Related papers: Regional fractional Laplacians: Boundary regularit…

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right.…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…

In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which…

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$-stable processes and exterior data, inhomogeneity in weighted $L^2$-spaces. This class of operators includes the fractional Laplacian.…

For nonlinear operators of fractional $p$-Laplace type, we consider two types of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on fractional Sobolev spaces and Perron solutions based on superharmonic functions. These…

For $\Omega\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(\Omega;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation}…

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu…

We consider stable solutions to the equation $ -\Delta_p u =f(u) $ in a smooth bounded domain $\Omega\subset\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain,…

Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-\Delta)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the…

In this work we introduce volume constraint problems involving the nonlocal operator $(-\Delta)_{\delta}^{s}$, closely related to the fractional Laplacian $(-\Delta)^{s}$, and depending upon a parameter $\delta>0$ called horizon. We study…

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the…

We consider the incompressible Navier-Stokes equations in a moving domain whose boundary is prescribed by a function $\eta=\eta(t,y)$ (with $y\in\mathbb R^2$) of low regularity. This is motivated by problems from fluid-structure…

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the…

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. A variant…

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…

We prove existence and up to the boundary regularity estimates in $L^{p}$ and H\"{o}lder spaces for weak solutions of the linear system $$ \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text{ in…

We study boundary regularity at the infinity point $\boldsymbol{\infty}$ for nonlinear elliptic equations of $p$-Laplace type in unbounded open sets $\Omega \subset \mathbf{R}^n$. We consider the case $p \ge n \ge 2$ and characterize the…

We establish gradient H\"older continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving…

We prove sharp boundary H{\"o}lder regularity for solutions to equations involving stable integro-differential operators in bounded open sets satisfying the exterior $C^{1,\text{dini}}$-property. This result is new even for the fractional…