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We consider the general second order elliptic equation in divergence form
n∑i=1∂∂xiai(x,u(x),Du(x))=b(x,u(x),Du(x)),x∈Ω, | (1.1) |
where Ω is an open set of Rn, n≥2, the vector field (ai(x,u,ξ))i=1,…,n and the right hand side b(x,u,ξ) are Carathéodory applications defined in Ω×R×Rn. We study the elliptic equations (1.1) under some general growth conditions on the gradient variable ξ=Du, named p,q− conditions, which we are going to state in the next Section 3.2. Under these assumptions we will obtain the local boundedness of the weak solutions, as stated in Theorem 3.2.
A strong motivation to study the local boundedness of solutions to (1.1) relies on the recent research in [53], where the local Lipschitz continuity of the weak solutions of the Eq (1.1) has been obtained under general growth conditions, precisely some p,q−growth assumptions, with the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable. In [53] the Sobolev class of functions where to start in order to get more regularity of the weak solutions was pointed out, precisely u∈W1,qloc(Ω)∩L∞loc(Ω). That is, in particular the local boundedness u∈L∞loc(Ω) of weak solutions is a starting assumption for more interior regularity; i.e., for obtaining u∈W1,∞loc(Ω) and more. When we refer to the classical cases this is a well known aspect which appears in the mathematical literature on a-priori regularity: in fact, for instance, under the so-called natural growth conditions, i.e., when q=p, then the a-priori boundedness of u often is a natural assumption to obtain the boundedness of its gradient Du too; see for instance the classical reference book by Ladyzhenskaya-Ural'tseva [45,Chapter 4,Section 3] and the C1,α−regularity result by Tolksdorf [60].
The aim of this paper is to derive the local boundedness of solutions to (1.1); i.e., to deduce the local boundedness of u only from the growth assumptions on the vector field (ai(x,u,ξ))i=1,…,n and the right hand side b(x,u,ξ) in (1.1). The precise conditions and the related results are stated in Section 3.
We start with a relevant aspect to remark in our context, which is different from what happens in minimization problems and it is peculiar for equations: although under p,q−growth conditions (with p<q) the Eq (1.1) is elliptic and coercive in W1,ploc(Ω), it is not possible a-priori to look for weak solutions only in the Sobolev class W1,ploc(Ω), but it is necessary to emphasize that the notion of weak solution is consistent if a-priori we assume u∈W1,qloc(Ω). This is detailed in Section 2.
Going into more detail, in this article we study the local boundedness of weak solutions to the p−elliptic equation (1.1) with q−growth, 1<p≤q<p+1, as in (3.2), (3.3) and (3.7)–(3.10). Starting from the integrability condition u∈W1,qloc(Ω) on the weak solution, under the bound on the ratio qp
qp<1+1n−1 |
we obtain u∈L∞loc(Ω). The proof is based on the powerful De Giorgi technique [29], by showing first a Caccioppoli-type inequality and then applying an iteration procedure. The result is obtained via a Sobolev embedding theorem on spheres, a procedure introduced by Bella and Schäffner in [3], that allows a dimensional gain in the gap between p and q. This idea has been later used by the same authors in [4], by Schäffner [58] and, particularly close to the topic of our paper, by Hirsch and Schäffner [43] and De Rosa and Grimaldi [30], where the local boundedness of scalar minimizers of a class of convex energy integrals with p,q−growth was obtained with the bound qp<1+qn−1.
Some references about the local boundedness of solutions to elliptic equations and systems, with general and p,q−growth conditions, start by Kolodīĭ [44] in 1970 in the specific case of some anisotropic elliptic equations. The local boundedness of solution to classes of anisotropic elliptic equations or systems have been investigated by the authors [18,19,20,21,22,23,24] and by Di Benedetto, Gianazza and Vespri [31]. Other results on the boundedness of solutions of PDEs or of minimizers of integral functionals can be found in Boccardo, Marcellini and Sbordone [7], Fusco and Sbordone [37,38], Stroffolini [59], Cianchi [14], Pucci and Servadei [57], Cupini, Leonetti and Mascolo [17], Carozza, Gao, Giova and Leonetti [12], Granucci and Randolfi [42], Biagi, Cupini and Mascolo [5].
Interior L∞−gradient bound, i.e., the local Lipschitz continuity, of weak solutions to nonlinear elliptic equations and systems under non standard growth conditions have been obtained since 1989 in [46,47,48,49,50]. See also the following recent references for other Lipschitz regularity results: Colombo and Mingione [16], Baroni, Colombo and Mingione [1], Eleuteri, Marcellini and Mascolo [34,35], Di Marco and Marcellini [32], Beck and Mingione [2], Bousquet and Brasco [9], De Filippis and Mingione [26,27], Caselli, Eleuteri and Passarelli di Napoli [13], Gentile [39], the authors and Passarelli di Napoli [25], Eleuteri, Marcellini, Mascolo and Perrotta [36]; see also [53]. For other related results see also Byun and Oh [10] and Mingione and Palatucci [55]. The local boundedness of the solution u can be used to achieve further regularity properties, as the Hölder continuity of u or of its gradient Du; we limit here to cite Bildhauer and Fuchs [6], Düzgun, Marcellini and Vespri [33], Di Benedetto, Gianazza and Vespri [31], Byun and Oh [11] as examples of this approach. For recent boundary regularity results in the context considered in this manuscript we mention Cianchi and Maz'ya [15], Bögelein, Duzaar, Marcellini and Scheven [8], De Filippis and Piccinini [28]. A well known reference about the regularity theory is the article [54] by Giuseppe Mingione. We also refer to [51,52,53] and to De Filippis and Mingione [27], Mingione and Rădulescu [56], who have outlined the recent trends and advances in the regularity theory for variational problems with non-standard growths and non-uniform ellipticity.
In order to investigate the consistency of the notion of weak solution, we anticipate the ellipticity and growth conditions of Section 3, in particular the growth in (3.3), (3.4),
{|ai(x,u,ξ)|≤Λ{|ξ|q−1+|u|γ1+b1(x)},∀i=1,…,n,|b(x,u,ξ)|≤Λ{|ξ|r+|u|γ2+b2(x)}. | (2.1) |
As well known the integral form of the equation, for a smooth test function φ with compact support in Ω, is
∫Ωn∑i=1ai(x,u,Du)φxidx+∫Ωb(x,u,Du)φdx=0. |
Let us discuss the summability conditions for the pairings above to be well defined. Since each ai in the gradient variable ξ grows at most as |ξ|q−1, more generally we can consider test functions φ∈W1,q0(Ω). In fact, starting with the first addendum and applying the Young inequality with conjugate exponents qq−1 and q, we obtain the L1 local summability
|ai(x,u,Du)φxi|≤Λ{|Du|q−1+|u|γ1+b1(x)}|φxi|≤Λq−1q{|Du|q−1+|u|γ1+b1(x)}qq−1+Λq|φxi|q∈L1loc(Ω) |
if u∈W1,qloc(Ω) and if qq−1γ1≤q∗, where q∗ is the Sobolev conjugate exponent of q, and b1∈Lqq−1loc(Ω). On γ1 equivalently we require (if q<n) γ1≤q∗q−1q=nqn−qq−1q=n(q−1)n−q, which essentially corresponds to our assumption (3.8) below (the difference being the strict sign "<" for compactness reasons). We also observe that the summability condition b1∈Lqq−1loc(Ω) is satisfied if b1∈Ls1loc(Ω), with s1>nq−1, as in (3.10).
Similar computations apply to |b(x,u,ξ)φ|, again if q<n and with conjugate exponents q∗q∗−1 and q∗,
|b(x,u,Du)φ|≤Λ{|Du|r+|u|γ2+b2(x)}|φ|≤Λq∗−1q∗{|Du|r+|u|γ2+b2(x)}q∗q∗−1+Λq∗|φ|q∗∈L1loc(Ω) |
and we obtain b2∈Lq∗q∗−1loc(Ω) (compare with (3.10), where b2∈Ls2loc(Ω) with s2>np, since q∗q∗−1≤p∗p∗−1≤p∗p∗−p=np) and the conditions for r and γ2 expressed by rq∗q∗−1≤q and γ2q∗q∗−1≤q∗; i.e., for the first one,
r≤qq∗−1q∗=qnqn−q−1nqn−q=q+qn−1, |
which correspond to the more strict assumption (3.9), with r<p+pn−1, with the sign "<" and where q is replaced by p. Finally for γ2 we obtain γ2≤q∗−1, which again corresponds to our assumption (3.8) with the strict sign.
Therefore our assumptions for Theorem 3.2 are more strict than that ones considered in this section and they are consistent with a correct definition of weak solution to the elliptic equation (1.1).
Let ai:Ω×R×Rn→R, i=1,...,n, and b:Ω×R×Rn→R be Carathéodory functions, Ω be an open set in Rn, n≥2. Consider the nonlinear partial differential equation
n∑i=1∂∂xiai(x,u,Du)=b(x,u,Du). | (3.1) |
For the sake of simplicity we use the following notation: a(x,u,ξ)=(ai(x,u,ξ))i=1,...,n, for all i=1,…,n.
We assume the following properties:
● p−ellipticity condition at infinity:
there exist an exponent p>1 and a positive constant λ such that
⟨a(x,u,ξ),ξ⟩≥λ|ξ|p, | (3.2) |
for a.e. x∈Ω, for every u∈R and for all ξ∈Rn such that |ξ|≥1.
● q−growth condition:
there exist exponents q≥p, γ1≥0, s1>1, a positive constant Λ and a positive function b1∈Ls1loc(Ω) such that, for a.e. x∈Ω, for every u∈R and for all ξ∈Rn,
|a(x,u,ξ)|≤Λ{|ξ|q−1+|u|γ1+b1(x)}; | (3.3) |
● growth conditions for the right hand side b(x,u,ξ):
there exist further exponents r≥0, γ2≥0, s2>1 and a positive function b2∈Ls2loc(Ω) such that
|b(x,u,ξ)|≤Λ{|ξ|r+|u|γ2+b2(x)}, | (3.4) |
for a.e. x∈Ω, for every u∈R and for all ξ∈Rn.
Without loss of generality we can assume Λ≥1 and b1,b2≥1 a.e. in Ω. We recall the definition of weak solution to (3.1).
Definition 3.1. A function u∈W1,qloc(Ω) is a weak solution to (3.1) if
∫Ω{n∑i=1ai(x,u,Du)φxi+b(x,u,Du)φ}dx=0 | (3.5) |
for all φ∈W1,q(Ω), suppφ⋐Ω.
Our aim is to study the local boundedness of weak solutions to (3.1). Since this regularity property is trivially satisfied for functions in W1,qloc(Ω) with q>n, from now on we only consider the case q≤n; more precisely
1<p<n,p≤q≤n, | (3.6) |
since if q>n then weak solutions are Hölder continuous as an application of the Sobolev-Morrey embedding theorem, see Remark 3.3.
Other assumptions on the exponents are
{q<1+pqp<1+1n−1 | (3.7) |
0≤γ1<n(q−1)n−p,0≤γ2<n(p−1)+pn−p, | (3.8) |
0≤r<p+pn−1, | (3.9) |
s1>nq−1,s2>np. | (3.10) |
Under the conditions described above the following local boundedness result holds.
Theorem 3.2 (Boundedness result). Let u∈W1,qloc(Ω), 1<q≤n, be a weak solution to the elliptic equation (3.1). If (3.2)–(3.4) and (3.6)–(3.10) hold true, then u is locally bounded. Precisely, for every open set Ω′⋐Ω there exist constants R0,c>0 depending on the data n,p,q,r,γ1,γ2,s1,s2 and on the norm ‖u‖W1,q(Ω′) such that ‖u‖L∞(BR/2(x0))≤c for every R≤R0, with BR0(x0)⊆Ω′.
Remark 3.3. We already observed that if q>n then the weak solutions to (3.1) are locally Hölder continuous. Let us now discuss why in (3.6) we do not consider the case p=q=n. If p=q (≤n), the same computations in the proof of Theorem 3.2 work with the set of assumptions (3.8)–(3.10). They can be written, coherently with the previous ones, as
0≤γ1<p∗p−1p,0≤γ2<p∗−1 | (3.11) |
0≤r<p−pp∗, | (3.12) |
s1>p∗p(p∗−p)(p−1),s2>p∗p∗−p. | (3.13) |
Here p∗ denotes the Sobolev exponent appearing in the Sobolev embedding theorem for functions in W1,p(Ω) with Ω bounded open set in Rn; i.e.,
p∗:={npn−p if p<nany real number >n, if p=n. | (3.14) |
Following the computations in [40,Theorem 2.1] and [41,Chapter 6] it can be proved that the weak solutions to (3.1) are quasi-minima of the functional
F(u):=∫Ω(|Du|p+|u|τ+bpp−11+bp∗p∗−12)dx, | (3.15) |
with τ:=max. It is known that if
\begin{equation} \tau < p^*\quad \text{and}\quad b_1^{\frac{p}{p-1}}+b_2^{\frac{p^*}{p^*-1}}\in L^{1+\delta} \ \text{ with } \delta > 0 \end{equation} | (3.16) |
then the gradient of quasi-minima of the functional (3.15) satisfies a higher integrability property; i.e., they belong to W^{1, p+\epsilon} , for some \epsilon > 0 .
Under our assumptions, (3.16) is satisfied; indeed, taking into account that we are considering p = q , by (3.10)
s_1 > \frac{n}{p-1}\ge \frac{p}{p-1} |
and, by (3.13)
s_2 > \frac{p^*}{p^*-p}\ge \frac{p^*}{p^*-1}. |
Analogously, by (3.11),
\gamma_1\frac{p}{p-1} < p*, \qquad \gamma_2\frac{p^*}{p^*-1} < (p^*-1)\frac{p^*}{p^*-1} = p^*. |
In particular, if p = q = n the quasi-minima of (3.15) are in W_{\rm{loc}}^{1, n+\epsilon}(\Omega) for some \epsilon > 0 , therefore the weak solutions to (3.1) are Hölder continuous. We refer to [41] Chapter 6 for more details.
If p\ge 1 and d\in \mathbb{N} , d\ge 2 , we define
(p_d)^*: = \left\{\begin{array}{lr} \frac{dp}{d-p}& \ \text{if }p < d \\ \text{any real number } > d , \ & \text{if }p = d .\end{array}\right. |
The Sobolev exponent appearing in the Sobolev embedding theorem for functions in W^{1, p}(\Omega) , p\ge 1 , with \Omega bounded open set in \mathbb{R}^n , is (p_n)^* and will be denoted, as usual, p^* .
Let t\in \mathbb{R} , t > 0 . We define t_* as follows:
\frac{1}{t_*}: = \min\left\{\frac{1}{t}+\frac{1}{n-1}, 1\right\}. |
We have, if n\ge 3 ,
t_* = \left\{\begin{array}{ll}\frac{t(n-1)}{t+n-1}& \text{if}\ t > \frac{n-1}{n-2}\\ 1 & \text{if}\ 1\le t\le \frac{n-1}{n-2}, \end{array}\right. |
and, if n = 2 , t_* = 1 for every t .
We notice that, if n\ge 3 ,
((t_*)_{n-1})^* = \left\{\begin{array}{ll} t& \text{if }\ t > \frac{n-1}{n-2}\\ \frac{n-1}{n-2} & \text{if }\ 1\le t\le \frac{n-1}{n-2}\end{array}\right. |
and, if n = 2 , for every t , ((t_*)_{n-1})^* stands for any real number greater than 1 .
Remark 4.1. Let us consider the exponents p, q satisfying (3.6) and (3.7) in Section 3. We notice that
\begin{equation} \frac{1}{\left(\frac{p}{p-q+1}\right)_{*}} = \left\{\begin{array}{ll}\frac{1}{\frac{p}{p-q+1}}+\frac{1}{n-1} & \text{if }q > 1+\frac{p}{n-1} \\ 1& \text{if } q\le 1+\frac{p}{n-1} .\end{array}\right. \end{equation} | (4.1) |
Due to assumption (3.7), if n = 2 , then \left(\frac{p}{p-q+1}\right)_{*} = 1 .
Moreover, if we denote t: = \left(\frac{p}{p-q+1}\right)_{*} then, if n\ge 3 ,
\begin{equation} (t_{n-1})^* = \left\{\begin{array}{ll}\frac{p}{p-q+1} &\text{if }q > 1+\frac{p}{n-1} \\ \frac{n-1}{n-2}& \text{if }q\le 1+\frac{p}{n-1} , \end{array}\right. \end{equation} | (4.2) |
if instead n = 2 than (t_{n-1})^* is any real number greater than 1 .
Let p, q satisfy (3.6) and (3.7). It is easy to prove that
\begin{equation} \frac{p}{p-q+1} < q^*. \end{equation} | (4.3) |
In the following it will be useful to introduce the following notation:
\nu: = \frac{1}{\left(\frac{p}{p-q+1}\right)_{*}}-\frac{1}{p}, |
or, more explicitly,
\begin{equation} \nu = \left\{\begin{array}{ll}\frac{p-1}{p} & \text{if }q\le 1+\frac{p}{n-1} \\ 1-\frac{q}{p}+\frac{1}{n-1} & \text{if } q > 1+\frac{p}{n-1} . \end{array}\right. \end{equation} | (4.4) |
Remark 4.2. Assume 1 < p\le q . Then easy computations give
\begin{equation} \nu > 0\Leftrightarrow q < \frac{pn}{n-1}, \qquad \nu = 0\Leftrightarrow q = \frac{pn}{n-1}. \end{equation} | (4.5) |
To get the sharp bound for q , we use a result proved in [43], see also [3,4,30,58]. Here we denote S_{\sigma}(x_0) the boundary of the ball B_{\sigma}(x_0) in \mathbb{R}^n .
Lemma 4.3. Let n\in \mathbb{N} , n\ge 2 . Consider B_{\sigma}(x_0) ball in \mathbb{R}^n and u\in L^1(B_{\sigma}(x_0)) and s > 1 . For any 0 < \rho < \sigma < +\infty , define
I(\rho, \sigma, u): = \inf\left\{\int_{B_{\sigma}(x_0)}|u||D\eta|^s\, dx\, :\, \eta\in C_0^1(B_{\sigma}(x_0)), \ 0\le \eta\le 1, \ \eta = 1\ \mathit{\text{ in }}\ B_{\rho}(x_0)\right\}. |
Then for every \delta\in]0, 1] ,
I(\rho, \sigma, v)\le (\sigma-\rho)^{s-1+\frac{1}{\delta}} \left(\int_{\rho}^{\sigma} \left(\int_{S_r(x_0)}|v|\, d\mathcal{H}^{n-1}\right)^{\delta}\, dr\right)^{\frac{1}{\delta}}. |
The following result is the Sobolev inequality on spheres.
Lemma 4.4. Let n\in \mathbb{N} , n\ge 3 , and \gamma\in [1, n-1[. Then there exists c depending on n and \gamma such that for every u\in W^{1, p}(S_1(x_0), d\mathcal{H}^{n-1})
\left(\int_{S_1(x_0)}|u|^{(\gamma_{n-1})^*}\, d\mathcal{H}^{n-1}\right)^{\frac{1}{(\gamma_{n-1})^*}} \le c\left(\int_{S_1(x_0)}(|Du|^{\gamma}+|u|^{\gamma})\, d\mathcal{H}^{n-1}\right)^{\frac{1}{\gamma}}. |
Lemma 4.5. Let n = 2 . Then there exists c such that for every u\in W^{1, 1}(S_1(x_0), d\mathcal{H}^{1}) and every r > 1 ,
\left(\int_{S_1(x_0)}|u|^{r}\, d\mathcal{H}^{1}\right)^{\frac{1}{r}} \le c\left(\int_{S_1(x_0)}(|Du|+|u|)\, d\mathcal{H}^{1}\right). |
Proof. By the one-dimensional Sobolev inequality
\|u\|_{L^{\infty}(S_1(x_0))}\le c\|u\|_{W^{1, 1}(S_1(x_0))}. |
Then, for every r > 1 ,
\left(\int_{S_1(x_0)}|u|^{r}\, d\mathcal{H}^{n-1}\right)^{\frac{1}{r}} \le c \|u\|_{L^{\infty}(S_1(x_0))}\le c\|u\|_{W^{1, 1}(S_1(x_0))}. |
We conclude this section, by stating a classical result; see, e.g., [41]. that will be useful to prove Theorem 3.2.
Lemma 4.6. Let \alpha > 0 and (J_h) a sequence of real positive numbers, such that
J_{h+1} \leq A\, \lambda^h J_h^{1+\alpha}, |
with A > 0 and \lambda > 1 .
If J_0 \leq A^{-\frac{1}{\alpha}}\lambda^{-\frac{1}{\alpha^2}} , then J_h \le \lambda^{-\frac{h}{\alpha}}J_0 and \lim_{h\to \infty} J_h = 0 .
Under the assumptions in Section 3 we have the following Caccioppoli-type inequality.
Given a measurable function u: \Omega\to \mathbb{R} , with \Omega open set in \mathbb{R}^n , and fixed x_0\in \mathbb{R}^n , k\in \mathbb{R} and \tau > 0 , we denote the super-level set of u as follows:
A_{k, \tau}(x_0): = \{x \in B_{\tau}(x_0)\, :\, u(x) > k \}; |
usually dropping the dependence on x_0 . We denote |A_{k, \tau}| its Lebesgue measure.
Proposition 5.1 (Caccioppoli's inequality). Let u\in W^{1, q}_{\rm{loc}}(\Omega) be a weak solution to (3.1). If (3.6)–(3.10) hold true, then there exists a constant c > 0 , such that for any B_{R_0}(x_0) \Subset \Omega , 0 < \rho < R\le R_0
\begin{align} & \int_{B_{\rho}}|D(u-k)_+|^p\, dx \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k)_+\|_{W^{1, p}(B_{R})}^{\frac{p}{p-q+1}}|A_{k, {R}}|^{\frac{p}{p-q+1}\nu} \\ & +c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p\gamma_1}{q-1}}|A_{k, R}|^{1-\frac{p\gamma_1}{p^*(q-1)}} + c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p}{p-r}}|A_{k, R}|^{1-\frac{1}{p^*\frac{p-r}{p}}} \\ & + c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2+1}|A_{k, R}|^{1-\frac{\gamma_2+1}{p^*}}+ c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2}|A_{k, R}|^{1-\frac{\gamma_2}{p^*}} \\ &+ ck^{\gamma_2}\|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{p^*}}+c \left(k^{\frac{p\gamma_1}{q-1}} + k^{\gamma_2} \right) |A_{k, R}|\\ & +c \|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}+c|A_{k, R}|^{1-\frac{p}{s_1(q-1)}} \end{align} | (5.1) |
with \nu as in (4.4) and c is a constant depending on n, p, q, r, R_0 , the L^{s_1} -norm of b_1 and the L^{s_2} -norm of b_2 in B_{R_0} .
Proof. Without loss of generality we assume that the functions b_1, b_2 in (3.3) are a.e. greater than or equal to 1 in \Omega . We split the proof into steps.
Step 1. Consider B_{R_0}(x_0) \Subset \Omega , 0 < \frac{R_0}{2}\le \rho < R\le R_0\le 1 .
We set
\begin{equation} \mathcal{A}(\rho, R): = \{\eta\in C_0^{\infty}(B_R(x_0))\, :\, \eta = 1\, \text{in}\ B_{\rho}(x_0), \ 0\le \eta\le 1\}. \end{equation} | (5.2) |
For every \eta \in \mathcal{A}(\rho, R) and fixed k > 1 we define the test function \varphi_k as follows
\varphi_{k}(x): = (u(x)-k)_+[\eta (x)]^{\mu}\quad \text{for a.e.}\ x\in B_{R_0}(x_0), |
with
\begin{equation} \mu: = \frac{p}{p-q+1} \end{equation} | (5.3) |
that is greater than 1 because q > 1 .
Notice that \varphi_{k}\in W_0^{1, q}(B_{R_0}(x_0)) , \operatorname{supp\, }\varphi_k\Subset B_R(x_0) .
Step 2. Let us consider the super-level sets:
A_{k, R}: = \{x \in B_R(x_0)\, :\, u(x) > k \}. |
In this step we prove that
\begin{align} &\int_{A_{k, \rho}} |Du|^p \, dx\le c \left\{\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\right.\\ & + \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+(u-k)^{\frac{p}{p-r}}+(u-k)^{\gamma_2+1}+(u-k)^{\gamma_2}\right)\, dx \\ &+ \left.c\int_{A_{k, R}} \left(k^{\gamma_2}(u-k)+b_2(u-k)+k^{\frac{p\gamma_1}{q-1}}+k^{\gamma_{2}}+b_1^{\frac{p}{q-1}}\right)\, dx\right\} \end{align} | (5.4) |
for some constant c independent of u and \eta .
Using \varphi_k as a test function in (3.5) we get
\begin{equation} \begin{aligned} I_1: = & \int_{A_{k, R}} \langle a(x, u, Du), Du\rangle \, \eta^{\mu}\, dx \\ = & -\mu \int_{A_{k, R}} \langle a(x, u, Du), D\eta\rangle \eta^{\mu-1}(u-k)\, dx \\&- \int_{A_{k, R}} b(x, u, Du) (u-k)\, \eta^{\mu}\, dx = :I_2+I_3. \end{aligned} \end{equation} | (5.5) |
Now, we separately consider and estimate I_i , i = 1, 2, 3 .
ESTIMATE OF I_3
Using (3.4) we obtain
I_3\le \Lambda\int_{A_{k, R}}\eta^{\mu} \left\{|Du|^{r}(u-k)+|u|^{\gamma_2}(u-k) +b_2(u-k)\right\} \, dx. |
We estimate the right-hand side using the Young inequality, with exponents \frac{p}{r} and \frac{p}{p-r} , and (3.2). There exists c , depending on \lambda , \Lambda , n , p , r , such that
\begin{align} & \Lambda|Du|^{r}(u-k) \le \frac{\lambda}{4}|Du|^p+c(u-k)^{\frac{p}{p-r}} \\ & \le \frac{1}{4} \langle a(x, u, Du), Du\rangle +c(u-k)^{\frac{p}{p-r}}\qquad \text{a.e. in }\{|Du|\ge 1\} . \end{align} | (5.6) |
and, recalling that b_2\ge 1 ,
\Lambda|Du|^{r}(u-k)\le \Lambda (u-k) \le \Lambda b_2(u-k)\qquad \text{a.e. in } \{|Du| < 1\} . |
Therefore,
\begin{align} I_3&\le \frac{1}{4}\int_{A_{k, R}\cap \{|Du|\ge 1\}} \langle a(x, u, Du), Du\rangle \eta^{\mu}\, dx \\ & +c \int_{A_{k, R}}\eta^{\mu} \left\{(u-k)^{\frac{p}{p-r}}+|u|^{\gamma_2}(u-k) +b_2(u-k)\right\}\, dx. \end{align} | (5.7) |
Collecting (5.5)–(5.7) we get
\begin{align*} \nonumber &\frac{3}{4}\int_{A_{k, R}\cap \{|Du|\ge 1\}} \langle a(x, u, Du), Du\rangle \eta^{\mu}\, dx\le I_2-\int_{A_{k, R}\cap \{|Du|\le 1\}} \langle a(x, u, Du), Du\rangle \eta^{\mu}\, dx \\& + c\int_{A_{k, R}}\eta^{\mu} \left\{(u-k)^{\frac{p}{p-r}}+ |u|^{\gamma_2}(u-k) +b_2(u-k) \right\}\, dx. \end{align*} |
Using (3.2) and (3.3) we get
\begin{align} &\frac{3\lambda}{4}\int_{A_{k, R}\cap \{|Du|\ge 1\}} |Du|^p \eta^{\mu}\, dx\le I_2+2\Lambda \int_{A_{k, R}\cap \{|Du|\le 1\}} (|u|^{\gamma_2}+b_1) \eta^{\mu}\, dx \\& + c\int_{A_{k, R}} \eta^{\mu} \left\{(u-k)^{\frac{p}{p-r}}+ |u|^{\gamma_2}(u-k) +b_2(u-k) \right\}\, dx. \end{align} | (5.8) |
ESTIMATE OF I_2 . For a.e. x\in A_{k, R}\cap \{\eta\ne 0\} we have
\begin{equation} \mu |\langle a(x, u, Du), D\eta\rangle|(u-k) \eta^{\mu-1}\le \mu\Lambda \left\{|Du|^{q-1}+|u|^{\gamma_1} +b_1\right\}|D\eta|(u-k)\eta^{\mu-1}. \end{equation} | (5.9) |
For a.e. x\in \{|Du|\ge 1\} \cap A_{k, R}\cap \{\eta\ne 0\} , by q < p+1 and the Young inequality with exponents \frac{p}{q-1} and \frac{p}{p-q+1} , and noting that \mu-1 = \mu\frac{q-1}{p} , we get
\begin{align} & \mu\Lambda|Du|^{q-1}|D\eta|(u-k)\eta^{\mu-1} \\ &\le \frac{\lambda}{4}|Du|^{p}\eta^{\mu}+ c(\lambda, \Lambda)\mu^\frac{p}{p-q+1} |D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}. \label{I3-2} \end{align} | (5.10) |
On the other hand we have
\begin{equation} \mu\Lambda|Du|^{q-1}|D\eta|(u-k)\eta^{\mu-1}\le \mu\Lambda |D\eta|(u-k)\eta^{\mu-1} \end{equation} | (5.11) |
a.e. in \{|Du| < 1\} \cap A_{k, R}\cap \{\eta\ne 0\} .
Therefore,
\begin{align*} &I_2\le \frac{\lambda}{4}\int_{A_{k, R}\cap \{|Du|\ge 1\}} |Du|^p \eta^{\mu}\, dx+ c(\lambda, \Lambda)\mu^\frac{p}{p-q+1} \int_{A_{k, R}\cap \{|Du|\ge 1\}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\\ &+ \int_{A_{k, R}} |D\eta|(u-k)\eta^{\mu-1}\, dx + c\int_{A_{k, R}}|D\eta|\eta^{\mu-1}\left\{ |u|^{\gamma_1} +b_1\right\}(u-k)\, dx. \end{align*} |
By (5.8) and the inequality above, we get
\begin{align*} \nonumber &\frac{\lambda}{2}\int_{A_{k, R}\cap \{|Du|\ge 1\}} |Du|^p \eta^{\mu}\, dx\le c(\lambda, \Lambda, p, q) \int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\\ & + \int_{A_{k, R}}|D\eta|\eta^{\mu-1}\left(|u|^{\gamma_1} +b_1\right)(u-k)\, dx \\ &+ c\int_{A_{k, R}}\eta^{\mu} \left((u-k)^{\frac{p}{p-r}}+ |u|^{\gamma_2}(u-k)+|u|^{\gamma_2} +b_2(u-k)+b_1\right)\, dx. \end{align*} |
Taking into account that b_1\ge 1
\begin{align*} &\int_{A_{k, R}} |Du|^p\eta^{\mu}\, dx = \int_{A_{k, R}\cap \{|Du|\ge 1\}} |Du|^p\eta^{\mu}\, dx + \int_{A_{k, R}\cap \{|Du| < 1\}} |Du|^p\eta^{\mu}\, dx\\ & \le \int_{A_{k, R}\cap \{|Du|\ge 1\}} |Du|^p\eta^{\mu}\, dx + \int_{A_{k, R}} b_1\eta^{\mu}\, dx, \end{align*} |
therefore
\int_{A_{k, R}} (|Du|^p-b_1)\eta^{\mu}\, dx\le \int_{A_{k, R}\cap \{|Du|\ge 1\}}|Du|^p\eta^{\mu}\, dx |
and we obtain
\begin{align} &\int_{A_{k, \rho}} |Du|^p \, dx\le c \int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\\ & + \int_{A_{k, R}}|D\eta|\eta^{\mu-1}\left( |u|^{\gamma_1} +b_1\right)(u-k)\, dx \\ &+ c\int_{A_{k, R}}\eta^{\mu} \left((u-k)^{\frac{p}{p-r}}+ |u|^{\gamma_2}(u-k)+|u|^{\gamma_2} +b_2(u-k)+b_1\right)\, dx. \end{align} | (5.12) |
We have
\int_{A_{k, R}}|D\eta|\eta^{\mu-1}|u|^{\gamma_1}(u-k)\, dx \le c(\gamma_1)\int_{A_{k, R}}|D\eta|\eta^{\mu-1}(u-k)^{\gamma_1+1}\, dx |
+ c(\gamma_1)\int_{A_{k, R}}|D\eta|\eta^{\mu-1}k^{\gamma_1}(u-k)\, dx. |
By Hölder inequality with exponents \frac{p}{q-1} and \frac{p}{p-q+1} , we get
\int_{A_{k, R}}|D\eta|\eta^{\mu-1}(u-k)^{\gamma_1+1}\, dx = \int_{A_{k, R}}|D\eta|(u-k)\eta^{\mu-1}(u-k)^{\gamma_1}\, dx |
\le c\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx+c\int_{A_{k, R}}\eta^{\frac{p(\mu-1)}{q-1}}(u-k)^{\frac{p\gamma_1}{q-1}}\, dx. |
Analogously,
\int_{A_{k, R}}|D\eta|\eta^{\mu-1}k^{\gamma_1}(u-k)\, dx \le c\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx |
+c\int_{A_{k, R}}\eta^{\frac{p(\mu-1)}{q-1}}k^{\frac{p\gamma_1}{q-1}}\, dx |
and
\int_{A_{k, R}}|D\eta|\eta^{\mu-1}b_1(u-k)\, dx\le c\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx |
+c \int_{A_{k, R}}\eta^{\frac{p(\mu-1)}{q-1}}b_1^{\frac{p}{q-1}}\, dx, |
obtaining
\begin{align*} \nonumber &\int_{A_{k, \rho}} |Du|^p \, dx\le c \left\{\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\right.\\ & \nonumber + \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+k^{\frac{p\gamma_1}{q-1}}+b_1^{\frac{p}{q-1}}\right)\, dx \\ &+ \left.c\int_{A_{k, R}} \left((u-k)^{\frac{p}{p-r}}+ |u|^{\gamma_2}(u-k)+|u|^{\gamma_2} +b_2(u-k)+b_1\right)\, dx.\right\}. \end{align*} |
Therefore,
\begin{align*} \nonumber &\int_{A_{k, \rho}} |Du|^p \, dx\le c \left\{\int_{A_{k, R}}|D\eta|^{\frac{p}{p-q+1}} (u-k)^{\frac{p}{p-q+1}}\, dx\right.\\ & \nonumber + \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+(u-k)^{\frac{p}{p-r}}+(u-k)^{\gamma_2+1}+(u-k)^{\gamma_2}\right)\, dx \\ &+ \left.c\int_{A_{k, R}} \left(k^{\gamma_2}(u-k)+k^{\gamma_2}+b_2(u-k)+b_1+k^{\frac{p\gamma_1}{q-1}}+b_1^{\frac{p}{q-1}}\right)\, dx.\right\}. \end{align*} |
Since b_1\ge 1 and q < p+1 , then
b_1+b_1^{\frac{p}{q-1}}\le 2b_1^{\frac{p}{q-1}}, |
and we get (5.4).
Step 3. In this step we prove that
\begin{align} & \int_{B_{\rho}}|D(u-k)_+|^p\, dx \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k)_+\|_{W^{1, p}(B_{R}(x_0))}^{\frac{p}{p-q+1}}|A_{k, {R}}|^{\frac{p}{p-q+1}\nu} \\ & + c \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+(u-k)^{\frac{p}{p-r}}+(u-k)^{\gamma_2+1}+(u-k)^{\gamma_2}\right)\, dx \\ &+ c\int_{A_{k, R}} \left(k^{\gamma_2}(u-k)+b_2(u-k)+k^{\frac{p\gamma_1}{q-1}}+k^{\gamma_2}+ b_1^{\frac{p}{q-1}}\right)\, dx. \end{align} | (5.13) |
We obtain this estimate starting by (5.4).
Consider \tau\in (\rho, R) and define the function
S_1(0)\ni y\mapsto w(y): = (u-k)_+(x_0+\tau y) |
where
S_1(0): = \{y\in \mathbb{R}^n\, :\, |y| = 1\}. |
This function w is in W^{1, \left(\frac{p}{p-q+1}\right)_{*}}(S_1, d\mathcal{H}^{n-1}) , with
\begin{equation} \frac{1}{\left(\frac{p}{p-q+1}\right)_{*}} = \min\left\{\frac{1}{\frac{p}{p-q+1}}+\frac{1}{n-1}, 1\right\}. \end{equation} | (5.14) |
Let us consider the case
q > 1+\frac{p}{n-1}. |
By (4.1) in Remark 4.1, we get
\begin{equation} \frac{1}{\left(\frac{p}{p-q+1}\right)_{*}} = \frac{1}{\frac{p}{p-q+1}}+\frac{1}{n-1}. \end{equation} | (5.15) |
By (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we get
\begin{equation} \left(\int_{S_1}|w|^{\frac{p}{p-q+1}}\, d\mathcal{H}^{n-1}\right)^{\frac{p-q+1}{p}}\le c(n, p, q) \left(\int_{S_1}(|Dw|^{\left(\frac{p}{p-q+1}\right)_{*}}+|w|^{\left(\frac{p}{p-q+1}\right)_{*}})\, d\mathcal{H}^{n-1}\right)^{1/\left(\frac{p}{p-q+1}\right)_{*}}. \end{equation} | (5.16) |
When
q\le 1+\frac{p}{n-1}, |
we distinguish among two cases: n\ge 3 and n = 2 . If n\ge 3 , by using Hölder's inequality, we get
\left(\int_{S_1}|w|^{\frac{p}{p-q+1}}\, d\mathcal{H}^{n-1}\right)^{\frac{p-q+1}{p}}\le c(n, p, q) \left(\int_{S_1}|w|^{\frac{n-1}{n-2}}\, d\mathcal{H}^{n-1}\right)^{\frac{n-2}{n-1}}, |
by (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we obtain the inequality (5.16).
If n = 2 , then \left(\frac{p}{p-q+1}\right)_{*} = 1 , then we obtain the inequality (5.16) by applying Lemma 4.5 with r = \frac{p}{p-q+1} .
Let \mathcal{A}(\rho, R) be as in (5.2). We apply Lemma 4.3, with
B_R(x_0) \ni y\mapsto v(y): = (u-k)_+^{\frac{p}{p-q+1}}(y), |
that is a function in L^{1}(B_R(x_0)) . Using (5.16) and recalling that \frac{R_0}{2}\le \rho < R\le R_0 , reasoning as in [30], we get
\begin{align} &\inf\limits_{\mathcal{A}(\rho, R)}\int_{B_R(x_0)}|D\eta|^{\frac{p}{p-q+1}}(u-k)_+^{\frac{p}{p-q+1}}\, dx \\ & \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \left(\int_{\rho}^R\int_{S_\tau(0)}\left(|D(u-k)_+(x_0+y)|^{\left(\frac{p}{p-q+1}\right)_{*}} \right.\right. \\ &\quad \left.\left. +|(u-k)_+(x_0+y)|^{\left(\frac{p}{p-q+1}\right)_{*}}\right) \, d\mathcal{H}^{n-1}(y)\, d\tau\right)^{\frac{p}{p-q+1}/\left(\frac{p}{p-q+1}\right)_{*}}. \end{align} | (5.17) |
By coarea formula, inequality (5.17) implies
\begin{align*} \nonumber &\inf\limits_{\mathcal{A}(\rho, R)}\int_{B_R(x_0)}|D\eta|^{\frac{p}{p-q+1}}(u-k)_+^{\frac{p}{p-q+1}}\, dx \\\nonumber & \le C(n, p, q, R_0) (R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k)_+\|_{W^{1, \left(\frac{p}{p-q+1}\right)_{*}}(B_R(x_0)\setminus B_{\rho}(x_0))}^{\frac{p}{p-q+1}} \end{align*} |
and, taking into account (3.7), Remark 4.1 and (4.5)
\left(\frac{p}{p-q+1}\right)_{*} < p\Leftrightarrow \frac{1}{\left(\frac{p}{p-q+1}\right)_{*}} > \frac{1}{p}\Leftrightarrow \nu > 0 \Leftrightarrow \frac{q}{p} < 1+\frac{1}{n-1}, |
by Hölder's inequality we get
\begin{align} &\inf\limits_{\mathcal{A}(\rho, R)}\int_{B_R(x_0)}|D\eta|^{\frac{p}{p-q+1}}(u-k)_+^{\frac{p}{p-q+1}}\, dx \\ & \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k)_+\|_{W^{1, p}(B_{R}(x_0))}^{\frac{p}{p-q+1}}|A_{k, {R}}|^{\frac{p}{p-q+1}\nu} \end{align} | (5.18) |
By (5.4) we get
\begin{align*} & \nonumber \int_{A_{k, \rho}}|Du|^p\, dx \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k)_+\|_{W^{1, p}(B_{R}(x_0))}^{\frac{p}{p-q+1}}|A_{k, {R}}|^{\frac{p}{p-q+1}\nu} \\ & \nonumber + c \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+(u-k)^{\frac{p}{p-r}}+(u-k)^{\gamma_2+1}+(u-k)^{\gamma_2}\right)\, dx \\ &+ c\int_{A_{k, R}} \left( k^{\gamma_2}(u-k)+b_2(u-k)+k^{\frac{p\gamma_1}{q-1}}+k^{\gamma_2}+b_1^{\frac{p}{q-1}}\right)\, dx. \end{align*} |
Since
\int_{B_{\rho}}|D(u-k)_+|^p\, dx = \int_{A_{k, \rho}}|D(u-k)_+|^p\, dx = \int_{A_{k, \rho}}|Du|^p\, dx |
we get (5.13).
Step 4. In this step we estimate the integrals at the right hand side of (5.13).
Consider
J_1: = \int_{A_{k, R}} \left((u-k)^{\frac{p\gamma_1}{q-1}}+(u-k)^{\frac{p}{p-r}}+(u-k)^{\gamma_2+1}+(u-k)^{\gamma_2}\right)\, dx. |
ESTIMATE OF J_1 .
By assumptions (3.8) and (3.9),
\max\{\frac{p\gamma_1}{q-1}, \gamma_2+1, \frac{p}{p-r}\} < p^*. |
Therefore, by using Hölder inequality with exponent \frac{p^*(q-1)}{p\gamma_1} we get
\int_{A_{k, R}} (u-k)^{\frac{p\gamma_1}{q-1}}\, dx\le \left(\int_{A_{k, R}} (u-k)^{p^*}\, dx\right)^{\frac{p\gamma_1}{p^*(q-1)}}|A_{k, R}|^{1-\frac{p\gamma_1}{p^*(q-1)}}; |
Hölder inequality with exponent p^*\frac{p-r}{p} implies
\int_{A_{k, R}} (u-k)^{\frac{p}{p-r}}\, dx\le \left(\int_{A_{k, R}} (u-k)^{p^*}\, dx\right)^{\frac{1}{p^*\frac{p-r}{p}}}|A_{k, R}|^{1-\frac{1}{p^*\frac{p-r}{p}}}. |
Moreover, by using Hölder inequality with exponent \frac{p^*}{\gamma_2+1} we get
\int_{A_{k, R}} (u-k)^{\gamma_2+1}\, dx\le \left(\int_{A_{k, R}} (u-k)^{p^*}\, dx\right)^{\frac{\gamma_2+1}{p^*}}|A_{k, R}|^{1-\frac{\gamma_2+1}{p^*}}; |
by using Hölder inequality with exponent \frac{p^*}{\gamma_2} we get
\int_{A_{k, R}} (u-k)^{\gamma_2}\, dx\le \left(\int_{A_{k, R}} (u-k)^{p^*}\, dx\right)^{\frac{\gamma_2}{p^*}}|A_{k, R}|^{1-\frac{\gamma_2}{p^*}}. |
Therefore, by using the Sobolev embedding theorem
\begin{align*} \nonumber J_1 \le & \|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p\gamma_1}{q-1}}|A_{k, R}|^{1-\frac{p\gamma_1}{p^*(q-1)}} + \|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p}{p-r}}|A_{k, R}|^{1-\frac{1}{p^*\frac{p-r}{p}}} \\ & + \|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2+1}|A_{k, R}|^{1-\frac{\gamma_2+1}{p^*}}+ \|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2}|A_{k, R}|^{1-\frac{\gamma_2}{p^*}}. \end{align*} |
Let us consider now the following integral in (5.13):
J_2: = \int_{A_{k, R}} \left(k^{\gamma_2}(u-k)+b_2(u-k)+k^{\frac{p\gamma_1}{q-1}}+k^{\gamma_2}+b_1^{\frac{p}{q-1}}\right)\, dx. |
Trivially,
\begin{align*} \int_{A_{k, R}} k^{\gamma_2}(u-k)\, dx&\le k^{\gamma_2}\|(u-k)_+\|_{L^{p^*}(A_{k, R})}^{\frac{1}{p^*}}|A_{k, R}|^{1-\frac{1}{p^*}}\\ &\le k^{\gamma_2}\|(u-k)_+\|_{W^{1, p}(A_{k, R})}|A_{k, R}|^{1-\frac{1}{p^*}}. \end{align*} |
By assumption b_2\in L^{s_2} , s_2 > \frac{n}{p} = \frac{p^*}{p^*-p} . Since \frac{p^*}{p^*-p} > \frac{p^*}{p^*-1} , then \frac{s_2}{s_2-1} < p^* . Therefore, by Hölder inequality
\begin{align*} \int_{A_{k, R}} b_2(u-k)\, dx&\le \|b_2\|_{L^{s_2}(A_{k, R})} \|(u-k)_+\|_{L^{\frac{s_2}{s_2-1}}} \\ &\le \|b_2\|_{L^{s_2}(B_R)} \|(u-k)_+\|_{L^{p^*}(A_{k, R})}|A_{k, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}, \end{align*} |
which implies
\int_{A_{k, R}} b_2(u-k)\, dx\le \|b_2\|_{L^{s_2}(B_R)} \|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}. |
Now, b_1\in L^{s_1} with s_1 > \frac{p}{q-1} ; by using Hölder inequality with exponent \frac{s_1(q-1)}{p} we get
\int_{A_{k, R}} b_1^{\frac{p}{q-1}} \, dx\le \left(\int_{A_{k, R}} b_1^{s_1} \, dx\right)^{\frac{p}{s_1(q-1)}}|A_{k, R}|^{1-\frac{p}{s_1(q-1)}}. |
We obtain
\begin{align*} \nonumber J_2&\le k^{\gamma_2}\|(u-k)_+\|_{W^{1, p}((B_R))}|A_{k, R}|^{1-\frac{1}{p^*}}+ \left(k^{\frac{p\gamma_1}{q-1}} +k^{\gamma_2}\right)|A_{k, R}|\\ & +\|b_2\|_{L^{s_2}(B_R)} \|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}+\|b_1\|_{L^{s_1}(B_R)}^{\frac{p}{q-1}}|A_{k, R}|^{1-\frac{p}{s_1(q-1)}}. \end{align*} |
Step 5. By Steps 3, 4 we get
\begin{align*} & \nonumber \int_{B_r}|D(u-k)_+|^p\, dx \le C(n, p, q, R_0)(R-\rho)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \nonumber \|(u-k)_+\|_{W^{1, p}(B_{R})}^{\frac{p}{p-q+1}}|A_{k, {R}}|^{\frac{p}{p-q+1}\nu} \\ & \nonumber +c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p\gamma_1}{q-1}}|A_{k, R}|^{1-\frac{p\gamma_1}{p^*(q-1)}} + c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\frac{p}{p-r}}|A_{k, R}|^{1-\frac{1}{p^*\frac{p-r}{p}}} \\ & \nonumber+ c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2+1}|A_{k, R}|^{1-\frac{\gamma_2+1}{p^*}}+ c\|(u-k)_+\|_{W^{1, p}(B_R)}^{\gamma_2}|A_{k, R}|^{1-\frac{\gamma_2}{p^*}} \\ &+ ck^{\gamma_2}\|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{p^*}}+c \left( k^{\frac{p\gamma_1}{q-1}}+ k^{\gamma_2}\right)|A_{k, R}|\\ & +c\|b_2\|_{L^{s_2}(B_R)} \|(u-k)_+\|_{W^{1, p}(B_R)}|A_{k, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}+ c\|b_1\|_{L^{s_1}(B_R)}^{\frac{p}{q-1}}|A_{k, R}|^{1-\frac{p}{s_1(q-1)}} \end{align*} |
and the inequality (5.1) follows.
Let u\in W_{\mathrm{loc}}^{1, q}(\Omega) , 1 < q\le n , be weak solution to (3.1). Consider \Omega' \Subset \Omega an open set.
I case q > p . Let B_{R_0}(x_0)\subseteq \Omega' .
For every k\ge 0
\begin{align} &\int_{B_{R_0}(x_0)}(u-k)_+^p\, dx +\int_{B_{R_0}(x_0)}|D(u-k)_+|^p\, dx\\ & \le \int_{B_{R_0}(x_0)}(|u|-k)_+^p\chi_{\left\{x\in B_{R_0}(x_0)\, :\, \left|u\right| > {k}\right\}}(x) \, dx +\int_{B_{R_0}(x_0)}|Du|^p\chi_{\left\{x\in B_{R_0}(x_0)\, :\, \left|u\right| > {k}\right\}}(x) \, dx \\ & \le \int_{B_{R_0}(x_0)}(|u|^p+|Du|^p)\chi_{\left\{x\in B_{R_0}(x_0)\, :\, \left|u\right| > {k}\right\}}(x) \, dx \\ & \le \left(\int_{B_{R_0}(x_0)}(|u|^q+|Du|^q)\, dx\right)^{p/q}\, |\left\{x\in B_{R_0}(x_0)\, :\, \left|u\right| > {k}\right\}|^{1-p/q } \\ & \le \|u\|_{W^{1, q}(B_{R_0}(x_0))}^{p} |B_{R_0}(x_0)|^{1-p/q}. \end{align} | (6.1) |
In particular, chosen R_0 such that
|B_{R_0}(x_0)|\le \|u\|_{W^{1, q}(\Omega')}^{-\frac{pq}{q-p}} |
we get
\begin{equation} \|(u-k)\|_{W^{1, p}(B_{R_{0}}(x_{0}))} < 1\quad \forall k\ge 0. \end{equation} | (6.2) |
II case q = p . By a well known result by Giaquinta and Giusti [40], the gradient of the weak solution satisfies a higher integrability property: its gradient is in L^{p+\varepsilon }(B_{R_{0}}(x_{0})) , for some \varepsilon > 0 sufficiently small. Moreover, u\in L^{p^{\ast }}(B_{R_{0}}(x_{0})) ; because p = q , we can repeat the above argument with q replaced by p+\varepsilon so obtaining (6.1). R_{0} > 0 depends on the norm \Vert u\Vert _{W^{1, p+\varepsilon }(B_{R_{0}}(x_{0}))} . Again, by the Giaquinta and Giusti result, the norm \Vert u\Vert _{W^{1, p+\varepsilon }(B_{R_{0}}(x_{0}))} can be estimated in terms of the \Vert u\Vert _{W^{1, p}(\Omega ^{\prime })} for B_{R_{0}}(x_{0})\subseteq\Omega ^{\prime }\Subset \Omega .
Finally, we can summarize: in both cases, either if q > p or if q = p , we can choose R_{0} such that (6.2) holds with R_{0} > 0 depending on the norm \Vert u\Vert _{W^{1, q}(\Omega ^{\prime })} . We also assume R_0 < 1 such that |B_{R_0}| < 1 , 0 < R \le R_0 .
Define the decreasing sequences
\rho_{h}: = \frac{R}{2}+\frac{R}{2^{h+1}} = \frac{R}{2}(1+\frac{1}{2^h}). |
Fixed a positive constant d\ge 2 , to be chosen later, define the increasing sequence of positive real numbers (k_h)
k_h: = d\left( 1-\frac{1}{2^{h+1}}\right), \, \, h \in \mathbb{N}. |
Define the decreasing sequence (J_h) ,
J_h: = \|(u-k_h)_+\|^p_{W^{1, p}(B_{\rho_h}(x_0))}. |
Notice that
\rho_0 = R, \qquad \lim\limits_{\rho\to +\infty}\frac{R}{2}(1+\frac{1}{2^h}) = \frac{R}{2}, |
k_0: = \frac{d}{2}, \qquad \lim\limits_{h\to +\infty}k_h = d. |
Moreover, by (6.2),
J_h\le J_0 = \|(u-\frac{d}{2})_+\|^p_{W^{1, p}(B_{R}(x_0))} < 1. |
Let us introduce the following notation:
\begin{equation} \tau: = \max\left\{\frac{pp^*}{p-q+1}\nu+\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}} {\left(\frac{p}{p-q+1}\right)_{*}}\right), p^*\right\}, \end{equation} | (6.3) |
\begin{align} \theta: = \min\Big\{&\frac{pp^*}{p-q+1}\nu, p^*-\frac{p\gamma_1}{q-1}, p^*-\frac{p}{p-r}, p^*-\gamma_2-1, p^*-p, \\ & p^*\left(1-\frac{1}{s_2}\right)-1, p^*\left(1-\frac{p}{s_1(q-1)}\right)\Big\} \end{align} | (6.4) |
and
\begin{equation} \sigma: = \min\left\{\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu, \frac{p^*}{p}-\frac{p^*}{s_1(q-1)}, \frac{p^*}{p}\left(1-\frac{1}{s_2}\right)\right\}, \end{equation} | (6.5) |
where \nu is defined in (4.4).
Proposition 6.1 (Estimate of J_{h+1} ). Let u\in W^{1, q}_{\rm{loc}}(\Omega) be a weak solution to (3.1). Assume (3.2)–(3.4) with the exponents satisfying the inequalities listed in Section 3.1. Then for every h\in \mathbb{N}
\begin{equation} J_{h+1}\le c\frac{(2^{\tau})^h}{d^{\theta}}J_h^{\sigma}, \end{equation} | (6.6) |
where c is a constant depending on n, p, q, r, R_0 , the L^{s_1} -norm of b_1 and the L^{s_2} -norm of b_2 in B_{R_0} .
We precede the proof with the following remark.
Remark 6.2. We remark that, by assumptions (3.6)–(3.10), then \tau, \theta > 0 and \sigma > 1 . As far as these inequalities are concerned, we remark that
p^* > p; |
\nu > 0\quad \text{(see (4.5))}; |
\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu > 1\Leftrightarrow p^*\nu > p-q |
that is satisfied, because p\le q
p^* > \frac{p}{p-r} \Leftrightarrow r < p-\frac{p}{p^*} \Leftrightarrow r < p+\frac{p}{n}-1; |
p^* > \frac{p\gamma_1}{q-1}\Leftrightarrow \gamma_1 < p^*\frac{q-1}{p} \Leftrightarrow \gamma_1 < \frac{n(q-1)}{n-p}; |
\gamma_2 < p^*-1; |
\frac{p^*}{p}-\frac{p^*}{s_1(q-1)} > 1\Leftrightarrow \frac{p}{s_1(q-1)} < 1-\frac{p}{p^*}\Leftrightarrow s_1 > \frac{n}{q-1} |
that is the first assumption in (3.10); this assumption also implies
s_1 > \frac{p}{q-1} > 0 |
that is equivalent to
1-\frac{p}{s_1(q-1)} > \frac{p}{p^*} > 0. |
By the second assumption in (3.10),
s_2 > \frac{n}{p}\Leftrightarrow s_2 > \frac{p^*}{p^*-p} \Leftrightarrow \frac{p^*}{p}\left(1-\frac{1}{s_2}\right) > 1. |
Proof of Proposition 6.1. By (5.1), used with k = k_{h+1} , \rho = \rho_{h+1} , R = \rho_{h} , we have
\begin{align} & \int_{B_{\rho_{h+1}}}|D(u-k_{h+1})_+|^p\, dx \le C(n, p, q, R_0)\left(\rho_h-\rho_{h+1}\right)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}\times \\ &\times \|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h}})}^{\frac{p}{p-q+1}}|A_{k_{h+1}, {\rho_{h}}}|^{\frac{p}{p-q+1}\nu} \\ & +c\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}^{\frac{p\gamma_1}{q-1}} |A_{k_{h+1}, R}|^{1-\frac{p\gamma_1}{p^*(q-1)}}\\ & + c\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}^{\frac{p}{p-r}}|A_{k_{h+1}, R}|^{1-\frac{1}{p^*\frac{p-r}{p}}} \\ & + c\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}^{\gamma_2+1}|A_{k_{h+1}, R}|^{1-\frac{\gamma_2+1}{p^*}}+ c\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}^{\gamma_2}|A_{k_{h+1}, R}|^{1-\frac{\gamma_2}{p^*}} \\ &+ ck_{h+1}^{\gamma_2}\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}|A_{k_{h+1}, R}|^{1-\frac{1}{p^*}}+ c \left(k_{h+1}^{\frac{p\gamma_1}{q-1}}+k_{h+1}^{\gamma_2}\right) |A_{k_{h+1}, R}|\\ & +c \|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_{h+1}})}|A_{k_{h+1}, R}|^{1-\frac{1}{s_2}-\frac{1}{p^*}}+ c|A_{k_{h+1}, R}|^{1-\frac{p}{s_1(q-1)}}. \end{align} | (6.7) |
Let us write the estimate above as
\begin{align} \int_{B_{\rho_{h+1}}}|D(u-k_{h+1})_+|^p\, dx\le& c\left(\rho_h-\rho_{h+1}\right)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}H_1 \\ & + c \left(H_2+H_3+ H_4+ H_5+ H_6+ H_7+ H_8+ H_9\right). \end{align} | (6.8) |
To estimate the sum at the right-hand side it is useful to remark that, for all h ,
\begin{equation} k_{h+1}-k_h = \frac{d}{2^{h+2}} \end{equation} | (6.9) |
and
k_{h+1}-k_h < u-k_h\quad \text{in}\ A_{k_{h+1}, \rho_h}. |
Since
| A_{k_{h+1}, \rho_h}|\le \int_{A_{k_{h+1}, \rho_h}}\left(\frac{u-k_h}{k_{h+1}-k_h}\right)^{p^*}\, dx\le \|(u-k)_+ \|_{L^{p^*}(B_{\rho_h})}^{p^*}\frac{1}{(k_{h+1}-k_h)^{p^*}}, |
by the Sobolev inequality we get
| A_{k_{h+1}, \rho_h}|\le c(n, p)\frac{J_{h}^{\frac{p^*}{p}}}{(k_{h+1}-k_h)^{p^*}}, |
that, together with (6.9), gives
\begin{equation} | A_{k_{h+1}, \rho_h}|\le c(n, p) J_{h}^{\frac{p^*}{p}}\left(\frac{2^{h}}{d}\right)^{p^*}. \end{equation} | (6.10) |
Moreover,
\begin{align} \|(u-k_{h+1})_+\|^p_{W^{1, p}(B_{\rho_h}(x_0))} & = \int_{A_{k_{h+1}, \rho_h}}(u-k_{h+1})^p\, dx +\int_{A_{k_{h+1}, \rho_h}}|D(u-k_{h+1})|^p\, dx \\ & \le \int_{A_{k_{h}, \rho_h}}(u-k_{h})^p\, dx +\int_{A_{k_{h}, \rho_h}}|D(u-k_{h})|^p\, dx \\ & \le J_{h}. \end{align} | (6.11) |
Inequalities (6.10) and (6.11) imply that
\|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_h}(x_0))}|A_{k_{h+1}, R}|^{-\frac{1}{p^*}}\le c(n, p)J_{h}^{\frac{1}{p}}\frac{J_{h}^{\frac{p^*}{p}\left(-\frac{1}{p^*}\right)}}{(k_{h+1}-k_h)^{p^*\left(-\frac{1}{p^*}\right)}} |
therefore, by (6.9),
\begin{equation} \|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_h}(x_0))}|A_{k_{h+1}, R}|^{-\frac{1}{p^*}}\le c(n, p)\left(\frac{2^{h}}{d}\right)^{-1}. \end{equation} | (6.12) |
This estimate, together with (6.10), implies:
\begin{equation} H_2\le c(n, p, q, \gamma_1)\left(\frac{2^{h}}{d}\right)^{-\frac{p\gamma_1}{q-1}}|A_{k_{h+1}, R}|\le c(n, p, q, \gamma_1) \left(\frac{2^{h}}{d}\right)^{p^*-\frac{p\gamma_1}{q-1}} J_{h}^{\frac{p^*}{p}}, \end{equation} | (6.13) |
and, analogously,
\begin{equation} H_3\le c(n, p, r)\left(\frac{2^{h}}{d}\right)^{p^*-\frac{p}{p-r}}J_{h}^{\frac{p^*}{p}}, \end{equation} | (6.14) |
\begin{equation} H_4\le c(n, p, \gamma_2)\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2-1}J_{h}^{\frac{p^*}{p}}, \end{equation} | (6.15) |
\begin{equation} H_5\le c(n, p, \gamma_2)\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2}J_{h}^{\frac{p^*}{p}}, \end{equation} | (6.16) |
\begin{equation} H_8\le c(n, p)\left(\frac{2^{h}}{d}\right)^{-1}|A_{k_{h+1}, R}|^{1-\frac{1}{s_2}} \le c(n, p, s_2) \left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{1}{s_2}\right)-1} J_{h}^{\frac{p^*}{p}\left(1-\frac{1}{s_2}\right)}, \end{equation} | (6.17) |
\begin{equation} H_9\le c(n, p, q, s_1) \left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{p}{s_1(q-1)}\right)}J_{h}^{\frac{p^*}{p}-\frac{p^*}{s_1(q-1)}}. \end{equation} | (6.18) |
Moreover, taking into account that
k_{h+1} = d\left( 1-\frac{1}{2^{h+2}}\right)\le d, |
\begin{equation} H_6\le c(n, p) d^{\gamma_2}\left(\frac{2^{h}}{d}\right)^{p^*-1}J_{h}^{\frac{p^*}{p}} = c(n, p)\frac{2^{h(p^*-1)}}{d^{p^*-\gamma_2-1}}J_{h}^{\frac{p^*}{p}} \end{equation} | (6.19) |
\begin{equation} H_7\le c\left( \frac{2^{hp^*}}{d^{p^*-\frac{p\gamma_1}{q-1}}} + \frac{2^{hp^*}}{d^{p^*-\gamma_2}} \right) J_{h}^{\frac{p^*}{p}}. \end{equation} | (6.20) |
Let us now estimate H_1 .
Inequalities (6.10) and (6.11) imply
\begin{align*} \nonumber & H_1: = \|(u-k_{h+1})_+\|_{W^{1, p}(B_{\rho_h}(x_0))}^{\frac{p}{p-q+1}} |A_{k_{h+1}, \rho_h}|^{\frac{p}{p-q+1}\nu} \\ \nonumber &\le c(n, p, q)J_{h}^{\frac{1}{p-q+1}} \left( \frac{J_{h}^{\frac{p^*}{p}}}{(k_{h+1}-k_h)^{p^*}}\right)^{\frac{p}{p-q+1}\nu} \end{align*} |
that gives
H_1\le c(n, p, q) \left(\frac{2^{h}}{d}\right)^{\frac{pp^*}{p-q+1}\nu} J_{h}^{\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu}. |
Taking into account that for every h
\frac{1}{4} \frac{R_0}{2^{h+1}}\le \rho_h-\rho_{h+1} = \frac{R}{2^{h+2}}\le \frac{1}{4} \frac{R_0}{2^{h}}, |
we conclude that
\begin{align} & \left(\rho_h-\rho_{h+1}\right)^{-\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}}{\left(\frac{p}{p-q+1}\right)_{*}}\right)}H_1\\ & \le c(n, p, q, R_0) \frac{(2^h)^{\frac{pp^*}{p-q+1}\nu+\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}} {\left(\frac{p}{p-q+1}\right)_{*}}\right)}}{d^{\frac{pp^*}{p-q+1}\nu}}J_{h}^{\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu}. \end{align} | (6.21) |
Collecting (6.13)–(6.21), by (6.8) we get
\begin{align} & \int_{B_{\rho_{h+1}}}|D(u-k_{h+1})_+|^p\, dx\le c\frac{(2^h)^{\frac{pp^*}{p-q+1}\nu+\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}} {\left(\frac{p}{p-q+1}\right)_{*}}\right)}}{d^{\frac{pp^*}{p-q+1}\nu}}J_{h}^{\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu} \\ &+c\left\{\left(\frac{2^{h}}{d}\right)^{p^*-\frac{p\gamma_1}{q-1}}+ \left(\frac{2^{h}}{d}\right)^{p^*-\frac{p}{p-r}} +\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2-1} +\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2}\right.\\ &\left. +\frac{2^{h(p^*-1)}}{d^{p^*-\gamma_2-1}} +\frac{2^{hp^*}}{d^{p^*-\frac{p\gamma_1}{q-1}}}+\frac{2^{hp^*}}{d^{p^*-\gamma_2}} \right\} J_{h}^{\frac{p^*}{p}} \\ & +c\left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{1}{s_2}\right)-1} J_{h}^{\frac{p^*}{p}\left(1-\frac{1}{s_2}\right)} +c\left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{p}{s_1(q-1)}\right)}J_{h}^{\frac{p^*}{p}-\frac{p^*}{s_1(q-1)}}. \end{align} | (6.22) |
Let us now add to both sides of (6.22) the integral \int_{B_{\rho_{h+1}}}|(u-k_{h+1})_+|^p\, dx .
By Hölder inequality
\int_{B_{\rho_{h+1}}}((u-k_{h+1})_+)^p\, dx \le \left(\int_{B_{\rho_{h+1}}}((u-k_{h+1})_+)^{p^*}\, dx\right)^{\frac{p}{p^*}}|A_{k_{h+1}, \rho_{h+1}}|^{1-\frac{p}{p^*}}. |
Since
\int_{B_{\rho_{h+1}}}((u-k_{h+1})_+)^{p^*}\, dx\le \int_{B_{\rho_{h+1}}}((u-k_{h})_+)^{p^*}\, dx\le \int_{B_{\rho_{h}}}((u-k_{h})_+)^{p^*}\, dx, |
the Sobolev embedding theorem gives
\begin{equation} \int_{B_{\rho_{h+1}}}((u-k_{h+1})_+)^p\, dx\le c\|(u-k_{h})_+\|_{W^{1, p}(B_{\rho_{h}})}^{p}|A_{k_{h+1}, \rho_{h+1}}|^{1-\frac{p}{p^*}}. \end{equation} | (6.23) |
Taking into account (6.10), we obtain
|A_{k_{h+1}, \rho_{h+1}}|^{1-\frac{p}{p^*}}\le |A_{k_{h+1}, \rho_{h}}|^{1-\frac{p}{p^*}}\le c(n, p)\left(\frac{2^{h}}{d}\right)^{p^*-p} J_{h}^{\frac{p^*}{p}-1}; |
therefore, the inequality (6.23) implies
\begin{equation} \int_{B_{\rho_{h+1}}}((u-k_{h+1})_+)^p\, dx\le c(n, p) \left(\frac{2^{h}}{d}\right)^{p^*-p} J_{h}^{\frac{p^*}{p}}. \end{equation} | (6.24) |
Inequalities (6.22) and (6.24) give
\begin{align} & J_{h+1}\le c\frac{(2^h)^{\frac{pp^*}{p-q+1}\nu+\left(\frac{p}{p-q+1}-1+\frac{\frac{p}{p-q+1}} {\left(\frac{p}{p-q+1}\right)_{*}}\right)}}{d^{\frac{pp^*}{p-q+1}\nu}}J_{h}^{\frac{1}{p-q+1}+\frac{p^*}{p-q+1}\nu} \\ &+c\left\{\left(\frac{2^{h}}{d}\right)^{p^*-\frac{p\gamma_1}{q-1}}+ \left(\frac{2^{h}}{d}\right)^{p^*-\frac{p}{p-r}} +\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2-1} +\left(\frac{2^{h}}{d}\right)^{p^*-\gamma_2}\right.\\ & \left.+\frac{2^{h(p^*-1)}}{d^{p^*-\gamma_2-1}} +\frac{2^{hp^*}}{d^{p^*-\frac{p\gamma_1}{q-1}}} +\frac{2^{hp^*}}{d^{p^*-\gamma_2}} +\left(\frac{2^{h}}{d}\right)^{p^*-p} \right\} J_{h}^{\frac{p^*}{p}} \\ & +c\left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{1}{s_2}\right)-1} J_{h}^{\frac{p^*}{p}\left(1-\frac{1}{s_2}\right)} +c\left(\frac{2^{h}}{d}\right)^{p^*\left(1-\frac{p}{s_1(q-1)}\right)}J_{h}^{\frac{p^*}{p}-\frac{p^*}{s_1(q-1)}}. \end{align} | (6.25) |
where c is a constant depending on n, p, q, r, R_0 , the L^{s_1} -norm of b_1 and the L^{s_2} -norm of b_2 in B_{R_0} .
By taking in account the notation in (6.3)–(6.5), we get, by (6.25), the inequality (6.6).
We are now ready to prove our regularity result.
Proof of Theorem 3.2. By Proposition 6.1, for every h\in \mathbb{N} ,
J_{h+1}\le c\frac{(2^h)^{\tau}}{d^{\theta}}J_h^{\sigma}, |
where c is a constant depending on n, p, q, R_0 , the L^{s_1} -norm of b_1 and the L^{s_2} -norm of b_2 in B_{R_0} and for every d\ge 2 . Thus, the following inequality holds:
J_{h+1}\le A\lambda^hJ_h^{1+\alpha}, |
with
A = \frac {c}{d^{\theta}}, \ \ \lambda = 2^{\tau}, \ \ \alpha = \sigma-1, |
where \theta , \tau and \sigma are defined in (6.4), (6.3), (6.5). We recall that \theta, \tau > 0, \sigma-1 > 0 , see Remark 6.2.
To apply Lemma 4.6, we need
\begin{equation} \|\big(u-\frac{d}{2})_+\|^p_{W^{1, p}(B_{R}(x_0))} = J_0 \leq A^{-\frac{1}{\alpha}}\lambda^{-\frac{1}{\alpha^2}} = c^{-\frac{1}{\sigma-1}}2^{-\frac{\tau}{(\sigma-1)^2}}d^{\frac{\theta}{\sigma-1}}. \end{equation} | (6.26) |
Since
\|\big(u-\frac{d}{2})_+\|^p_{W^{1, p}(B_{R}(x_0))} \leq \|u\|^p_{W^{1, p}(B_{R}(x_0))}, |
if we choose d\ge 2 satisfying
\begin{equation} d^{\frac{\theta}{\sigma-1}} = 2+c^{\frac{1}{\sigma-1}}2^{\frac{\tau}{(\sigma-1)^2}}\|u\|^p_{W^{1, p}(B_{R}(x_0))}, \end{equation} | (6.27) |
we get 0 = \lim_{h\to +\infty}J_h = \|\big(u-d)_+\|^p_{W^{1, p}(B_{\frac{R}{2}})} and we conclude that
u(x)\le d\qquad \text{a.e. in } B_{\frac{R}{2}}(x_0) . |
To prove that u is locally bounded from below, we proceed as follows. The function -u is a weak solution to
\sum\limits_{i = 1}^{n}\frac{\partial }{\partial x_{i}} \overline{a}^i(x, u, Du) = \overline{b}( x, u, Du). |
where
\overline{a}(x, u, \xi): = a(x, -u, -\xi)\qquad \text{and } \qquad \overline{b} (x, u, \xi): = b(x, -u, -\xi). |
Notice that, by (3.2)–(3.4) the following properties hold:
● p- ellipticity condition at infinity:
for a.e. x\in \Omega and for every u\in \mathbb{R} ,
\langle \overline{a}(x, u, \xi), -\xi\rangle \ge \lambda|\xi|^{p} \qquad \forall \xi\in \mathbb{R}^n, |\xi| > 1, |
● q- growth condition:
for a.e. x\in \Omega and every u\in \mathbb{R} and \xi\in \mathbb{R}^{n}
\left|\overline{a}(x, u, \xi)\right| \le \Lambda\left\{ |\xi|^{q-1}+|u|^{\gamma_1} +b_1(x)\right\}, |
● growth condition for the right hand side b\left(x, u, \xi \right) :
|\overline{b}(x, u, \xi)|\le \Lambda \left\{|\xi|^{r}+|u|^{\gamma_2} +b_2(x)\right\}. |
To prove the analogue of Proposition 5.1 we now consider the test function \varphi_{k}(x): = (k-u(x))_+[\eta (x)]^{\mu} where \eta is a cut-off function. Let us consider the sub-level sets:
B_{k, R}: = \{x \in B_R(x_0)\, :\, u(x) < k \}, \qquad k\in \mathbb{R}. |
Then we obtain, in place of (5.5),
\begin{align*} \nonumber \int_{B_{k, R}} \langle \overline{a}(x, u, Du), -Du\rangle \, \eta^{\mu}\, dx & = -\mu \int_{B_{k, R}} \langle \overline{a}(x, u, Du), D\eta\rangle \eta^{\mu-1}(k-u)\, dx \\&+ \int_{B_{k, R}} \overline{f}(x, u, Du) (k-u)\eta^{\mu}\, dx. \end{align*} |
The proof goes on with no significant changes with respect the previous case, arriving to the conclusion that there exists d' such that we obtain that B_{\frac{R}{2}}\subseteq \{u\ge d'\} , and
u(x)\ge d'\qquad \text{a.e. in } B_{\frac{R}{2}}(x_0) . |
Collecting the estimates from below and from above for u , we conclude.
The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The authors declare no conflict of interest.
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