arXiv research

The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.

In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by …

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the …

Paper solves curvature prescription problem on surfaces with boundary.

In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solution…

In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for…

Study finds solutions to curvature equation with boundary conditions.

Study uniquely determines Riemannian metric derivatives from boundary data.

Study inverse boundary value problem for Monge-Ampère equation on convex domains.

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

Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

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

In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and p…

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

In this paper, we consider the global regularity for Monge-Ampère type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained under some necessary assumptions. Our main result extends the main theorem from the …

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

Sharp inequality outside ball proved using Neumann method.

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order …

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

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

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

After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $…

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

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

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.

Study Neumann problem for special Lagrangian type equations.

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

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

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

In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

Stokes equations help uniquely identify manifold metrics from boundary data.

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

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges toward…

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

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

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…

Anisotropic metric on manifolds uniquely determined by boundary data.

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…

Neumann eigenmaps improve landmark-based diffusion map embeddings.