Related papers: Fractional Dirichlet problems with an overdetermin…

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $\Omega \subset \mathbb R^n$. More precisely, we prove that if the fractional torsion…

We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $\Omega$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $\Omega$ to be a…

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u = f(u)…

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N+1}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\,…

We consider the solution of the torsion problem $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial \Omega$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_\nu$ is constant on $\partial \Omega$, then $\Omega$…

Given a bounded regular domain $\omega \subset \mathbb{R}^{N-1}$ and the half-cylinder $\Sigma = \omega \times (0,+\infty)$, we consider the relative overdetermined torsion problem in $\Sigma$, i.e. \[\left\{ \begin{array}{ll} \Delta…

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem…

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega. \end{cases} $$…

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $\Delta + V$ on an arbitrary domain $\Omega$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in \Omega$ and…

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -\Delta_{\mathbb{H}^N}…

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $\Gamma$ of the boundary…

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Lapacians $(-\Delta)^s_\Omega $, with $\Omega\subset \mathbb{R}^N$. Recall that the regional fractional Laplacians are generated…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation $$ u_{tt}-\mu(t)\Delta u+\omega(t)u_t=f(u),\ x\in\Omega,\ t\in\mathbb{R}, $$ subject to Dirichlet boundary condition…

Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $\Omega \subset \mathbb{R}^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$ with constant source…

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…

We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_\Omega…

The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $\Omega\subset \mathbb R^N$, we consider…