arXiv research

Gradient estimate for harmonic functions with boundary condition proved.

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.

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…

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

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

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

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…

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

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…

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

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…

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…

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

Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.

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 …

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…

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

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…

The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.

Enhances neural network solvers for PDEs with complex boundary conditions.

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…

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

Fundamental gap vanishes for convex domains in hyperbolic space.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

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…

Proves Hölder continuity of complex Monge-Ampère solutions.

New method reveals corners of drum shapes.

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker t…

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of…

New neural network approach solves Poisson equations efficiently.

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…

Paper solves Dirichlet problem for $p$-convex hypersurfaces with curvature constraints.

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

Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.

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

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…

We consider a Monge-Ampère functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the right hand side of the equation is nonpositive. We remove this assumption, to set…

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

Heat kernel estimates on manifolds with mixed boundary conditions.

Estimates prove existence of curvature flow in curved spaces.

New PINNs method improves accuracy in computing Mean Escape Time from bounded domains.

Study of Dirac operator with chiral boundary conditions on spin manifolds.

New boundary conditions improve Hamiltonian analysis in GR.

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…

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is o…