Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type, and in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $Ω$ is not simply connected, or if $u$ is $C^{\infty}$ but not real analytic.
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constr…
We consider the classical "Serrin symmetry result" for the overdetermined boundary value problem related to the equation $Δu=-1$ in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable "compatibility" assumption between the solution and the geometry of the model.
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 consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.
Delaunay type domains for an overdetermined elliptic problem in S^n x R and H^n x R
证明了在M^n x R中存在可数族Delaunay型域Ω j，其中M^n是黎曼流形S^n或H^n，n至少为2，它从圆柱B^n x R（其中B^n是M^n中半径为1的测地球）分支出来，对于这类域，具有零Dirichlet边界条件的Laplace-Beltrami算子的第一特征函数在边界处也具有常数Neumann数据.域Ω_j相对于圆柱的R轴是旋转对称的和周期性的，并且当j收敛到0时，域Ω_j收敛到圆柱B^n x R。
We show uniqueness for overdetermined elliptic problems defined on topological disks $Ω$ with $C^2$ boundary, i.e., positive solutions $u$ to $Δu + f(u)=0$ in $Ω\subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte $ along $\partial Ω$, $\vecη$ the unit outward normal along $\partialΩ$ under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. When $(M^2,g)$ is the standard sphere $\mathbb S^2$ and $f$ is a $C^1$ function so that $f(x)>0$ and $f(x)\ge x \,…
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 problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain $Ω\subset \mathbb{H} ^2$ with $C^2$ boundary whose complement is connected and supports a bounded positive solution $u$ …
Let $Ω$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $Ω$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $Ω$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $Ω$ has the constant flow property if and only …
