arXiv research

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

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…

Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.

The paper studies elliptic operators on manifolds with boundary.

Abstracts a construction of boundary triplets for self-adjoint elliptic problems.

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

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

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

New calculus solves boundary value problems for elliptic operators.

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.

The paper finds sign-changing solutions for a specific type of elliptic equation.

The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard …

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. …

We develop a general method of proving the ellipticity of boundary value problems for the stationary vacuum space time, by showing that the stationary vacuum field equations are elliptic subjected to a geometrically natural collection of boundary conditions in the projection formalism. Using this we prove the manifold …

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…

We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …

In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator $D$ on smooth manifold $M$ with boundary $\p M$, $D$ possesses an elliptic boundary condition if and only if $\partial [D]$ = 0 in $K_1(\partial M)$, where $[D]$ is the relative $K$-cycle in $K_…

We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as ass…

The general theory of boundary value problems for linear elliptic wedge operators (on smooth manifolds with boundary) leads naturally, even in the scalar case, to the need to consider vector bundles over the boundary together with general smooth fiberwise multiplicative group actions. These actions, essentially trivial…

Extends a theorem for first-order elliptic operators on manifolds.

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…

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.

Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.

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…

Rigidity theorem for spherical sectors in Riemannian manifolds.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $Π$ is Fredholm by showing that the stationary vacuum equations (combined with p…

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer bo…

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…

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

Solves a 60-year-old compatibility problem on manifolds with boundary.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as…

Estimates gaps between eigenvalues for elliptic operators on manifolds.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

Develops Hodge theory for boundary-value problems on general geometric structures.

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues…

New boundary condition for weak inverse mean curvature flow in bounded domains.

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the bound…

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. …

Rigidity for 4D Willmore submanifolds with boundary.

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones an…

Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves …

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value …

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.

Study proves all free boundary CMC annuli are of finite type.