Related papers: A Serrin-type problem with partial knowledge of th…
In 1971 J. Serrin proved that, given a smooth bounded domain $\Omega \subset \mathbb{R}^N$ and $u$ a positive solution of the problem: \begin{equation*} \begin{array}{ll} -\Delta u = f(u) &\mbox{in $\Omega$, } u =0 &\mbox{on…
We show that in every dimension $N\geq3$ there are many bounded domains $\Omega\subset\mathbb{R}^{N}$, having only finite symmetries, in which the Bahri-Coron problem \[-\Delta u=|u| ^{4/(N-2)}u\text{\in}\Omega,\text{\ \}u=0\text{\…
We establish a rigidity theorem for annular sector-like domains in the setting of overdetermined elliptic problems on model Riemannian manifolds. Specifically, if such a domain admits a solution to the inhomogeneous Helmholtz equation…
We investigate the overdetermined problem given by \begin{equation*} \Delta u=0 \text{ in } \Omega,\quad \frac{\partial u}{\partial\nu} =\sigma_1 u \text{ on } \partial \Omega, \quad |\nabla u|=\text{constant on } \partial \Omega,…
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…
In this note, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia (Calc. Var. Partial Differential Equations 58: no. 160, 2019. https://doi.org/10.1007/s00526-019-1603-3, and prove a rigidity result that…
We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overline{\Omega}$ in $\mathbb{R}^n$($n\geq2$). First, we establish the…
We establish a quantitative version of the isoperimetric inequality for the torsion of multiply connected domains, among sets with given area and with given joint area of the holes. Since the optimal shape is the annulus, we investigate how…
In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to…
We study the Stokes problem in a bounded planar domain $\Omega$ with a friction type boundary condition that switches between a slip and no-slip stage. Unlike our previous work [6], in the present paper the threshold value may depend on the…
We study the thermal insulation of a bounded body $\Omega\subset\mathbb{R}^n$, under a prescribed heat source $f>0$, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the…
Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined…
In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…
We consider a steady-state heat conduction problem in a multidimensional bounded domain Omega for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Gamma_1 of…
We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains $\Omega\times[0, T)$, where $\Omega\subset \mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We show existence for…
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 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…
In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body…
In this paper, we prove a Serrin-type result for an elliptic system of equations, overdetermined with both Dirichlet and a generalized Neumann conditions. With this tool, we characterize the critical shapes under volume constraint of some…
We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the…