arXiv research

Rotationally symmetric hypersurfaces converge to cylinders under area-preserving flow.

Mean curvature flow extended with free boundary condition.

In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index…

Classifies $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$.

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of…

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…

Study on minimal surfaces with free boundary in a half-space, improving index estimates.

Rigidity of spectral data for spherical manifolds with boundary.

Existence of YMH fields with boundary conditions proven.

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $μ$ and for $K$ a constant,…

Study uniquely determines Riemannian metric derivatives from boundary data.

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…

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

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.

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

Study on axially symmetric surfaces' flow, showing all singularities are of type I.

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

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

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 …

Sharp inequality outside ball proved using Neumann method.

Estimates derived for solutions of Neumann problems on Riemannian manifolds.

Solves H-system on annulus with Neumann conditions.

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.

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

Study on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.

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

New findings on Obata equation with Robin boundary conditions on manifolds.

Study proves solutions concentrate on a capillary surface in a manifold.

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.

Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.

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.

Paper proves translating solutions for a specific flow in a product manifold.

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.

Estimates for eigenfunctions on manifolds with boundary.