Optimal Regularity Results Related to a Partition Problem Involving the Half-Laplacian

Abstract

For a class of optimal partition problems involving the half-Laplacian operator and a subcritical cost functionals, we derive the optimal regularity of the density-functions which characterize the partitions, for the entire set of minimizers. We present a numerical scheme based on the arguments of the proof and we collect some numerical results related to the problem.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 321186: “Reaction-Diffusion Equations, Propagation and Modelling” held by Henri Berestycki, and under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems” held by Susanna Terracini.

Appendix A: The Brezis-Kato Inequality

In this last section, we will give a proof of Corollary 2.7, using in fact the following version of the Brezis-Kato inequality

Lemma A.1

Let $\Omega \subset \mathbb {R}^N$ be a smooth and bounded domain and let us consider $\mathbf {u}\in H^{1/2}_\Omega (\mathbb {R}^N, \mathbb {R}^k)$ to be solutions to the system

$$\begin{aligned} (- \Delta )^{1/2} u_i = a_i (1+|u_i|) - \beta u_i \sum \limits _{j \ne i} u_j^2. \end{aligned}$$

(A.1)

where $a_i \in L^N(\mathbb {R}^N)$. Then $u_i \in L^\infty (\mathbb {R}^N)$ for all $i = 1, \ldots , k$ and the norm can be bounded uniformly in $\beta$ with a constant that depends only on the $H^{1/2}$-norm of $\mathbf {u}$ and the $L^N$-norm of $a_i$.

Remark A.2

In order to apply the previous result to the setting of Corollary 2.7, it is sufficient to introduce the functions

$$ a_{i,\beta } : = \frac{(\gamma _{i,\beta } - Ke'(u_{i,\beta } - \bar{u}_i))u_{i,\beta } - f_i(x,u_{i,\beta }) }{ 1 + |u_{i,\beta }|} $$

and to observe that, thanks to the sub-criticality of $f_i$ and the uniform boundedness of Lagrange multipliers, we have $\Vert a_{i,\beta }\Vert _{L^N(\mathbb {R}^N)} \le C$ uniformly in $\beta$.

Proof

In order to simplify the proof, we resort to the extensional formulation of the half-Laplacian, relating the system (A.1) to

$$ {\left\{ \begin{array}{ll} - \Delta v_i = 0 &{}\text {in}\ \mathbb {R}^{N+1}_+\\ \partial _{\nu } v_i = a_i (1+|v_i|) - \beta v_i \sum \limits _{j \ne i} v_j^2 &{}\text {on}\ \Omega \subset \partial \mathbb {R}^{N+1}_+ \\ v_i = 0 &{}\text { on}\ \mathbb {R}^N \setminus \Omega \end{array}\right. } $$

where $v_i \in H^1(\mathbb {R}^{N+1}_+)$ satisfy $v_i(\cdot , 0) = u_i$. Let $g_\varepsilon \in \mathcal {C}^{\infty }(\mathbb {R})$ be a smooth approximation of the modulus functions, that is, $g_\varepsilon (t) = \sqrt{\varepsilon +t^2}$. The Stampacchia’s lemma and the Lebesgue’s theorem ensure that

$$ g_\varepsilon (v_i) \rightarrow |v_i| \text { in}\ H^1(\mathbb {R}^{N+1}_+), \quad g_\varepsilon '(v_i)v_i \rightarrow |v_i| \text { in}\ L^2(\mathbb {R}^{N}) $$

For any test function $\phi \in H^1(\mathbb {R}^{N+1}_+)$ such that $\phi \ge 0$, we have

$$\begin{aligned}&\int _{\mathbb {R}^{N+1}_+} \nabla g_\varepsilon (v_i) \nabla \phi + \int _{\mathbb {R}^N} \beta g_\varepsilon '(v_i) v_i \sum \limits _{j \ne i} v_j^2 \phi \\&= \int _{\mathbb {R}^{N}} g_\varepsilon '(v_i) a_i (1+|v_i|)\phi - \int _{\mathbb {R}^{N+1}_+} g_\varepsilon ''(v_i) |\nabla v_i|^2 \phi \end{aligned}$$

and letting $\varepsilon \rightarrow 0^+$ we obtain

$$ \int _{\mathbb {R}^{N+1}_+} \nabla |v_i| \nabla \phi + \int _{\mathbb {R}^N} \beta |v_i| \sum \limits _{j \ne i} v_j^2 \phi \le \int _{\mathbb {R}^{N}} \mathrm {sgn}(v_i) a_i (1+|v_i|) \phi . $$

(similar computations are present in [12, Lemma 5.5]). As a result, each $|v_i| \in H^1(\mathbb {R}^{N+1}_+)$ is a subsolution of the equation in $w_i \in H^1(\mathbb {R}^{N+1}_+)$

$$ {\left\{ \begin{array}{ll} - \Delta w_i = 0 &{}\text {in}\ \mathbb {R}^{N+1}_+\\ \partial _{\nu } w_i = |a_i| (1+w_i) &{}\text {on}\ \Omega \subset \partial \mathbb {R}^{N+1}_+\\ w_i = 0 &{}\text {on}\ \mathbb {R}^N \setminus \Omega \end{array}\right. } $$

Thus, if we show a uniform bound for the functions $w_i$ in $L^\infty$, by the comparison principle we could evince that the same bounds holds for the functions $|v_i|$. To conclude it is then sufficient to recall the Brezis-Kato estimate for the half-Laplacian, shown in [2, Theorem 5.2], which implies the sought $L^\infty$ bound.   $\square$