arXiv research

The paper proves rigidity results for Serrin-type problems in manifolds.

Proves rotational symmetry for Serrin-type problems in doubly connected domains.

The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.

The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to $+\infty$ or to $-\infty.$ We give a survey on the development of Jenkins-Serr…

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

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.

We give a proof of the classical Schwarz reflection principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(κ, τ)$ for $κ\leqslant 0$ and $τ\geqslant 0$. In our previous paper we proved a reflection principle in Riemannian manifolds. The statements and techniques in the two papers are d…

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equ…

Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.

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 b…

New classification of Serrin domains using algebraic curves and periodic solutions.

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

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.

Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an ingredient in the pro…

Paper solves curvature equations in Minkowski space for non-convex domains.

Motivated by Ilmanen's correspondence, we present an explicit solution to the prescribed Hoffman-Osserman Gauss map problem for non-minimal translators to the mean curvature flow in Euclidean 4-space. We propose a conjecture on the non-existence of Jenkins-Serrin type unit-speed graphical translators.

Stability results for geometric equations in warped product spaces.

Solves Jenkins-Serrin problem in 3-manifolds with Killing vector fields.

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$.

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…

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions …

Study proves symmetry of bounded domains in Riemannian manifolds.

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are …

We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over domains of $M$ bounded by ideal geodesic polygons and show the existence of a se…

Paper solves Serrin problem for ring-shaped domains, showing velocity has finitely many maxima.

Study proves radial symmetry in convex cones using subharmonic functions.

In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometr…

Given a complete $n$-dimensional Riemannian manifold $M$, we study the existence of vertical graphs in $M\times\mathbb{R}$ with prescribed mean curvature $H=H(x,z)$. Precisely, we prove that the Dirichlet problem for the vertical mean curvature equation in a smooth bounded domain $Ω\subset M$ has solution for arbitrary…

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $\{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>\varphi (x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}.$ We prove that up to isometry the ep…

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds

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

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 …

A version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol$_3$ is proved. Moreover, we construct examples of admissible domains where the results may be applied.

We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.

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.

In this paper we find functions over bounded domains in the 2-dimensional Euclidean space, whose graphs (in the Heisenberg space) has constant mean curvature different from zero and taking on (possibly) infinite boundary values over the boundary of the domain.

A classical problem in constant mean curvature hypersurface theory is, for given $H\geq 0$, to determine whether a compact submanifold $Γ^{n-1}$ of codimension two in Euclidean space $\R_+^{n+1}$, having a single valued orthogonal projection on $\R^n$, is the boundary of a graph with constant mean curvature $H$ over a …

Survey on smooth function and form density in Riemannian Sobolev spaces.

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 describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfac…

In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that use is to solve a Jenkins-Serrin problem over a strip…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sect…

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 r…

Alternative solvability criterion for minimal surface equations and mean curvature flow.