Averaging methods for piecewise-smooth ordinary differential equations

1.   Introduction

For $N \geq 2$, we let $\Omega_0$ be a bounded open set of ${\mathbb{R}}^N$ with boundary $\Gamma _0$. Consider the heat equation with initial data the indicator function of the set $\Omega_0$:

for some time $t_1 > 0$. In 1992, Bence-Merriman-Osher ^[5] provided a computational algorithm for tracking the evolution in time of the set $\Omega_0$ whose boundary $\Gamma _0$ moves with normal velocity proportional to its classical mean curvature. At time $t_1 > 0$, they considered

Bence-Merriman-Osher ^[5] applied iteratively this procedure to generate a sequence of sets $\left(\Omega_j\right)_{j\geq 0}$ and conjectured in ^[5] that their boundaries $\Gamma _j$ evolved by mean curvature flow.

Later Evans ^[16] provided a rigorous proof for the Bence-Merriman-Osher algorithm by means of the level-set approach to mean curvature flow developed by Osher-Sethian ^[32], Evans-Spruck ^{[17,18,19,20]} and Chen-Giga-Goto ^[10]. For related works in this direction, we refer the reader to ^{[4,25,28,29,31,34,36]} and references therein.

Recently Caffarelli and Souganadis considered in ^[9] nonlocal diffusion of open sets $E\subset {\mathbb{R}}^N$ given by

where

We consider the fractional heat kernel $K_s$ with Fourier transform given by $\widehat K(\xi, t) = e^{-t|\xi|^{2s}}$. It satisfies

It follows that the unique bounded solution to (1.2) is given by

By solving a finite number of times (1.2) for a small fixed time step $\sigma _s(h)$, the authors in ^[9] find a discrete family of sets

for a suitable scaling function $\sigma _s$ to be defined below. It is proved in ^[9] that as $nh\to t$, $\partial E^h_{nh}$ converges, in a suitable sense, to $\Gamma _t$. Here, the family of hypersurface $\{\Gamma _t\}_{t > 0}$, with $\Gamma _0 = \partial E$, evolves under generalized mean curvature flow for $s\in [\frac{1}{2}, 1)$ and under generalized fractional mean curvature flow for $s \in (0, \frac{1}{2})$. We refer the reader to ^[9,16,27] for the notion generalized (nonlocal) mean curvature flow which considers the level sets of viscosity solutions to quasilinear parabolic integro-differential equations.

In the present paper, we are interested in the normal velocity of the sets

as they depart from a sufficiently smooth initial set $E_0: = E$. We consider here and in the following

We make the following assumptions:

and

for some constants $C_{N, s}, \mathcal{C}_{N, s} > 0$. In the Section 1.1 below, we provide examples of valuable kernels ${\mathcal K}_s$ satisfying the above properties.

Now, as we shall see below (Lemma 2.1), for $t > 0$ small, $\nabla_xu(x, \sigma _s(t)) \not = 0$ for all $x\in B(y, \sigma _s(t)^{\frac{1}{2s}})$ and $y\in \partial E$. Hence $\partial E_t$ is a $C^1$ hypersurface, for small $t > 0$. For $t > 0$ and $y\in \partial E$, we let $v = v(t, y)$ be such that

where $\nu(y)$ is the unit exterior normal of $E$ at $y$. In the spirit of the work of Evans ^[16] on diffusion of smooth sets, we provide in this paper an expansion of $v(t, y)$ as $t\to 0$. It turns out that $v(0, y)$ is proportional to the fractional mean curvature of $\partial E$ at $y$ for $s\in (0, 1/2)$ and $v(0, y)$ is proportional to the classical mean curvature of $\partial E$ at $y$ for $s\in [1/2, 1)$.

We notice that it is not a priori clear from (1.8), that $v$ remains finite as $t\to0$. This is where the (unique) appropriate choice of $\sigma _s(t)$ enters during our estimates. Here and in the following, we define

and for $s = 1/2$, $\sigma _s(t)$ is the unique positive solution to

Before stating our main result, we recall that for $s \in (0, \frac{1}{2})$ and $\partial E$ is of class $\mathcal{C}^{1, \beta }$ for some $\beta > 2s$, the fractional mean curvature of $\partial E$ is defined for $x\in \partial E$ as

On the other hand, if $\partial E$ is of class $\mathcal{C}^{2}$ then the normalized mean curvature of $\partial E$ is given, for $x\in \partial E$, by

see also (2.4) and ^[21]. Having fixed the above definitions, we now state our main result.

Theorem 1.1. We let $s\in (0, 1)$ and $E \subset{\mathbb{R}}^N$, $N\geq 2$. We assume, for $s\in (0, 1/2)$, that $\partial E$ is of class $\mathcal{C}^{1, \beta }$ for some $\beta > 2s$ and that $\partial E$ is of class $\mathcal{C}^{3}$, for $s \in[{1}/{2}, 1)$. Then, as $t\to 0$, the expansion of $v(t, y)$, defined in $(1.8)$, is given, locally uniformly in $y\in \partial E$, by

where $H_s$ and $H$ are respectively the fractional and the classical mean curvatures of $\partial E$ and the positive constants $a_{N, s}$, $b_{N, 1/2}$ and $c_{N, s}$ are given by

and $P_s(y): = \kappa _s(y, 1)$.

Some remarks are in order. The assumption of $E$ being of class $\mathcal{C}^3$ in Theorem 1.1 is motivated by the result of Evans in ^[16], where in the case $s = 1$ and ${\mathcal K}_1$, the heat kernel, he obtained $v = (N-1)H(0)+O(t^{\frac{1}{2}})$. We notice that from our argument below, we cannot improve the error term $o_t(1)$ in the case $s\in (0, 1/2)$ even if $E$ is of class $\mathcal{C}^\infty$. This is due to the definition of the fractional mean curvature $H_s$ as a principal value integral. Using polar coordinates and the estimates in (1.6), it easy follows that

This then justify the choice of the scaling when $s = \frac{1}{2}$. In the particular case, that $\kappa _{1/2}~~(y, t) = C_{N}\frac{t}{(t^2+|y|^2)^{\frac{N+1}{2}}},$ we have that

For the general case, we get the following

1.1.   Some applications of Theorem 1.1

We next put emphasis on two valuable examples where Theorem 1.1 applies.

$1)$ Fractional heat diffusion of smooth sets. We recall, see e.g., ^[6,35], that the fractional heat kernel $K_s$ satisfies (1.5), (1.6) and (1.7) with

We recall that $K_s$ is known explicitly only in the case $s = 1/2$, where $K_{1/2}~~(y, t) = C_{N, 1/2}\frac{t}{(t^2+|y|^2)^{\frac{N+1}{2}}}.$ In this case Theorem 1.1 provides an approximation of the (fractional) mean curvature motion by fractional heat diffusion of smooths sets, thereby extending, in the fractional setting, Evan's result in ^[16] on heat diffusion of smooth sets.

$2)$ Diffusion of smooth sets by Harmonic extension. We consider the Poisson kernel on the half space ${\mathbb{R}}^{N+1}_+: = {\mathbb{R}}^N\times (0, \infty)$, given by

where $p_{N, s}: = \frac{1}{\int_{{\mathbb{R}}^N}~ (1+|y|^2)^{-\frac{N+2s}{2}}\, dy}$. Thanks to the result of Caffarelli and Silvestre in ^[8], the function

solves

It is clear that ${\mathcal K}_s(x, t): = \overline K_s(x, t^{\frac{1}{2s}})$ satisfies (1.5), (1.6) and (1.7) with $C_{N, s} = p_{N, s}$. Hence, Theorem 1.1 provides an expansion of the normal velocities of the boundary of the sets

where $\sigma _s(t)$ is given by (1.9) and (1.10). Therefore this Harmonic extension yields an approximation of (fractional) mean curvature motion of smooth sets.

We conclude Section 1 by noting that the notion of nonlocal curvature appeared for the first time in ^[9]. Later on, the study of geometric problems involving fractional mean curvature has attracted a lot of interest, see ^[1,7], the survey paper ^[21] and the references therein. While the mean curvature flow is well studied, see e.g., ^{[2,3,15,24,26]}, its fractional counterpart appeared only recently in the literature, see e.g., ^{[11,12,13,14,27,30,33]}.

We finally remark that the changes of normal velocity of the nonlocal diffused sets as $s$ varies in (0, 1/2) and [1/2, 1), appeared analogously in phases transition problems, see e.g., ^[22,23,33].

2.   Preliminary results and notations

Unless otherwise stated, we assume for the following that $E$ is an open set of class $\mathcal{C}^{1, \beta }$, with $0\in \partial E$ and the unit normal of $\partial E$ at 0 coincides with $e_N$. We denote by $Q_r = B^{N-1}_r \times (-r, r)$ the cylinder of ${\mathbb{R}}^N$ centred at the origin with $B^{N-1}_r$ the ball of ${\mathbb{R}}^{N-1}$ centred at the origin with radius $r > 0$. Decreasing $r$, if necessary, we may assume that

with $\gamma \in \mathcal{C}^{1, \beta }(B_r^{N-1})$ satisfying

In the following, for $f, g: {\mathbb{R}} \to {\mathbb{R}}$, we write $g(t): = O(f(t))$ if

We also write $g(t) = o(f(t))$ if $g(t) = O(f(t))$ and moreover when $f(t) \neq 0$, we have

We denote by $o_t(1)$ any function that tends to zero when $t \to 0$.

If in addition, $\partial E$ is of class $\mathcal{C}^3$, then for $y'\in B_r^{N-1}$, we have

and the normalized mean curvature of $\partial E$ at 0 is given by

Recall that the unit exterior normal $\nu(y'): = \nu(y', \gamma (y'))$ of $E$ and the volume element $d\sigma (y')$ on $\partial E\cap Q_r$ are given by

We finally note, in view of (1.5) and (1.6), that we have

for some positive constant $C = C(N, s)$. We start with the following result.

Lemma 2.1. Let $s\in (0, 1)$ and $E$ be a set of class $\mathcal{C}^{1, \beta }$ satisfying $(2.1)$. Define

Then there exist $t_0, C > 0$, only depending on $N, s, \beta$ and $E$, such that for all $t \in (0, t_0)$ and $z\in B_{t^{\frac{1}{2s}}}$,

As a consequence, for all $t \in (0, t_0)$, the set

Proof. We fix $t > 0$ small so that $t^{\frac{1}{2s}} < \frac{r}{8}$ and let $z\in B_{t^{\frac{1}{2s}}}$. We write

By a change of variable, (1.5) and (1.6), we have

Integrating by parts, we have

By a change of variable, (1.5), (1.6) and the fact that $Q_{r/8}\subset B_{r/4}\subset B_{\frac{r}{2}}(z) \subset Q_r$, we have

provided $\frac{r}{8} > t\frac{^1}{2s}$. Next, using (2.6) and recalling that $z\in B_{t^{\frac{1}{2s}}}$, we then have

From this and (2.11), we deduce that

Combining this with (2.9) and (2.10), we get

Therefore (2.7) follows. Finally (2.8) follows from the inverse function theorem and the fact that $w$ is of class $C^1$ on ${\mathbb{R}}^N\times (0, \infty)$.

In the sequel, we will need the following lemmas to estimate some error terms.

Lemma 2.2. For $s\in(0, 1)$, we let $E\subset{\mathbb{R}}^N$ be a set of class $\mathcal{C}^{1, \beta }$, for some $\beta > 2s$, as in Section 2. For $r > 0$, we set

Then we have

where $C$ is a positive constant depending only on $N$, $\beta,$ $s$ and $E$.

Proof. Since $\tau_E = \mathbb{1}_E- \mathbb{1}_{{\mathbb{R}}^N\setminus \overline E}$, we get

Since the map $y \longmapsto K_s(y, {t})$ is radial, we have $\kappa _s(y', y_N, {t}) = \kappa _s(y', -y_N, {t})$ so that

Therefore

Then, by (2.6),

where $C$ is a positive constant depending on $N$, $\beta$ and $s$ and which may change from a line to another. Next, using (2.6), (1.7) and the dominate convergence theorem, we obtain

This then ends the proof.

Lemma 2.3. Let $s\in(0, 1)$ and let $x = vte_N\in\partial E_t\cap B_{\sigma _s(t)^{\frac{1}{2s}}}$, with $E_t$ given by $(1.4)$. Then

Proof. Since $x = vte_N\in\partial E_t$, we have that $u(x, \sigma _s(t)) = 0.$ By the fundamental theorem of calculus, we have

so that

We write

Then by Lemma 2.2 and (2.6), we have

for some constant $C$ depending on $r$. Furthermore by (2.7), we have

Therefore, by (2.12), (2.13) and (2.14), we obtain

This then ends the proof.

Lemma 2.4. Under the assumptions of Lemma 2.3, we have

Proof. Let $\theta \in [0, 1]$. By (1.5), (1.6) and a change of variable, we have

Applying Lemma 2.3, we get

as $t \to 0$. This then ends the proof.

3.   Proof of Theorem 1.1 in the case $s\in(0, \frac{1}{2})$

In this section, we start by the following preliminary result.

Lemma 3.1. Let $s \in (0, 1/2)$. We assume that $E$ is of class $\mathcal{C}^{1, \beta }$ for some $\beta > 2s$ satisfying $(2.1)$. Then, for all $\theta \in [0, 1]$, we have

Proof. We have

where $B_r$ is the ball of ${\mathbb{R}}^N$ centered at the origin and of radius $r > 0$. By integration by parts, we have

Therefore

Then by a change of variable and (2.5), we have

By the Fundamental Theorem of calculus, we can write

In the following, we let

Then we have

Therefore By a change of variable, (1.5) and (1.6), we have

By a change of variable, (1.6) and (3.4), we have

We use (2.2), (3.4) and Lemma 2.3 to get

Then by (1.6), we have

Hence

It follows from (3.5), (3.6) and (3.7) that

By a change of variable and the fact that $|\tau _E(y)|\leq 1$, we have

We use (1.7) to get, as $t\to0$,

Therefore, the expansion (3.1) follows immediately from (3.2), (3.8), (3.9) and the above estimate. This ends the proof.

The following result completes the proof of Theorem 1.1 in the case $s\in (0, 1/2)$.

Proposition 3.2. Under the assumptions of Lemma 3.1, we have

where $H_s(0)$ is the fractional mean curvature of $\partial E$ at the point $0$ and the positive constant $a_{N, s}$ is given by

Proof. We put with $x = vte_N$ and we recall that

By the fundamental theorem of calculus, we have

Then

where $\widetilde J_r(t) = J_r(\sigma _s(t))$, $\widetilde I_r(t) = I_r(\sigma _s(t))$, while $I_r(t)$ and $J_r(t)$ are given by Lemma 2.2. Moreover, by Lemma 3.1, we have

Therefore

Putting (3.12) in (3.11), we obtain that

Recalling that $\sigma _s(t) = t^{\frac{2s}{1+2s}}$ and using the fact that $u(x, \sigma _s(t)) = 0$, we have

As a consequence,

Therefore by Lemma 2.2, taking the limsups as $t\to 0$ and as $r \to 0$ respectively, we obtain

Hence

This then ends the proof.

4.   Proof of Theorem 1.1 in the case $s\in(\frac{1}{2}, 1)$

We have the following result.

Proposition 4.1. We consider $E$ a set of class $C^3$ satisfying the condition in Section 2. For $s\in(1/2, 1),$ we have

where $H(0)$ is the normalized mean curvature of $\partial E$ at $0$ and the positive constant $c_{N, s}$ is given by

Proof. We let $x = vt e_N\in \partial E_t$ and we expand

By (2.6) and Lemma 2.3, we have

Therefore

By a change of variable, the fact that $\tau _E = \mathbb{1}_E(x)- \mathbb{1}_{{\mathbb{R}}^N\setminus \overline E}(x)$ and $x = vt e_N$, we have

The last line is due to the fact that the map $y_N \to \kappa _s(y, \sigma _s(t))$ is even so that

Therefore we have

By (2.6) and the fact that $vt = o_t(1)$, we have

By a change of variable, the Fundamental Theorem of Calculus, (1.5) and (2.3), we have

Therefore, recalling (2.4),

By a change of variable and (1.5), we have

and

Moreover by (1.6), we get

and

By Lemma 2.4, we get

Combining (4.4), (4.6), (4.7), (4.8), (4.9) and (4.11), we obtain

By (4.3) and (4.12), we obtain

Since $x = vt \nu \in \partial E_t$, we have $u(x, \sigma _s(t)) = 0.$ Now, from the definition of $\sigma _s(t) = t^s$, we deduce that

Thus

where

This then ends the proof.

5.   Proof of Theorem 1.1 in the case $s=\frac{1}{2}$

As usual, we consider the function

and recall that

To alleviate the notations, for the following of this section, we write $\sigma _{1/2}~~(t): = \sigma (t).$

Proposition 5.1. For $s = 1/2,$ we have

where $H(0)$ is the mean curvature of $\partial E$ at $0$.

Proof. Recall that $x = vt\nu\to 0$ as $t\to0$, thanks to Lemma 2.3. We write

where $Q_r = B^{N-1}_r \times (-r, r)$. By (2.6), we have

Then, we have

By a change of variable and (2.3), we have

The last line is due to the fact that the map $y_N \to \kappa _{1/2}~~(y, \sigma (t))$ is even so that

Therefore we have

Using (2.6), we find that

By a change of variable, the fundamental theorem of calculus, (1.5) and (2.3), we have

Therefore

By (1.5), (1.6) and a change of variable, we have

Now Lemma 2.3 yields $vt = O(\sigma (t)^2)$ and thus

We get from (5.3), (5.4), (5.5), (5.6) and (5.7) that

Thanks to (1.6), we have

This implies that

Using polar coordinates and (1.6), we then have

where $\omega_{N-2}: = |S^{N-2}|$. By the change of variable $\rho = \frac{1}{m}$, we have

Letting $b_{N}(t): = \frac{ \int_{B^{N-1}_{\sigma (t)^{-1}}}~~|y^\prime|^2 P_{1/2}~~(y^\prime, 0) dy^\prime }{-2\log(\sigma (t)) \int_{{\mathbb{R}}^{N-1}}~~ P_{1/2}~~(y^\prime, 0) dy^\prime}$, by (5.8), (5.9) and the above estimate, we obtain, as $t\to0$,

Since $x = vt\nu \in \partial E_t$, we have $u(x, \sigma (t)) = 0$. Recalling that $t = \sigma ^2(t)|\log(\sigma (t))|$, we finally get

Hence

The proof is then ended.

Acknowledgments

This work is supported by the Alexander von Humboldt foundation and the German Academic Exchange Service (DAAD). Part of this work was done while the authors were visiting the International Center for Theoretical Physics (ICTP) in December 2019 within the Simons associateship program.

Conflict of interest

The authors declare no conflict of interest.