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Manel Sanchon

Publications and source records attributed to Manel Sanchon.

8 recordsLinked to original sources
arXiv2022

Optimal regularity of stable solutions to nonlinear equations involving the $p$-Laplacian

We consider the equation $-Δ_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $Δ_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when $n<p+4p/(p-1)$, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on $f$. In this article we solve a long-standing open problem: we prove an interior $C^α$ bound for stable solutions which holds for every nonnegative $f\in C^1$ whenever $p\geq2$ and the optimal condition $n<p+4p/(p-1)$ holds. When $p\in(1,2)$, we obtain the same result under the non-sharp assumption $n<5p$. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the $p$-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when $p=2$ in the optimal range $n<10$.

math.AP↗
arXiv2017

Antisymmetry of solutions for some weighted elliptic problems

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.

math.AP↗
arXiv2014

A priori estimates for semistable solutions of semilinear elliptic equations

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound follows from a $L^p$-estimate for $f'(u)$ for every $p<3$ and $n\geq 2$. Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.

math.AP↗
arXiv2013

Regularity of stable solutions to semilinear elliptic equations on Riemannian models

We consider the reaction-diffusion problem $-Δ_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian manifolds includes the classical \textit{space forms}, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish $L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend on the nonlinearity $f$. As an application, under standard assumptions on the nonlinearity $λf(u)$, we prove that the corresponding extremal solution $u^*$ is bounded whenever $n\leq9$. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.

math.AP↗
arXiv2012

Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-Δ_p u= g(u)$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(Ω)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(Ω)\cap W^{1,p}_0(Ω)$ if $n>p+2$.

math.AP↗
arXiv2011

Geometric-type Sobolev inequalities and applications to the regularity of minimizers

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-Δu=g(u)$ in a bounded domain $Ω$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(Ω)$ bound for the extremal solution of $-Δu=λf(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\leq 4$, and that their boundedness is expected ---but still unkwown--- for $n\leq 9$.

math.AP↗
arXiv2007

Regularity of radial minimizers of reaction equations involving the p-Laplacian

We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.

math.AP↗