Related papers: Overdetermined boundary value problems for the $\i…
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…
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal…
The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a…
In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -\Delta_1u=\lambda f(u) &\hbox{in }\Omega\,;\\[2mm] u=0…
In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic…
In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we…
In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-\Delta)^s u &= u^{-q} + f(x,u), \;u >0\;…
For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…
We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-\Delta_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function…
We consider the following problem $ -\Delta_{p}u= h(x,u) \mbox{ in }\Omega$, $u\in W^{1,p}_{0}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that…
We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of…
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…
Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx…
Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…
Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which…
Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions…
This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…
Let $P$ be a convex polygon in ${\mathbb C}$ and let $\Delta_{D, P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $\Delta=-4\partial_z\partial_{\bar z}$ in $P$. We derive a variational formula for the logarithm…
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}…
Let \Omega and \tilde{\Omega} be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: \Omega \to \tilde{\Omega} such that the graph \Sigma = \{(x,f(x)): x \in \Omega\} is a minimal…