We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega)$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-H\"older continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/psN/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a H\"older continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.
We establish the optimal stability of Dirichlet boundary value problem for the regional (fractional) $p$-Laplacian $(-\Delta)^s_{p,\Omega}$ with $0<s\leq 1$, $\frac{1}{s}<p<\infty$ and $\Omega\subset \mathbb{R}^d$ bounded Lipschitz. More precisely, if $u_s \in W^{s,p}(\Omega)$ satisfies $(-\Delta)^s_{p,\Omega} u_s = f_...
In a Lipschitz domain
$$\Omega \subset \mathbb {R}^N$$
, we investigate the boundary behavior of positive superharmonic functions
u
satisfying the nonlinear inequality
$$-\Delta u(x)\le cd(x,\partial \Omega )^{-\tau }u(x)^p$$
. Based on the fact that, in some ranges of
p
and
$$\tau $$
, the po...
Kentaro Hirata· The Journal of Analysis· 0 citations
Let $1 \leq q<p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positive...
Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,\alpha}$, for $\alpha \in (0,1)$. Let $p \geq 2$ and $\beta>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shap...
In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally H\"older continuous. Suppose that $Du\in C^{0,\a...
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
