arXiv research

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations whi…

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

Paper proves inequality for capillary hypersurfaces with new proof.

On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data o…

New methods solve complex PDEs with mixed boundary conditions.

We study elliptic theory on manifolds with boundary represented as a covering space. Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. We show that elliptic elements define Fredholm operators and prove an index formula. …

A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition and Poisson's equation with mixed boundary conditions. The principal advantage o…

We prove an Obata-type rigidity result for the spherical cap and apply it for an eigenvalue problem with mixed boundary condition.

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function…

Study proves radial symmetry in convex cones using subharmonic functions.

The paper studies elliptic operators on manifolds with boundary.

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with pa…

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …

In the thesis at hand we give a comprehensive discussion of basic problems for generalized Maxwell equations with mixed boundary conditions using the calculus of alternating differential forms on Riemannian manifolds of arbitrary dimension. We prove compactness results, Hodge decompositions and Poincare type estimates.…

The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.

Develops conformal Bayes for two-sided censored Gaussian regression under label shift.

Heat kernel estimates on manifolds with mixed boundary conditions.

Study on elasticity with mixed boundary conditions, proving spectral asymptotics.

Unified Bayesian Optimisation for mixed variables improves performance.

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $rel \partial$ case the surgery on…

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

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calcul…

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential…

We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines differ…

The spectral asymptotics for linear elasticity with mixed boundary conditions are shown to be old results.

Study on singularities of specific polynomial functions.

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…

Compactifies maximal component of surface group representations into a closed ball.

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…

Study improves Poisson equation solutions on various manifolds.

Paper describes links of mixed polynomials with specific properties.

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we pres…

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…

A guide for solving first-order elliptic boundary value problems.

New spectral estimates for minimal surfaces with boundary conditions.

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …

Formula for Z_2-valued index of symmetric operators on manifolds.

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $\partial M \smallsetminus \partial_D M$. We prove the regularity and well-p…

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.

Let $M$ be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator $Δ\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$, invertible?" We consider also the case of mixed boundary conditions. The study of this main question lea…

We discuss the $ζ-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.

New PDEs of mixed type emerge in fluid mechanics and geometry.

New boundary conditions improve Hamiltonian analysis in GR.

Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.

Study of elliptic boundary value problems on non-compact manifolds.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

Local invertibility of higher order tensor transforms on compact manifolds.

Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formu…