Related papers: Minimal Graphs in the Hyperbolic Space with Singul…
In this paper, we define natural capacities using a relative volume of graphs over manifolds, which can be characterized by solutions of bounded variation to Dirichlet problems of minimal hypersurface equation. Using the capacities, we…
We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that…
In this article, we investigate the rate at which the first Dirichlet eigenvalue of geodesic balls decreases as the radius approaches infinity. We prove that if the conformal infinity of an asymptotically hyperbolic Einstein manifold is of…
In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant $\sigma_{n-1}$ curvature, i.e. the existence of a complete hypersurface in $\mathbb{H}^{n+1}$ satisfying $\sigma_{n-1}(\kappa)=\sigma\in (0,n)$ with a…
We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for…
We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.
The aim of this work is to study how the asymptotic boundary of a minimal hypersurface in H^nxR determines the behavior of the hypersurface at finite points, in several geometric situations.
We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to…
We consider the Dirichlet problem $-\Delta u=\lambda f(u)$ with $\lambda<0$ and $f$ non-negative and non-decreasing. We show existence and uniqueness of solutions $u_\lambda$ for any $\lambda$ and discuss their asymptotic behavior as…
In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the special…
We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are…
We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space H^2xR; where H^2 is the hyperbolic plane. Under some geometric conditions on the asymptotic boundary of an oriented stable minimal…
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…
We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…
In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in…
We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an…
We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-\Delta u=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a…
We study two types of asymptotic problems whose common feature - and difficulty- is to exhibit oscillating Dirichlet boundary conditions : the main contribution of this article is to show how to recover the Dirichlet boundary condition for…
The aim of this paper is to give two uniqueness results for the Dirichlet problem associated to the constant mean curvature equation. We study constant mean curvature graphs over strips of R^2. The proofs are based on height estimates and…
We prove several results on the number of solutions to the asymptotic Plateau problem in $\mathbb H^3$. Firstly we discuss criteria that ensure uniqueness. Given a Jordan curve $\Lambda$ in the asymptotic boundary of $\mathbb H^3$, we show…