arXiv research

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…

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…

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…

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

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.

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.

In this paper, we study the well-posedness of boundary value problems for a special class of degenerate elliptic equations coming from geometry. Such problems is intimately tied to rigidity problem arising in infinitesimal isometric deformation, The characteristic form of this class of equations is changing its signs i…

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

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

We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of solutions on the second boundary value problem for a family of special Lagrangian equations

This article is a continuation of earlier work [R.L. Huang and Y.H. Ye, On the second boundary value problem for a class of fully nonlinear flows I, to appear in International Mathematics Research Notices], where the long time existence and convergence were given on some general parabolic type special Lagrangian equati…

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.

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…

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 …

New boundary conditions improve Hamiltonian analysis in GR.

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 …

We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.

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

New calculus solves boundary value problems for elliptic operators.

In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between …

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…

Study translating solitons in Minkowski space with prescribed Gauss image.

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…

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 …

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-We…

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…

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. …

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

We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version …

We show that the $G_{2}$ holonomy equation on a manifold with boundary, with prescribed 3-form on the boundary, is elliptic. The main point is to set up a suitable linear elliptic boundary value problem. This result leads to a deformation theory. In particular we establish the existence of certain $G_{2}$ cobordisms be…

We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our result extends an index theorem of D. Freed to non-compact manifolds, thus provi…

We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This le…

In the paper we introduce a boundary value problem for a G_{2} structure on a 7-manifold with boundary, with prescribed 3-form on the boundary. We make some general observations about this problem and then study in more detail reductions to 4 and 3 dimensions by imposing symmetry. The 3-dimensional reduction admits a d…

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property …

Smooth solutions found for hydrodynamic equations.

Study investigates non-existence of bounded solutions on curved spaces.

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 …

Study on spectral asymptotics in elasticity on smooth manifolds.

Paper finds unique solutions for curved surfaces with specific gradient.

Fast algorithm solves BVPs in linear time with probabilistic uncertainty.

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…

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometri…

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed …

Proof of local well-posedness for a specific boundary condition in general relativity.

Study proves symmetry of bounded domains in Riemannian manifolds.

Paper introduces magnetic Steklov operator on differential forms and its properties.

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. …