Related papers: Quantitative symmetry in a mixed Serrin-type probl…

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…

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…

We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is…

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 $\Omega$…

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $\sigma$ and $q$ of the equation $-\mbox{div}(\sigma \nabla u)+qu = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given…

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap…

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to…

We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain $\Omega $, when the equation is known on $\Omega \setminus…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, put…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial \Omega$ and $\Sigma$ be a closed submanifold contained on $\partial \Omega$ and containing $0$. We are interesting in the existence of…

Let $\Omega \subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega…

We study the semilinear elliptic equation --$\Delta$u + g(u)$\sigma$ = $\mu$ with Dirichlet boundary condition in a smooth bounded domain where $\sigma$ is a nonnegative Radon measure, $\mu$ a Radon measure and g is an absorbing…

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^su=\lambda u^q+u^{2_s^*-1},\ u>0\quad\text{in…

Let $\Omega \subset \mathbb{R}^N$, $N\ge 2$, be an open, connected, bounded set with $C^2$ boundary. In this paper we consider the torsion problem with Robin boundary conditions and we study the symmetry of the solutions when suitable extra…

Given $\epsilon \in (0,1)$ and $\lambda > 1$, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value condition $$ \begin{cases} \Delta u+\epsilon^2 (e^{u} - e^{-u})=0 &\mbox{in }\Omega\\ \frac{\partial…

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $\Omega \subseteq \mathbb{R}^n$. Specifically, we show that if…

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of…