arXiv research

Solves overdetermined boundary problems for semilinear equations.

Study proves radial symmetry of solutions to certain nonlinear equations in space forms.

Paper solves a mixed boundary value problem in space forms with umbilical boundaries.

Rigidity theorem for spherical sectors in Riemannian manifolds.

Study proves radial symmetry in convex cones using subharmonic functions.

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

Develops Hodge theory for boundary-value problems on general geometric structures.

Study solves overdetermined problem on compact surfaces.

Solves a 60-year-old compatibility problem on manifolds with boundary.

The paper optimizes domains for the first nontrivial eigenvalue in Riemannian manifolds.

Method of moving planes used for proving symmetry in PDEs and geometric analysis.

Study proves symmetry of bounded domains in Riemannian manifolds.

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $Ω: = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $Δu + f(u) = 0$ in $Ω$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neum…

Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.

Analyzes constant heat flow properties on Riemannian manifolds.

The paper connects minimal surfaces to cones in a ball.

The paper proves stability of Wulff shapes using anisotropic curvature functionals.

Mathematicians study surfaces with constant nonlocal mean curvature.

Extends Serrin's symmetry result to model manifolds.

Study proves a sharp upper bound for the zero set area of a static manifold's potential.

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 tha…

The paper classifies domains with constant boundary temperature in various 3D spaces.

New domains found in hyperbolic space solve a specific elliptic problem.

The paper characterizes gauge balls in the Heisenberg group and solves overdetermined problems.

We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and gi…

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 uniqu…

We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Unde…

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.

Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.

We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.

Paper solves overdetermined $k$-Hessian equation in exterior domains.

This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux.

The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.

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.

Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.

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…

Solves division problem for L. Hörmander's systems.

Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.

The study examines special domains in S^2 supporting specific solutions to a PDE.

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Be…

The paper proves rigidity for capillary graphs with specific curvature properties.

The purpose of this present paper is to investigate the geometric structure of regular overdetermined systems of second order with two independent and one dependent variables from the point of view of rank 2 prolongations. Utilizing this notion of prolongations, we characterize the type of these overdetermined systems.…

We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero …

Let $M$ be a Riemannian manifold and $Ω$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $Ω$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the nor…

In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds w…

Paper constructs solutions for a class of overdetermined systems.

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…

We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs. First, we study narrow properties of such domains in a Hada…