u}, and \varepsilon>0, depending on {\rm{dist}}(x_0, \partial\Omega) and \underline{u}, such that the following hold:
1. \tilde w_k(x_0) \searrow u(x_0);
2. \tilde w_k < 0 and F(D^2\tilde w_k^-) = f_2 in B_{\varepsilon h}(x_0);
3. the sequence \{\tilde w_k\} is uniformly Lipschitz in B_{\varepsilon h/2}(x_0), with Lipschitz constant L_0 depending on {\rm{dist}}(x_0, \partial\Omega).
4. \tilde w_k \searrow u uniformly on B_{h\varepsilon ilon/4}
Proof. Let u(x_0) = -h < 0, \{w_k\}\subset{\rm{class}} be such that w_k\searrow u in some neighborhood of x_0 and \{\tilde w _k\}\subset{\rm{class}} be the corresponding double replacements, as in Lemma 4.2. Then first three points are direct consequence of the lemma above, and we are left to prove that \tilde w_k \searrow u uniformly on B_{h\varepsilon ilon/4}. By equicontinuity, \tilde{w}_k \to \tilde{w} in B_{\varepsilon ilon h/2}(x_0), and suppose by contradiction that \tilde{w}(x_1)>u(x_1) for some x_1\in B_{\varepsilon ilon h/4}(x_0). Then consider a new sequence \{v_k\}_{k} converging to u at x_1 and define \{\tilde{u}_k\}_k as the double replacement of \{\min\{\tilde{v}_k, \tilde{w}_k\} \}_{k} in B_{\varepsilon ilon h/2}(x_0). Then \tilde{u}_k\to\tilde{u}, \tilde{u}\leq\tilde{w} in B_{\varepsilon ilon h /2}(x_0), \tilde{u}(x_0) = \tilde{w}(x_0) and \tilde{u}(x_1) < \tilde{w}(x_1). Since F(D^2\tilde{w}) = F(D^2\tilde{u}) = f_2 in B_{\varepsilon ilon h/2}(x_0), this contradicts the strong maximum principle.
Corollary 4.4. For any \overline{D}\subset\Omega there exists \{w_k\}_{k}\subset {\rm{class}} such that w_k\searrow u uniformly in \overline{D}. Furthermore, if \overline{D}\subset \Omega^-(u), then each w_k may be taken non-positive in \overline{D}.
Proof. The first part follows from the previous corollary. By compactness, it is enough to prove the second part for balls \overline{B}_{\varepsilon}(x_0) \subset \Omega^-(u), with \varepsilon small. Let w_k \searrow u uniformly in \overline B_{2\varepsilon}(x_0) \subset \Omega^-(u), and let
w_k^\varepsilon(x) = \frac{w_k(x_0+\varepsilon x)}{\varepsilon}\searrow u_\varepsilon\qquad \text{in }B_2. |
Let \phi be such that
\begin{cases} \Delta \phi = -c\varepsilon \|f\|_\infty & \text{in }B_2\setminus \overline{B}_{1}\\ \phi = a & \text{on }\partial B_2\\ \phi = 0 & \text{on }\partial B_{1}, \end{cases} |
with a and \varepsilon positive and sufficiently small so that
\nabla\phi(e_1) \cdot e_1 \lt \inf\limits_{x, \nu}G(0, x, \nu) |
(this is possible by explicit calculations, see for instance Lemma A.1); notice that this condition insure that \phi, extended to zero in B_1, is a supersolution in B_2 (when c universal is suitably chosen). Since u_\varepsilon \leq 0 in \overline B_2, for k sufficiently large w_k\leq a/2 in \overline B_2. Let us define
\bar{w}_k^\varepsilon = \begin{cases} \min\{w_k^\varepsilon, \phi\}&\text{in }\quad\overline B_2, \\ w^\varepsilon &\text{otherwise.} \end{cases} |
Then, by Lemma 4.1, the function
\bar w_k(x) = \varepsilon \bar w_k^\varepsilon\left(\frac{x-x_0}{\varepsilon}\right) |
satisfies \bar w_k \in {\rm{class}}, \bar w_k \leq 0 in \overline B_\varepsilon(x_0) and \bar w_k \searrow u in \overline B_{\varepsilon}(x_0), as required.
Corollary 4.5. u is locally Lipschitz in \Omega, continuous in \overline \Omega, u = g on \partial \Omega. Moreover u solves
\mathcal L u = f_2 \chi_{\{u \lt 0\}}, \quad in\; \Omega^-(u). |
In this section we will show that u^+ is non-degenerate, in the sense of the following result.
Lemma 5.1. Let x_0\in {\mathcal{F}}(u) and let A be a connected component of \Omega^+(u)\cap (B_{r}(x_0)\setminus \overline{B}_{r/2}(x_0)) satisfying
\overline{A}\cap \partial B_{r/2}(x_0)\not = \emptyset, \quad\overline{A}\cap\partial B_{r}(x_0)\not = \emptyset, |
for r \leq r_0 universal. Then
\sup\limits_{A}u\geq Cr. |
Moreover
\frac{|A\cap B_{r}(x_0)|}{|B_r(x_0)|}\geq C \gt 0, |
where all the constants C depend on d(x, \partial \Omega) and on \underline{u}.
Corollary 5.2. {\mathcal{F}}(w_k) \to {\mathcal{F}}(u) locally in Hausdorff distance and \chi_{\{w_k>0\}}\to \chi_{\{u>0\}} in L^1_{\mathit{\boldsymbol{loc}}}.
The proof of the above result will follow by the two following lemmas.
Lemma 5.3. Let u be a Lipschitz function in \overline{\Omega}\cap B_1(0), with 0\in\partial \Omega, satisfying
\begin{cases} F(D^2 u) = f & in \; \Omega\cap B_1\\ u = 0 & on \;\partial\Omega\cap B_1. \end{cases} |
If there exists c>0 such that
u(x)\geq c{\rm{dist}}(x, \partial \Omega) \qquad for\; every \;x\in \Omega\cap B_{1/2} | (5.1) |
then there exists a constant C>0 such that
\sup\limits_{B_r(0)}u\geq Cr, |
for all r \leq r_0 universal.
Proof. Let x_0 \in \Omega\cap B_1, \varepsilon = {\rm{dist}}(x_0, \partial\Omega), and let L denote the Lipschitz constant of u. Then
c\varepsilon\le u(x_0) \le L \varepsilon. |
We will show that, for \delta>0 to be fixed, there exists x_1\in B_\varepsilon(x_0) such that
u(x_1) \ge (1+\delta) u(x_0). | (5.2) |
Then, iterating the procedure, one can conclude as in [9,Lemma 5.1].
Assume by contradiction that (5.2) does not hold. Then, defining the elliptic, homogeneous operator F^*(M) = - F(-M), we infer that
v(x): = (1+\delta)u(x_0) - u(x) \gt 0 \quad \text{in }B_\varepsilon(x_0) \qquad\text{satisfies }F^*(D^2 v) = -f. |
Let r(L) = 1-c/(4L); using the Harnack inequality we have that there exists C(L) such that
v\le C(L) (\delta u(x_0) + \varepsilon^2\|f\|_\infty) \le \frac12 u(x_0)\qquad \text{in }\overline B_{r(L)\varepsilon}(x_0), |
provided both \delta and \varepsilon are sufficiently small (depending on c, L and \|f\|_\infty). In terms of u, the previous inequality writes as
u \ge \frac{c\varepsilon}{2}\quad \text{in }\overline B_{r(L)\varepsilon}(x_0). |
On the other hand, there exists y_0\in\partial B_{r(L)\varepsilon}(x_0) such that {\rm{dist}}(y_0, \partial\Omega) = (1-r(L))\varepsilon and hence
\min\limits_{\overline B_{r(L)\varepsilon}(x_0)} u \le u(y_0) \le L {\rm{dist}}(y_0, \partial\Omega) = \frac{c\varepsilon}{4}. |
This is a contradiction, therefore (5.2) holds true.
Lemma 5.4. There exist universal constants \bar r, \bar C such that
u(x_0) \ge \bar C {\rm{dist}}(x_0, {\mathcal{F}}(u)) \qquad for \;every \; x_0\in\{x\in\Omega^+(u):{\rm{dist}}(x, {\mathcal{F}}(u))\le \bar r\}. |
Proof. Let x_0\in\{x\in\Omega^+(u):{\rm{dist}}(x, {\mathcal{F}}(u))\le \bar r\}, with \bar r universal to be specified later, and let r: = {\rm{dist}}(x_0, {\mathcal{F}}(u)). We distinguish two cases.
First let us assume that
{\rm{dist}}(x_0, \Omega^+(\underline u))\le \frac{r}{2}. |
In this case, for any x\in{\mathcal{F}}(\underline u) we define
\rho(x): = \max\{r \gt 0:\text{for some }z, \ x\in\partial B_r(z)\text{ and }B_r(z)\subset\Omega^+ (\underline u)\}. |
Notice that \rho(x)>0 for every x, since any point in {\mathcal{F}}(\underline u) is regular from the right by assumption. Thus, recalling that \underline u^+ has linear growth bounded below by \inf_{x, \nu}G(0, x, \nu), and noticing that B_{3r/4}(x_0)\cap \Omega^+(\underline u) contains a ball of radius comparable with r (at least for a suitable choice of \bar r):
\sup\limits_{B_{3r/4}(x_0)} u^+ \ge \sup\limits_{B_{3r/4}(x_0)} \underline u^+ \ge \bar C r, |
where \bar C only depends on \underline u.
On the other hand, in case
{\rm{dist}}(x_0, \Omega^+(\underline u))\ge \frac{r}{2}, |
we have \underline u\le 0 in B_{r/2}(x_0). By Corollary 4.4 we can find \{w_k\}_k\subset{\rm{class}} converging uniformly to u on some D\supset B_{r}(x_0). By scaling
u_r(x) = \frac{u(x_0+rx)}{r}, \qquad w_k^r(x) = \frac{w_k(x_0+rx)}{r}, |
we need to find \bar C universal such that u_r(0)\ge \bar C. Let us assume by contradiction that
u_r(0) \lt \bar C. |
Then by Harnack inequality
u_r \le C(\bar C + r \|f_1\|_\infty)\qquad\text{in }B_{1/2} |
and, for k sufficiently large,
w_k^r \le C'(\bar C + r \|f_1\|_\infty)\qquad\text{in }B_{1/2}. |
Now, reasoning as in the proof of Corollary 4.4, let \phi be such that
\begin{cases} \Delta \phi = -c r \|f\|_\infty & \text{in }B_{1/2}\setminus \overline{B}_{1/4}\\ \phi = a & \text{on }\partial B_{1/2}\\ \phi = 0 & \text{on }\partial B_{1/4}, \end{cases} |
with a and r positive and sufficiently small so that \nabla\phi(e_1/4) \cdot e_1 < \inf_{x, \nu}G(0, x, \nu), in such a way that \phi, extended to zero in B_{1/4}, is a supersolution in B_{1/2}. Then, choosing \bar C < (a-r \|f_1\|_\infty)/C' we obtain that w_k^r < \phi on \partial B_{1/2} and then the functions
\bar{w}_k^r = \begin{cases} 0 &\text{in }\overline B_{1/4}, \\ \min\{w_k^r, \phi\}&\text{in }\overline B_{1/2}\setminus B_{1/4}, \\ w^r_k &\text{otherwise} \end{cases} |
are continuous, while
\bar w_k(x) = r\bar w_k^r\left(\frac{x-x_0}{r}\right) |
satisfy \bar w_k \in {\rm{class}}, \bar w_k \equiv 0 in \overline B_{r/4}(x_0). This is in contradiction with the fact that u(x_0)>0, and the lemma follows.
This section is devoted to the proof that u satisfies the supersolution condition (ⅰ) in Definition 1.1. Thanks to Lemma 2.4 we only need to prove that, whenever u admits asymptotic developments at x_0\in{\mathcal{F}}(u), with coefficients \alpha and \beta, then \alpha\le G(\beta, x_0, \nu_{x_0}). To do that, we need to distinguish the two cases \beta>0 and \beta = 0.
Lemma 6.1. Let x_0\in{\mathcal{F}}(u), and
\begin{split} u^+(x)& = \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|), \\ u^-(x)& = \beta \left\langle x-x_0, \nu \right\rangle^- + o(|x-x_0|), \end{split} |
with
\beta \gt 0. |
Then \alpha\le G(\beta, x_0, \nu_{x_0}).
Proof. Since \beta >0, then {\mathcal{F}}\left(u\right) is tangent at x_{0} to the hyperplane
\pi :\left\langle x-x_{0}, \nu \right\rangle = 0 |
in the following sense: for any point x\in {\mathcal{F}}\left(u\right) , dist\left(x, {\mathcal{F}}\left(u\right) \right) = o\left(\left\vert x-x_{0}\right\vert \right). Otherwise we get a contradiction to the asymptotic development of u.
Let \{w_k\}_k\subset {\rm{class}} be uniformly decreasing to u, as in Corollary 4.4. By the non-degeneracy of u^+ we have that, for k large, w_k can not remain strictly positive near x_0. Let d_{k} = d_{H}\left({\mathcal{F}}\left(w_{k}\right), {\mathcal{F}}\left(u\right) \right) be the Hausdorff distance between the two free boundaries. In the ball B_{2\sqrt{d_{k}}}\left(x_{0}\right) , {\mathcal{F}}\left(u\right) is contained in a strip parallel to \pi of width o\left(\sqrt{d_{k}} \right) and, since d_{k}\rightarrow 0, {\mathcal{F}}\left(w_{k}\right) is contained in a strip S_{k} of width d_{k}+o\left(\sqrt{d_{k}}\right) = o\left(\sqrt{d_{k}}\right) .
Consider now the points x_{k} = x_{0}-\sqrt{d_{k}}\nu and let B_{k} = B_{r_{k}}\left(x_{k}\right) be the largest ball contained in \Omega ^{-}\left(w_{k}\right) with touching point z_{k}\in {\mathcal{F}}\left(w_{k}\right) . Then z_{k}\in S_{k} and, since w_{k}\geq u, from the asymptotic developments of w_{k} and u we have
\beta \sqrt{d_{k}}+o\left( \sqrt{d_{k}}\right) = u^{-}\left( x_{k}\right) \geq w^{-}\left( x_{k}\right) = \beta _{k}r_{k}+o(r_{k}), |
since
\sqrt{d_{k}}+o\left( \sqrt{d_{k}}\right) \leq r_{k}\leq \sqrt{d_{k}}. |
Passing to the limit we infer
\lim \sup \beta _{k}\leq \beta . |
Reasoning in the same way on the other side th the points y_{k} = x_{0}+\sqrt{ d_{k}} (and the same z_{k}, which are regular from the left), we get
\alpha \leq \lim \inf \alpha _{k}\text{.} |
From \alpha _{k}\leq G\left(\beta _{k}, z_k, \nu_{k}\right) , where \nu _{k} = \left(x_{k}-z_{k}\right) /\left\vert x_{k}-z_{k}\right\vert , we get \alpha \leq G\left(\beta, x_0, \nu_{x_0} \right) .
To treat the case \beta = 0 we need the following preliminary lemma.
Lemma 6.2. Let v\geq 0 continous in B_1(x_0) be such that \Delta v\geq -M. Let
\Psi _{r}\left( x_0, v\right) = \frac{1}{r^{2}} \int_{B_{r}(x_0)}\frac{\left\vert \nabla v\right\vert ^{2}}{\left\vert x-x_0\right\vert ^{n-2}}dx. |
Then, for r small,
\Psi _{r}\left( x_0, v\right) \leq c\left( n\right) \left\{ \sup\limits_{B_{2r}(x_0)}\left( \frac{v}{r}\right) ^{2}+M\sup\limits_{B_{2r}(x_0)}v\right\} . | (6.1) |
Proof. We may assume x_0 = 0 and write \Psi _{r}\left(0, v\right) = \Psi _{r}\left(v\right). Rescale setting v_{r}\left(x\right) = v\left(rx\right) /r; we have \Delta v_{r}\geq -rM and
\Psi _{r}\left( v\right) = \Psi _{1}\left( v_{r}\right) \text{.} |
Let \eta \in C_{0}^{\infty }\left(B_{2}\right) , \eta = 1 in B_{1}. Since 2\left\vert \nabla v_{r}\right\vert ^{2}\leq 2rMv_{r}+\Delta v_{r}^{2}, we have:
\begin{eqnarray*} \Psi _{1}\left( v_{r}\right) &\leq &C\int_{B_{2}}\eta \frac{\left\vert \nabla v_{r}\right\vert ^{2}}{\left\vert x\right\vert ^{n-2}}\leq C\int_{B_{2}}\eta \frac{2Mv_{r}+\Delta v_{r}^{2}}{\left\vert x\right\vert ^{n-2}} \\ & = &C\int_{B_{2}}\left[\frac{2Mv_{r}}{\left\vert x\right\vert ^{n-2}} +v_{r}^{2}\Delta \left( \frac{\eta }{\left\vert x\right\vert ^{n-2}}\right)\right], \end{eqnarray*} |
so that
\begin{eqnarray*} \Psi _{r}\left( v\right) & = &\Psi _{1}\left( v_{r}\right) \leq c\left( n\right) \left( \left\vert v_{r}\right\vert _{L^{\infty }\left( B_{2}\right) }^{2}+rM\left\vert v_{r}\right\vert _{L^{\infty }\left( B_{2}\right) }\right) \\ & = &c\left( n\right) \left\{ \sup\limits_{B_{2r}}\left( \frac{v}{r}\right) ^{2}+M\sup\limits_{B_{2r}}v\right\} , \end{eqnarray*} |
which is (6.1).
Lemma 6.3. Let x_0\in{\mathcal{F}}(u), and
\begin{split} u^+(x)& = \alpha \left\langle x-x_0, \nu \right\rangle^+ + o(|x-x_0|), \\ u^-(x)& = o(|x-x_0|). \end{split} |
Then \alpha\le G(\beta, x_0, \nu_{x_0}).
Proof. As before, let \{w_k\}_k\subset {\rm{class}} be uniformly decreasing to u, with w_k that is not strictly positive near x_0, for k large. The first part of the proof is exactly as in Lemma 6.3 of [9], until equation (6.2) below. For the reader's convenience, we recall such argument here.
For each k we denote with
B_{m, k} = B_{\lambda _{m, k}}\left(x_{0}+\frac{1}{m}\nu \right) |
the largest ball centered at x_{0}+\nu/{m} contained in \Omega ^{+}(w_{k}), touching {\mathcal{F}}(w_{k}) at x_{m, k} where \nu _{m, k} is the unit inward normal of {\mathcal{F}}(w_{k}) at x_{m, k}. Then up to proper subsequences we deduce that
\lambda _{m, k}\rightarrow \lambda _{m}, \quad x_{m, k}\rightarrow x_{m}, \quad \nu _{m, k}\rightarrow \nu _{m} |
and B_{\lambda _{m}}(x_{0}+\nu/m) touches {\mathcal{F}}(u) at x_{m}, with unit inward normal \nu _{m}. From the behavior of u^{+}, we get that
|x_{m}-x_{0}| = o\left(\frac{1}{m}\right), |
\frac{1}{m}+o\left(\frac{1}{m}\right)\leq \lambda _{m}\leq \frac{1}{m} |
and
|\nu _{m}-\nu | = o(1). |
Now since w_{k}\in \mathcal{F}, near x_{m, k} in B_{m, k}:
w_{k}^{+}\leq \alpha _{m, k}\langle x-x_{m, k}, \nu _{m, k}\rangle ^{+}+o(|x-x_{m, k}|) |
and in \Omega \setminus B_{m, k}
w_{k}^{-}\geq {\beta _{m, k}}\langle x-x_{m, k}, \nu _{m, k}\rangle ^{-}+o(|x-x_{m, k}|) |
with
0\leq \alpha _{m, k}\leq G(\beta _{m, k}, x_{m, k}, \nu _{m, k}), |
(by Lemma 2.5 the touching occurs at a regular point, for m, k large.) We know that
w_{k}^{+}\geq u^{+}\geq \alpha \langle x-x_{0}, \nu \rangle ^{+}+o(|x-x_{0}|), |
hence
\underline{\alpha }_{m} = \liminf\limits_{k\rightarrow \infty }\alpha _{m, k}\geq \alpha -\varepsilon ilon _{m} |
and \varepsilon ilon _{m}\rightarrow 0, as m\rightarrow \infty. We have to show that
\underline{\beta } = \liminf\limits_{m, k\rightarrow +\infty }\beta _{m, k} = 0. |
We assume by contradiction that \bar\beta>0. Acting as in [9,Lemma 6.3] we obtain, for r small,
(1+\omega (r))\Phi _{r}(x_{m, k}, w_{k})+C\omega (r)\geq c_{n}\;\alpha _{m, k}^{2}\beta _{m, k}^{2}, | (6.2) |
where
\Phi _{r}\left( x_{m, k}, w_{k}\right) = \Psi _{r}\left( x_{m, k}, w_{k}^{+}\right) \Psi _{r}\left( x_{m, k}, w_{k}^{-}\right) . |
By concavity we have that \Delta w_{k}^{\pm }\geq -M where M = c\min \left(\| f_{1}\|_{\infty }, \|f_{2}\|_{\infty }\right) . Lemma 6.2 implies
\begin{split} c_n\alpha_{m, k} ^{2}{\beta}^{2}_{m, k} &\leq \left( 1+\omega \left( r\right) \right) \Psi _{r}\left( x_{m, k}, w_{k}^{+}\right) \Psi _{r}\left( x_{m, k}, w_{k}^{-}\right) +C\omega \left( r\right) \\ &\leq c^{2}\left( n\right) \left( 1+\omega \left( r\right) \right) \left\{ \sup\limits_{B_{2r}\left( x_{m, k}\right) }\left( \frac{w_{k}^{+}}{r}\right) ^{2}+M\sup\limits_{B_{2r}\left( x_{m, k}\right) }w_{k}^{+}\right\} \times \\ & \qquad \times\left\{ \sup\limits_{B_{2r}\left( x_{m, k}\right) }\left( \frac{w_{k}^{-}}{r}\right) ^{2}+M\sup\limits_{B_{2r}\left( x_{m, k}\right) }w_{k}^{-}\right\} +C\omega \left( r\right) \\ &\leq C_{1}\left( n, M, L)\right) \left\{ \sup\limits_{B_{2r}\left( x_{m, k}\right) }\left( \frac{w_{k}^{-}}{r}\right) ^{2}+M\sup\limits_{B_{2r}\left( x_{m, k}\right) }w_{k}^{-}\right\} +C\omega \left( r\right) , \end{split} |
where L is the uniform Lipschitz constant of \{w_k^+\}_k (recall Lemma 3.4). Taking the \liminf as m, k\to\infty and using the uniform convergence of w_{k} to u we infer
0 \lt c_n \alpha ^{2}\bar{\beta}^{2}\leq C_{1}\left( n, M, L\right) \left\{ \sup\limits_{B_{2r}\left( x_0\right) }\left( \frac{u^{-}}{ r}\right) ^{2}+M\sup\limits_{B_{2r}\left( x_0\right) }u^{-}\right\} +C\omega \left( r\right) . |
Recalling that, by assumption, u^-(x) = o(|x-x_0|) as x\to x_0, we have
\sup\limits_{B_{2r}\left( 0\right) }\left( \frac{u^{-}}{r}\right) ^{2} = o \left(1\right) \qquad\text{as }r\rightarrow 0, |
and we get a contradiction.
In this section we want to show that u is a subsolution according to Definition 1.1. Note that, if x_{0}\in {\mathcal{F}}(u) is a regular point from the left with touching ball B\subset \Omega ^{-}(u), then near to x_{0}
u^{-}(x) = \beta \langle x-x_{0}, \nu \rangle ^{-}+o(| x-x_{0}| ), \quad \beta \geq 0, |
in B, and
u^{+}(x) = \alpha \langle x-x_{0}, \nu \rangle ^{+}+o(| x-x_{0}| ), \quad \alpha \geq 0 |
in \Omega \backslash B. Indeed, even if \beta = 0, then \Omega ^{+}(u) and \Omega ^{-}(u) are tangent to \{\langle x-x_{0}, \nu \rangle = 0\} at x_{0} since u^{+} is non-degenerate. Thus u has a full asymptotic development as in the next lemma. We want to show that \alpha \geq G(\beta, x_{0}, \nu). We follow closely [3] and [9].
Lemma 7.1. Assume that near x_{0}\in {\mathcal{F}}(u),
u(x) = \alpha \langle x-x_{0}, \nu \rangle ^{+}-\beta \langle x-x_{0}, \nu \rangle ^{-}+o(| x-x_{0}|), |
with \alpha >0, \beta \geq 0. Then
\alpha \geq G(\beta , x_{0}, \nu ). |
Proof. Assume by contradiction that \alpha < G(\beta, x_{0}, \nu). We construct a supersolution w\in {\rm{class}} which is strictly smaller than u at some point, contradicting the minimality of u. Let u_{0} be the two-plane solution, i.e.
u_{0}(x): = \lim\limits_{r\rightarrow 0}\frac{u(x_{0}+rx)}{r} = \alpha \langle x, \nu \rangle ^{+}-\beta \langle x, \nu \rangle ^{-}. |
Suppose that \alpha \leq G(\beta, x_{0}, \nu)-\delta _{0} with \delta _{0}>0. Fix \zeta = \zeta (\delta _{0}), to be chosen later. By Corollary 4.4, we can find w_{k}\in F\searrow u locally uniformly and, for r small, k large, the rescaling w_{k, r} satisfies the following conditions:
if \beta >0, then
w_{k, r}(x)\leq u_{0}+\zeta \min \{\alpha , \beta \}\ \text{on }\partial B_{1}; |
if \beta = 0, then
w_{k, r}(x)\leq u_{0}+\alpha \zeta \;\text{ on }\partial B_{1} |
and
w_{k, r}(x)\leq 0, \quad \text{in}\; \{\langle x, \nu \rangle \lt -\zeta \}\cap \overline{B}_{1}. |
In particular,
w_{k, r}(x)\leq u_{0}(x+\zeta \nu )\quad \text{on}\; \partial B_{1}. |
If \beta >0, let v satisfy
\begin{cases} {F(D}^{2}v) = rf_{1}^{r}, &\text{in } \{\langle x, \nu \rangle \gt -\zeta +\varepsilon ilon \phi (x)\} \\ {F(D}^{2}v^{-}) = rf_{2}^{r}, &\text{in } \{\langle x, \nu \rangle \lt -\zeta +\varepsilon ilon \phi (x)\} \\ v(x) = 0, &\text{on } \{\langle x, \nu \rangle = -\zeta +\varepsilon ilon \phi (x)\} \\ v(x) = u_{0}(x+\zeta \nu ), &\text{on } \partial B_{1}, \end{cases} | (7.1) |
where \phi \geq 0 is a cut-off function, \phi \equiv 0 outside B_{1/2}, \phi \equiv 1 inside B_{1/4}.
For \beta = 0, replace the second equation with v = 0.
Along the new free boundary, {\mathcal{F}}(v) = \{\langle x, \nu \rangle = -\zeta +\varepsilon ilon \phi (x)\} we have the following estimates:
|v_{\nu }^{+}-\alpha |\leq c(\varepsilon +\zeta )+Cr, \quad |v_{\nu }^{-}-\beta |\leq c(\varepsilon +\zeta )+Cr, |
with c, C universal.
Indeed,
v^{+}-\alpha \langle x, \nu \rangle ^{+} |
is a solution of
{F}(D^2(v-\alpha \langle x, \nu \rangle ^{+})) = rf_{1}^{r}. |
Thus, by standard C^{1, \gamma } regularity estimates (see [16,Theorem 1.1])
|v_{\nu }^{+}-\alpha |\leq C\left( \Vert v-\alpha \langle x, \nu \rangle ^{+}\Vert _{\infty }+[-\zeta +\varepsilon ilon \phi ]_{1, \gamma }+r\Vert f_{1}\Vert _{\infty }\right) , |
which gives the desired bound. Similarly, one gets the bound for v_{\nu }^{-}.
Hence, since \alpha \leq G(\beta, x_{0}, \nu (x_{0}))-\delta _{0}, say for \varepsilon = 2\zeta and \zeta, r small depending on \delta _{0}
v_{\nu }^{+} \lt G(v_{\nu }^{-}, x_{0}, \nu ), |
and the function,
\bar{w}_{k} = \left\{ \begin{array}{l} \min \{w_{k}, \lambda v(\frac{x-x_{0}}{\lambda })\}\quad \text{ in } B_{\lambda }(x_{0}), \\ w_{k}\quad \text{in }\Omega \setminus B_{\lambda }(x_{0}), \end{array} \right. |
is still in {\rm{class}}. However, the set
\{\langle x, \nu \rangle \leq -\zeta +\varepsilon ilon \phi \} |
contains a neighborhood of the origin, hence rescaling back x_{0}\in \Omega ^{-}(\bar{w}_{k}). We get a contradiction since x_{0}\in F(u) and \Omega ^{+}(u)\subseteq \Omega ^{+}(\bar{w}_{k}).
In this section we prove the weak regularity properties of the free boundary. Both statements and proofs are by now rather standard and follows the papers [3] and [9] for problems governed by homogeneous and inhomogeneous divergence equations, respectively. Thus we limit ourselves to the few points in which differences from the previous cases emerge. Denote by \mathcal{N}_{\varepsilon }\left(A\right) an \varepsilon-neighborhood of the set A. The following lemma provides a control of the {\mathcal{H}}^{n-1} measure of {\mathcal{F}}\left(u\right) and implies that \Omega ^{+}\left(u\right) is a set of finite perimeter.
Lemma 8.1. Let u be our Perron solution. Let x_{0}\in {\mathcal{F}}\left(u\right) \cap B_{1}. There exists a positive universal \delta_{0} < 1 such that, for every 0 < \varepsilon < \delta \leq \delta _{0}, the following quantities are comparable:
1. \frac{1}{\varepsilon }\left\vert \left\{ 0 < u < \varepsilon\right\} \cap B_{\delta }\left(x_{0}\right) \right\vert ,
2. \frac{1}{\varepsilon }\left\vert \mathcal{N}_{\varepsilon }\left({\mathcal{F}}\left(u\right) \right) \cap B_{\delta }\left(x_{0}\right) \right\vert ,
3. N\varepsilon ^{n-1}, where N is the number of any family of balls of radius \varepsilon, with finite overlapping, covering {\mathcal{F}}\left(u\right) \cap B_{\delta }\left(x_{0}\right) ,
4. {\mathcal{H}}^{n-1}\left({\mathcal{F}}\left(u\right) \cap B_{\delta }\left(x_{0}\right) \right).
Proof. From [3], it is sufficient to prove the following two equivalences:
c_{1}\varepsilon ^{n}\leq \int_{B_{\varepsilon }\left( x_{0}\right) }\left\vert \nabla u\right\vert ^{2}\leq C_{1}\varepsilon ^{n} | (8.1) |
and
c_{3}\varepsilon \delta ^{n-1}\leq \int_{\left\{ 0 \lt u \lt \varepsilon \right\} \cap B_{\delta }\left( x_{0}\right) }\left\vert \nabla u\right\vert ^{2}\leq C_{2}\varepsilon \delta ^{n-1} | (8.2) |
with universal constants c_{1}, c_{2}, C_{1}, C_{2}.
Since F \left(D^{2}u\right) = \inf_{\alpha }L_{\alpha }u where L_{\alpha } is a uniformly elliptic operator with constant coefficients and ellipticity constant \lambda, \Lambda , we have L_{\alpha }u^{+}\geq f_{1} in \Omega ^{+}\left(u\right) . Fix \alpha = \alpha _{0} and set
L_{\alpha _{0}} = L = \sum\limits_{i, j = 1}^{n}a_{ij}\partial _{_{i}{}_{j}}\text{, }\quad A = \left( a_{ij}\right) . |
The upper bound in (8.1) follows by the Lipschitz continuity of u. The lower bound follows from \sup_{B_{\varepsilon }\left(x_{0}\right) }u^{+}\geq c\varepsilon , c universal, \inf_{B_{\varepsilon }\left(x_{0}\right) }u^{+} = 0, the Lipschitz continuity of u, and the Poincaré inequality (see [1,Lemma 1.15]).
To prove (8.2), rescale by setting
u_{\delta }\left( x\right) = \frac{u\left( x_{0}+\delta x\right) }{\delta }, \text{ }f_{1}^{\delta }\left( x\right) = f_{1}\left( x_{0}+\delta x\right) \quad x\in B_{1} = B_{1}\left( 0\right) . |
Then Lu_{\delta }\geq \delta f_{1}^{\delta } in \Omega ^{+}\left(u^{\delta }\right) \cap B_{1}. For 0 < \varepsilon < \delta, let
u_{\delta, s, \varepsilon } = u_{s, \varepsilon }: = \max \left\{ s/\delta , \min \left\{ u_{\delta }, \varepsilon /\delta \right\} \right\} . |
We have:
\begin{split} & -\delta \int_{B_{1}}f_{1}^{\delta }u_{\varepsilon , s} = -\int_{B_{1}}u_{\varepsilon , s}Lu_{\delta }^{+} \\ & = \int_{B_{1}}\langle A\nabla u_{\delta }^{+}, \nabla u_{\varepsilon , s}^{+}\rangle dx-\int_{\partial B_{1}}\langle A\nabla u_{\delta }^{+}, \nu \rangle u_{\varepsilon , s}d\mathcal{H}^{n-1} \\ & = \int_{B_{1}\cap \{0 \lt s/\delta \lt u_{\delta } \lt \varepsilon /\delta \}}\langle A\nabla u_{\delta }, \nabla u_{\delta }\rangle dx-\int_{\partial B_{1}}\langle A\nabla u_{\delta }^{+}, \nu \rangle u_{\varepsilon , s}d \mathcal{H}^{n-1} \end{split} |
since \nabla u_{\varepsilon, s} = \nabla u_{\delta }\cdot \chi _{\{s/\delta < u_{\delta } < \varepsilon /\delta \}}.
By uniform ellipticity, since u^{+} is Lipschitz and f_{1} is bounded, we get (\delta < 1)
\int_{B_{1}\cap \{0 \lt s/\delta \lt u_{\delta } \lt \varepsilon /\delta \}}| \nabla u_{\delta }| ^{2}dx\leq C\frac{\varepsilon }{\delta }, |
with C universal. Letting s\rightarrow 0 and rescaling back, we obtain the upper bound in (8.2).
For the lower bound, let V be the solution to
\left\{ \begin{array}{l} LV = -\frac{\chi _{B_{\sigma }}}{| B_{\sigma }| }, \quad \mbox{in}\quad B_{1} \\ V = 0, \quad \mbox{on}\quad \partial B_{1} \end{array} \right. | (8.3) |
with \sigma to be chosen later. By standard estimates, see for example [12], V\leq C\sigma ^{2-n} and -\langle A\nabla V, \nu \rangle \sim C^{\ast } on \partial B_{1}, with C^{\ast } independent of \sigma. By Green's formula
\int_{B_{1}}(LV)\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon }-\left(L \frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon }\right)V = \int_{\partial B_{1}}\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon }\langle A\nabla V, \nu \rangle d\mathcal{H}^{n-1} | (8.4) |
since V = 0 on \partial B_{1}. We estimate
\delta \left| \int_{B_{1}}(LV)\frac{u_{\delta }^{+}u_{\varepsilon }}{ \varepsilon }dx\right| = \frac{\delta }{| B_{\sigma }| } \left|\int_{B_{\sigma }}\frac{u_{\delta }^{+}u_{\varepsilon }}{\varepsilon }dx\right| \leq \bar{C}\sigma , | (8.5) |
since u is Lipschitz and 0\leq u_{\varepsilon, 0}\leq \varepsilon /\delta . From (8.4) and (8.5) and the fact that \langle A_{\delta }\nabla V, \nu \rangle \sim -C^{\ast } on \partial B_{1} we deduce that
\begin{split} \delta \int_{B_{1}}\left(L\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon } \right)Vdx& \geq -\bar{C}\sigma -\delta \int_{\partial B_{1}}\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon }\langle A\nabla V, \nu \rangle d \mathcal{H}^{n-1} \\ & \geq -\bar{C}\sigma +C^{\ast }\delta \int_{\partial B_{1}}\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon }d\mathcal{H}^{n-1}. \end{split} |
Thus using that u^{+} is non-degenerate and choosing \sigma small enough (universal) we get that (\delta >\varepsilon )
\delta \int_{B_{1}}\left(L\frac{u_{\delta }^{+}u_{\varepsilon , 0}}{\varepsilon } \right)Vdx\geq \tilde{C}. | (8.6) |
On the other hand in \{0 < u_{\delta }^{+} < \varepsilon /\delta \}\cap B_{1},
Lu_{\delta }^{+}u_{\varepsilon , 0} = 2\delta u_{\varepsilon }f_{1}^{\delta }+\langle A\nabla u_{\delta }, \nabla u_{\delta }\rangle . | (8.7) |
Combining (8.6), (8.7) and using the ellipticity of A we get that
\frac{2\delta ^{2}}{\varepsilon }\int_{B_{1}}u_{\varepsilon }f_{1}^{\delta }V+\frac{\delta \Lambda }{\varepsilon }\int_{B_{1}}|\nabla u_{\delta }|^{2}V\geq \bar{C}. |
From the estimate on V we obtain that for \delta small enough
\frac{\delta }{\varepsilon }\int_{B_{1}}|\nabla u_{\delta }|^{2}\geq C |
for some C universal. Rescaling, we obtain the desired lower bound.
Lemma 8.1 implies that \Omega ^{+}(u)\cap B_{r}(x), x\in {\mathcal{F}}(u), is a set of finite perimeter. Next we show that in fact this perimeter is of order r^{n-1}.
Theorem 8.2. Let u be our Perron solution. Then, the reduced boundary of \Omega ^{+}(u) has positive density in \mathcal{H}^{n-1}-measure at any point of {\mathcal{F}}(u), i.e. for r < r_{0}, r_{0} universal,
\mathcal{H}^{n-1}({\mathcal{F}}^{\ast }(u)\cap B_{r}(x))\geq cr^{n-1} |
for every x\in {\mathcal{F}}(u).
Proof. The proof follows the lines of Corollary 4 in [3] and Theorem 8.2 in [9]. Let w_{k}\in {\rm{class}}, w_{k}\searrow u in \overline{ B}_{1} and L as in Lemma 8.1. Then \Omega ^{+}\left(u\right) \subset \subset \Omega ^{+}\left(w_{k}\right) and Lw_{k}\geq F\left(D^{2}w_{k}\right) = f_{1} in \Omega ^{+}(u). Let x_{0}\in {\mathcal{F}}(u). We rescale by setting
u_{r}(x) = \frac{u(x_{0}+rx)}{r}, \quad w_{k, r} = \frac{w_{k}(x_{0}+rx)}{r}\quad x\in B_{1}. |
Let V be the solution to (8.3). Since \nabla w_{k, r} is a continuous vector field in \overline{\Omega _{r}^{+}(u_{r})\cap B_{1}}, we can use it to test for perimeter. We get
\begin{split} & \int_{B_{1}\cap \Omega _{r}^{+}(u_{r})}\left( Vrf_{1}^{r}-w_{k, r}LV\right) \leq \int_{B_{1}\cap \Omega _{r}^{+}(u_{r})}\left( VLw_{kr}-w_{k, r}LV\right) \\ & = \int_{{\mathcal{F}}^{\ast }(u_{r})\cap B_{1}}\left( V\langle A\nabla w_{k, r}, \nu \rangle -w_{kr}\langle A\nabla V, \nu \rangle \right) d\mathcal{H} ^{n-1}-\int_{\partial B_{1}\cap \Omega _{r}^{+}(u_{r})}w_{kr}\langle A\nabla V, \nu \rangle d\mathcal{H}^{n-1}. \end{split} | (8.8) |
Using the estimates for V and the fact that the w_{k} are uniformly Lipschitz, we get that
\left| \int_{{\mathcal{F}}^{\ast }(u_{r})\cap B_{1}}V\langle A\nabla w_{k, r}, \nu \rangle d \mathcal{H}^{n-1}\right| \leq C(\sigma )\mathcal{H}^{n-1}({\mathcal{F}}^{\ast }(u_{r})\cap B_{1}). | (8.9) |
As in [3] we have, as k\rightarrow \infty ,
\int_{{\mathcal{F}}^{\ast }(u_{r})\cap B_{1}}w_{k, r}\langle A\nabla V, \nu \rangle d \mathcal{H}^{n-1}\rightarrow 0, |
\int_{\partial B_{1}\cap \Omega _{r}^{+}(u_{r})}w_{kr}\langle A\nabla V, \nu \rangle d\mathcal{H}^{n-1}\rightarrow \int_{\partial B_{1}}u_{r}^{+}\langle A\nabla V, \nu \rangle d\mathcal{H}^{n-1} |
and
-\int_{B_{1}\cap \Omega _{r}^{+}(u_{r})}w_{k, r}LV\rightarrow {{\rlap{-} \smallint }_{B_{\sigma }}u_{r}^{+}}. |
Passing to the limit in (8.8) and using all of the above we get
\begin{split} &\left |r\int_{B_{1}\cap \Omega ^{+}(u_{r})}Vf_{1}^{r}+{{\rlap{-} \smallint }_{B_{\sigma }}u_{r}^{+}}+\int_{\partial B_{1}}u_{r}^{+}\langle A\nabla V, \nu \rangle d \mathcal{H}^{n-1}\right| \\ & \leq C(\sigma )\mathcal{H}^{n-1}({\mathcal{F}}^{\ast }(u_{r})\cap B_{1}). \end{split} | (8.10) |
Since u is Lipschitz and non-degenerate, for \sigma small
\frac{1}{\left\vert B_{\sigma }\right\vert }\int_{B_{\sigma }}u_{r}^{+}\leq \bar{C}\sigma, |
and using the estimate for \langle A\nabla V, \nu \rangle
-\int_{\partial B_{1}}u_{r}^{+}\langle A\nabla V, \nu \rangle d\mathcal{H} ^{n-1}\geq \bar{c} \gt 0. |
Also, since f_{1}^{r} is bounded,
\int_{B_{1}\cap \Omega _{r}^{+}(u_{r})}Vf_{1}^{r}\leq \bar{C}(\sigma ). |
Hence choosing first \sigma and then r sufficiently small we get that
\mathcal{H}^{n-1}({\mathcal{F}}^{\ast }(u_{r})\cap B_{1})\geq \tilde{C}, |
\tilde{C} universal.
For the reader's convenience we collect here some explicit barrier functions which arise frequently in our arguments. Their proof is based on comparison arguments, together with the well known chain of inequalities
\mathcal{P}^-_{\lambda/n, \Lambda} u \le F(D^2u) \le c \Delta u, | (A.1) |
where \mathcal{P}^-_{\lambda/n, \Lambda} denotes the lower Pucci operator, and c = c(\lambda, \Lambda, n)> 0 since F is concave (see [5] for further details).
Lemma A.1 (Barrier for subsolutions). Let u satisfy
\begin{cases} F(D^2u)\ge f & in \;B_{2}(0)\setminus \overline{B}_{1}(0)\\ u\le a & on \;\partial B_{2}(0)\\ u\le 0 & on \;\partial B_{1}(0). \end{cases} |
Then
u(x)\le \alpha(x_1-1) + o(|x-e_1|)\qquad where \;\alpha\le c_1 a + c_2\|f\|_\infty, |
as x\to e_1, where the positive constants c_1, c_2 only depend on \lambda, \Lambda, n.
Proof. By comparison and (A.1) we infer that u\ge \phi in B_{2} \setminus \overline{B}_{1}, where \phi solves
\begin{cases} \Delta \phi = -c\|f\|_\infty & \text{in }B_{2}\setminus \overline{B}_{1}\\ \phi = a & \text{on }\partial B_{2}\\ \phi = 0 & \text{on }\partial B_{1}, \end{cases} |
for a universal c. Then direct calculations show that, for n\ge3,
\phi(x) = A(|x|^2-1)+B(|x|^{-n+2}-1), |
where
A = -\frac{c}{2n}\|f\|_\infty, \qquad B = \frac{3}{1-2^{-n+2}} A - \frac{1}{1-2^{-n+2}} a. |
Then the Lemma follows by choosing
\alpha : = \nabla \phi (e_1) \cdot e_1 = 2 A - (n-2) B. |
The proof in dimension n = 2 is analogous.
Lemma A.2 (Barrier for supersolutions). Let u satisfy
\begin{cases} F(D^2u)\le r f & in \;B_{2}(0)\setminus B_{1}(0)\\ u\ge 0 & on \;\partial B_{2}(0)\\ u\ge a \gt 0 & on \;\partial B_{1}(0). \end{cases} |
Then
u(x)\ge \alpha(x_1+2) + o(|x+2e_1|)\qquad \text{where }\alpha\ge c_1 a - c_2r\|f\|_\infty, |
as x\to -2e_1, whenever r\le \bar r, where the positive constants c_1, c_2 and \bar r only depend on \lambda, \Lambda, n.
Proof. By comparison and (A.1) we infer that u\ge \phi in B_{2} \setminus \overline{B}_{1}, where \phi solves
\begin{cases} \mathcal{P}^-_{\lambda/n, \Lambda} \phi = r\|f\|_\infty & \text{in }B_{2}\setminus \overline{B}_{1}\\ \phi = 0 & \text{on }\partial B_{2}\\ \phi = a & \text{on }\partial B_{1}. \end{cases} |
Then direct calculations show that
\phi(x) = A(|x|^2-4)+B(|x|^{-\gamma}-2^{-\gamma}), \qquad\text{where } \gamma = \frac{\Lambda n (n-1)}{\lambda} -1 \ge 1 |
and
A = \frac{n}{2(\gamma+2)\lambda}r\|f\|_\infty \gt 0, \qquad B = \frac{1}{1-2^{-\gamma}} a + \frac{3}{1-2^{-\gamma}} A \gt 0. |
To check this, one needs to choose r\le\bar r = \bar r(\gamma), in such a way that D^2\phi(x) has exactly one positive eigenvalue, for 1\le|x|\le2. Then the Lemma follows by choosing
\alpha : = \nabla \phi (-2e_1) \cdot e_1 = -4 A + \gamma 2^{-\gamma-1} B. |
Work partially supported by the INDAM-GNAMPA group. G. Verzini is partially supported by the project ERC Advanced Grant 2013 n. 339958: "Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT"and by the PRIN-2015KB9WPT Grant: "Variational methods, with applications to problems in mathematical physics and geometry".
The authors declare no conflict of interest.
[1] |
Dai X and Gao Z (2013) From model, signal to knowledge: A data-driven perspective of fault detection and diagnosis. IEEE T Ind Inform 9: 2226-2238. doi: 10.1109/TII.2013.2243743
![]() |
[2] |
Dai X, Gao Z, Breikin T, et al. (2012) High-gain observer-based estimation of parameter variations with delay alignment. IEEE T Automat Contr 57: 726-732. doi: 10.1109/TAC.2011.2169635
![]() |
[3] | Grewal MS (2011) Kalman filtering, Springer. |
[4] |
Guo L and Wang H (2006) Minimum entropy filtering for multivariate stochastic systems with non-gaussian noises. IEEE T Automat Contr 51: 695-700. doi: 10.1109/TAC.2006.872771
![]() |
[5] |
Guo L, Yin L, Wang H, et al. (2009) Entropy optimization filtering for fault isolation of nonlinear non-gaussian stochastic systems. IEEE T Automat Contr 54: 804-810. doi: 10.1109/TAC.2008.2009599
![]() |
[6] |
Reif K, Gunther S, Yaz E, et al. (1999) Stochastic stability of the discrete-time extended kalman filter. IEEE T Automat Contr 44: 714-728. doi: 10.1109/9.754809
![]() |
[7] |
Ren M, Zhang J, Jiang M, et al. (2015) Minimum ({h, φ})- entropy control for non-gaussian stochastic networked control systems and its application to a networked dc motor control system. IEEE Transactions on Control Systems and Technology 23: 406-411. doi: 10.1109/TCST.2014.2324978
![]() |
[8] | Ren M, Zhang Q and Zhang J (2019) An introductory survey of probability density function control. Systems Science & Control Engineering 7: 158-170. |
[9] | Wang S, Wang H, Fan R, et al. (2018) Objective pdf-shaping-based economic dispatch for power systems with intermittent generation sources via simultaneous mean and variance minimization. 2018 IEEE 14th International Conference on Control and Automation (ICCA), 927-934. |
[10] | Yang J and Chai T (2016) Data-driven demand forecasting method for fused magnesium furnaces. 2016 12th World Congress on Intelligent Control and Automation (WCICA), 2015-2022. |
[11] |
Yang J, Chai T, Luo C, et al. (2019) Intelligent demand forecasting of smelting process using data-driven and mechanism model. IEEE T Ind Electron 66: 9745-9755. doi: 10.1109/TIE.2018.2883262
![]() |
[12] |
Yao L, Qin J, Wang H, et al. (2012) Design of new fault diagnosis and fault tolerant control scheme for non-gaussian singular stochastic distribution systems. Automatica 48: 2305-2313. doi: 10.1016/j.automatica.2012.06.036
![]() |
[13] | Yin X, Zhang Q, Wang H, et al. (2019) Rbfnn-based minimum entropy filtering for a class of stochastic nonlinear systems. IEEE T Automat Contr, 1-1. |
[14] |
Yu M, Liu C, Li B, et al. (2016) An enhanced particle filtering method for gmti radar tracking. IEEE T Aero Elec Sys 52: 1408-1420. doi: 10.1109/TAES.2016.140561
![]() |
[15] | Zhang Q and Hu L (2018) Probabilistic decoupling control for stochastic non-linear systems using ekf-based dynamic set-point adjustment. 2018 UKACC 12th International Conference on Control (CONTROL), 330-335. |
[16] | Zhang Q and Sepulveda F (2017) Entropy-based axon-to-axon mutual interaction characterization via iterative learning identification. EMBEC & NBC 2017, 691-694, Springer. |
[17] | Zhang Q and Sepulveda F (2017) A model study of the neural interaction via mutual coupling factor identification. 2017 39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), 3329-3332. |
[18] | Zhang Q and Yin X (2018) Observer-based parametric decoupling controller design for a class of multi-variable non-linear uncertain systems. Systems Science & Control Engineering 6: 258-267. |
[19] | Zhou Y, Zhang Q and Wang H (2016) Enhanced performance controller design for stochastic systems by adding extra state estimation onto the existing closed loop control. 2016 UKACC 11th International Conference on Control (CONTROL), 1-6. |
[20] |
Zhou Y, Zhang Q, Wang H, et al. (2018) Ekf-based enhanced performance controller design for nonlinear stochastic systems. IEEE T Automat Contr 63: 1155-1162. doi: 10.1109/TAC.2017.2742661
![]() |
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