arXiv research

Universal inequalities found for Laplacian eigenvalues on convex domains.

The paper defines Dirichlet domains for Anosov subgroups in Lie groups.

In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions the approximate Dirichlet domain can work equally well as an exact Dirichlet domain. In particular, we consider a problem of tiling a hyperbolic ball with copies of the Dirichlet domain. This problem a…

Solves a specific Dirichlet problem for Lagrangian mean curvature equations.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

New approach to solving minimal surface system Dirichlet problem on smooth domains.

Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.

Study on complex hyperbolic bidisk isometries and their Dirichlet domains.

Algorithm finds Dirichlet domains for hyperbolic surfaces.

The paper solves the Dirichlet problem for minimal surfaces on unbounded helicoidal domains.

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

Fundamental gap vanishes for convex domains in hyperbolic space.

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

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $Ω$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result …

Proves Hölder continuity of complex Monge-Ampère solutions.

Heat flow fails to preserve concavity in curved spaces.

In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in term of lines of divergence in the domain. Using this second result, we build so…

In this paper, it is shown that a Fuchsian group, acting on the upper half-plane model for $\mathbb{H}^2$, admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflect…

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $Ω$ in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bdΩ)/{\rm Vol}(Ω)$.

In this paper we study the Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds with dimension $n$. Imitating the generalized solution theory of Miranda-Giusti, we define a new conformal area functional and a generalized solution to this Dirichlet problem. The existence of gene…

New upper bound found for nodal sets of Laplace eigenfunctions.

Formula for Laplacian determinants on polygonal domains with slits.

Estimates the mass gap for domains with integral Ricci curvature bounds.

Paper solves a complex equation for smooth domains.

We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

In this paper we generalize in Lorentz-Minkowski space $ł^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike extension to the domain. We also classify all singular maximal surfaces of $ł^3$ …

New heat trace coefficients reveal curvature effects in polygonal domains.

New approach to extremal hyperbolic surfaces using NEC groups.

Reconstructing a planar domain from its Dirichlet-to-Neumann data

Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an extension to Riemannian manifolds of an existence result of G. H. Williams ( J. Re…

Paper proves new inequalities for hyperbolic space Laplacian eigenvalues.

Radial graphs with constant mean curvature found in Euclidean space.

Estimates for eigenvalues on Riemannian manifolds using classical inequalities.

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

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem

PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

In this paper we study the non-existence of solutions to the Dirichlet problem for minimal graphs of codimension $\geq 2$, including certain situations over domain $Ω$ even with non-$C^1$ boundary $\partial Ω$.

Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.

Study proves Maximum Principles for unbounded Riemannian domains.

Let $Ω\subset\r^n$ be a bounded mean convex domain. If $α<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $Ω$ for the $α$-singular minimal surface equation with arbitrary continuous boundary data.

The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.

Study estimates eigenvalues for concave Hessian operators on convex domains.

Study of invariant solutions for certain PDEs on Riemannian manifolds.

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb S^n$ is $\geq 3 \frac{π^2}{D^2}$ when $n \geq 3$. We prove the same result when $n=2$. In fact our proof works for all dimension. We also…

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniquenes…

The restricted class of quasicircles sometimes called the "Weil-Petersson-class" has been a subject of interest in the last decade. In this paper we establish a Sokhotski-Plemelj jump formula for WP-class quasicircles, for boundary data in a certain conformally invariant Besov space. We show that this Besov space is pr…