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Joaquim Serra

Publications and source records attributed to Joaquim Serra.

45 records · Page 3Linked to original sources
arXiv2014

Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric Lévy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^α$ then $u$ belong to $C^{α+2s}$ whenever $α+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-ε}$ for all $ε>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $\mathbb R^n\setminusΩ$, in $C^{1,1}$ domains $Ω$. We show that solutions $u$ satisfy $u/d^s\in C^{s-ε}(\overlineΩ)$ for all $ε>0$, where $d$ is the distance to $\partialΩ$. Finally, we show that our results are sharp by constructing two counterexamples.

math.AP↗
arXiv2014

Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(σ)$ of Caffarelli and Silvestre, $σ\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$, then $u$ is $C^β$ in space and $C^{β/σ}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $β< \min\{σ, 1+α\}$, where $α>0$. Our proof combines a Liouville type theorem ---relaying on the nonlocal parabolic $C^α$ estimate of Chang and Dávila--- and a blow up and compactness argument.

math.AP↗
arXiv2013

Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels $K$ satisfying that $K(y)|y|^{n+σ}$ is nondecreasing along rays from the origin, for some $σ\in(0,2)$ in case $a_{ij}\equiv0$ and for $σ=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for $L$ in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to $n$ and $σ$). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian $(-Δ)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.

math.AP↗
arXiv2013

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem $(-Δ)^s u=λf(u)$ in $Ω$, $u\equiv0$ in $\R^n\setminusΩ$, where $λ>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $n<4s$. We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $n<10s$. In the limit $s\uparrow1$, $n<10$ is optimal. In addition, we show that the extremal solution is $H^s(\R^n)$ in any dimension whenever the domain is convex. To obtain some of these results we need $L^q$ estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $L^p$ data. We prove optimal $L^q$ and $C^β$ estimates, depending on the value of $p$. These estimates follow from classical embedding results for the Riesz potential in $\R^n$. Finally, to prove the $H^s$ regularity of the extremal solution we need an $L^\infty$ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.

math.AP↗
arXiv2013

Sharp isoperimetric inequalities via the ABP method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.

math.AP↗
arXiv2012

Euclidean balls solve some isoperimetric problems with nonradial weights

In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990.

math.AP↗
arXiv2012

The Dirichlet problem for the fractional Laplacian: regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-Δ)^s u = g$ in $Ω$, $u \equiv 0$ in $\R^n\setminusΩ$, for some $s\in(0,1)$ and $g \in L^\infty(Ω)$, then $u$ is $C^s(\R^n)$ and $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$ for some $α\in(0,1)$, where $δ(x)={\rm dist}(x,\partialΩ)$. For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on $g$ we obtain higher order Hölder estimates for $u$ and $u/δ^s$. Namely, the $C^β$ norms of $u$ and $u/δ^s$ in the sets $\{x\inΩ: δ(x)\geqρ\}$ are controlled by $Cρ^{s-β}$ and $Cρ^{α-β}$, respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian \cite{RS-CRAS,RS}.

math.AP↗
arXiv2011

Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities

We prove a radial symmetry result for bounded nonnegative solutions to the $p$-Laplacian semilinear equation $-Δ_p u=f(u)$ posed in a ball of $\mathbb R^n$ and involving discontinuous nonlinearities $f$. When $p=2$ we obtain a new result which holds in every dimension $n$ for certain positive discontinuous $f$. When $p\ge n$ we prove radial symmetry for every locally bounded nonnegative $f$. Our approach is an extension of a method of P. L. Lions for the case $p=n=2$. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.

math.AP↗