In this paper, we prove that a domain which verifies some integral inequality is either (strictly) contained in the solution of some free boundary problem, or it coincides with an $N$-ball. We also present new overdetermined value problems which have an $N$-ball as a solution. To reach our results, we use an integral identity which involves the domain derivative of the solution of Dirichlet problem.
This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of dimension no less than three. Secondly, we prove an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett's symmetry theorem. Finally, we prove a symmetry result for an overdetermined problem of second order by integral identities.
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.
We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth…
In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions that would have been out of reach otherwise. Another benefit of this method lies in the improvement of the regularity gap between the free boundary and the given one. Finally, some g…
In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain $D$ on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.
In this paper we study a partially overdetermined mixed boundary value problem for domains $Ω$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain $Ω\subset \mathcal C$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $Γ$ inside $\mathcal C$. In the case when $\mathcal C$ is a cylinder we obtain further results whenever the relative boundary of $Ω$ or the surface $Γ$ is a graph on …
In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze-Karcher-Ros type inequality for embedded hypersurfaces with free boundary lying on a horosphere …
We consider overdetermined problems related to the fractional capacity. In particular we study $s$-harmonic functions defined in unbounded exterior sets or in bounded annular sets, and having a level set parallel to the boundary. We first classify the solutions of the overdetermined problems, by proving that the domain and the solution itself are radially symmetric. Then we prove a quantitative stability counterpart of the symmetry results: we assume that the overdetermined condition is slightly perturbed and we measure, in a quantitative way, how much the domain is close to a symmetric set.
We deal with Monge-Ampère type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on $H$, on the gradient image of the domain. The Wulff shape symmetry associated with $H$ of the solutions is established.
Positive solutions to general semilinear overdetermined boundary problems
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究委员会数学科学研究所）；Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid（马德里康普顿斯大学数学学院分析与数学应用系）；ICREA（加泰罗尼亚高级研究院）；Universitat de Barcelona, Departament de Matemàtiques i Informàtica（巴塞罗那大学数学与计算机科学系）；Centre de Recerca Matemàtica（数学研究中心）
In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.
We study overdetermined problems for fully nonlinear elliptic equations in subdomains $Ø$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolic space $\mathbb{H}^{N}$. We prove, the existence of a classical solution to the underlined equation forces $Ø$ to be a geodesic ball in the ambient space. Our result extends to fully nonlinear equations, a similar result in the case of semilinear equations with the Laplace operator due to Kumaresan and Prajapat.
We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function $P$ and integral identities. In dimension $2$, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.
We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neu…
We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $Σ$ intersected with the exterior of a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $Σ\capΩ$ must be the intersection of the Wulff shape and $Σ$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.
We study the existence of non-trivial unbounded domains of $Ω\subset \mathbb{R}^2$ where the equation \begin{align} - λu_{xx} -u_{tt} &= u \qquad \text{in $Ω$,}\nonumber u &=0 \qquad \text{on $\partial Ω$,}\nonumber \end{align} is solvable subject to the conditions \begin{align} \frac{\partial u}{\partial η} =-1\quad \text{on $\partial Ω^+$} \quad \textrm{and}\quad \frac{\partial u}{\partial η} =+1\quad \text{on $\partial Ω^-$.} \end{align} For every integer $m\geq 0$, we prove the existence of a family of unbounded domains $Ω\subset \mathbb{R}^2$ indexed by $0 \leqslant\ell\leqslant 2m$, wher…
In this paper, we prove the existence of nontrivial unbounded domains $Ω\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} Δu +f(u)=0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=\mbox{constant} &\mbox{on $\partialΩ$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +\infty) \to \mathbb{R}$. Roughly speaking, we only ask that the D…
