arXiv research

Study uniquely determines Riemannian metric derivatives from boundary data.

Gradient estimate for harmonic functions with boundary condition proved.

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.

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

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

The paper establishes gradient estimates for harmonic and heat equation solutions on manifolds with boundary.

Proves well-posedness for Einstein equations with specific boundary conditions.

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field me…

Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.

Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.

The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.

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

Solves Dirichlet problem for elliptic equations on Hermitian manifolds.

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

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

In this paper we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addit…

Proves well-posedness for Einstein equations with specific boundary data.

Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …

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

The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…

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…

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the do…

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

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

Stokes equations help uniquely identify manifold metrics from boundary data.

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

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

The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.

Solves a specific Dirichlet problem for Lagrangian mean curvature equations.

New upper bound found for nodal sets of Laplace eigenfunctions.

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion un…

Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.

We study isospectrality for manifolds with mixed Dirichlet-Neumann boundary conditions and express the well-known transplantation method in graph- and representation-theoretic terms. This leads to a characterization of transplantability in terms of monomial relations in finite groups and allows for the generating of ne…

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

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 …

Anisotropic metric on manifolds uniquely determined by boundary data.

Willmore flow converges globally for surfaces with rotational symmetry below a specific energy threshold.

Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.

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

This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed…

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…

Proves Hölder continuity of complex Monge-Ampère solutions.