arXiv research

Survey on minimal surface equation on Cartan-Hadamard manifolds.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

Study asymptotic behavior of Weingarten surfaces at infinity.

We show, by modifying Borbély's example, that there are $3$-dimen\-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $\le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

The paper solves a specific Dirichlet problem for constant mean curvature surfaces in a particular manifold.

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.

We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or…

Study investigates non-existence of bounded solutions on curved spaces.

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)}$ and a pointwise pinching condition $|K(P)|\le C_K |K(P')|$ for some const…

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean cur…

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with pre…

The article proves the existence of a smooth hypersurface with constant scalar curvature and a specified boundary in hyperbolic space.

Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.

We study the Dirichlet problem for the following prescribed mean curvature PDE $\begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }Ω\\ v=\varphi \text{ on }\partialΩ. \end{cases}$ where $Ω$ is a domain contained in a complete Riemannian manifold $M,$ $f:Ω\times\mathbb{R\rig…

Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.

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 the study of the asymptotic behaviour of solutions to the Dirichlet problem.

Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this cas…

Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.

In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given admissible boundary data and prescribed asymptotic behavior at infinity.

Estimates prove existence of curvature flow in curved spaces.

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}$ It…

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on…

The paper studies magnetic field effects on surface eigenvalues and spectral properties.

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We …

Sharp bounds derived for eigenvalues on specific geometric spaces.

Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.

Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $K(P)\le - \frac{φ(φ-1)}{r(x)^2}$ and a pointwise pinching condition $|K(P)|\le C_K|K(P')|$ for some constants $φ>1$ and $C_K\ge 1…

Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.

We study an asymptotic Dirichlet problem for Weyl structures on asymptotically hyperbolic manifolds. By the bulk-boundary correspondence, or more precisely by the Fefferman-Graham theorem on Poincaré metrics, this leads to a natural extension of the notion of Branson's $Q$-curvature to Weyl structures on even-dimension…

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting…

Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature conditions $-C e^{2-η}r(x) \leq K_M(x)\leq -1$ with $η>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack i…

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometr…

We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.

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 derive two-term asymptotics for eigenvalues in the limit when the distance between…

Study reveals how boundary wave operator determines metric properties on anti-de Sitter spacetimes.

We formulate the variational problem for AdS gravity with Dirichlet boundary conditions and demonstrate that the covariant counterterms are necessary to make the variational problem well-posed. The holographic charges associated with asymptotic symmetries are then rederived via Noether's theorem and `covariant phase sp…

Sharp bounds for Dirichlet sums lead to improved Bayesian algorithm analysis.

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator $Δ+V$ with $V\in C^2(M_0)$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 …

It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold $M$ under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth conditi…

Study finds solitons on curved spaces with varying behavior.

A robust bandit algorithm uses Dirichlet sampling to minimize regret under various distributional assumptions.

Recent research connects Hörmander's old work to modern boundary Laplacian analysis.

Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.