Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

On the uniqueness of the solution of the first two boundary value problems for the heat equation without initial conditions

We consider boundary value problems for the heat equation without initial data in the class of functions of polynomial growth at infinity. We prove the unique solvability of the first and second boundary value problems and show that the conditions at infinity are important; i.e., their weakening results in the nonuniqueness of the solution.     

Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds

This paper studies several aspects of asymptotically hyperbolic (AH) Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness.     

Désingularisation de métriques d’Einstein. I

We find a new obstruction for a real Einstein 4-orbifold with an A ₁-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.     

Reduction from a Semi-Infinite Interval to a Finite Interval of a Nonlinear Boundary Value Problem for a System of Second-Order Equations with a Small Parameter

We consider a boundary value problem over a semi-infinite interval for a nonlinear autonomous system of second-order ordinary differential equations with a small parameter at the leading derivatives. We impose certain constraints on the Jacobian under which a solution to the problem exists and is unique. To transfer the boundary condition from infinity, we use the well-known approach that rests on distinguishing the variety of solutions satisfying the limit condition at infinity. To solve an auxiliary Cauchy problem, we apply expansions of a solution in the parameter.     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.     

Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with specified asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.     

Global solutions to bubble growth in porous media

We study a moving boundary problem modeling an injected fluid into another viscous fluid. The viscous fluid is withdrawn at infinity and governed by Darcy?s law. We present solutions to the free boundary problem in terms of time-derivative of a generalized Newtonian potentials of the characteristic function of the bubble. This enables us to show that the bubble occupies the entire space as the time tends to infinity if and only if the internal generalized Newtonian potential of the initial bubble is a quadratic polynomial. Howison (1985) [7], and DiBenedetto and Friedman (1986) [2], studied such behavior, but for bounded bubbles. We extend their results to unbounded bubbles.     

On a nonlocal diffusion model with Neumann boundary conditions

We study a nonlocal diffusion model analogous to heat equation with Neumann boundary conditions. We prove the existence and uniqueness of solutions and a comparison principle. Furthermore, we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable.     

Martin points on open manifolds of non-positive curvature

The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of full harmonic measure. The result of this paper provides a partial answer to an open problem of S. T. Yau.

     

The canonical-boundary representation for automorphism groups of locally compact countable state Markov shifts

We introduce the canonical-boundary representation and study its range. This conjugacy invariant homomorphism captures information about the symmetry of the Markov shift near its (canonical) boundary and exhibits which actions on the boundary can be realized by automorphisms. The path-structure at infinity — a relation on the set of orbits of the canonical boundary — is a new conjugacy invariant, which is stronger than the canonical boundary and the periodic data at infinity. Moreover we determine its influence on the range of the canonical-boundary representation and the extendability of automorphisms from subsystems (ascending sequences of shifts os finite type (SFTs) and infinite subsets of periodic points) to the entire Markov shift.     

Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation

Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions.     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

Riemann–Hilbert approach and $$N$$ -soliton solutions of the generalized mixed nonlinear Schrödinger equation with nonzero boundary conditions

Theoretical and Mathematical Physics - We apply the inverse scattering transformation to the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition at infinity. The...     

Asymptotic behavior of the unbounded solutions of some boundary layer equations

We give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics.Received: 24 May 2004     

Exponential decay for the solutions of nonlinear elliptic systems posed in unbounded cylinders

We study the asymptotic behavior at infinity of the solutions of a nonlinear elliptic system posed in a cylinder of infinite length. The problem is written in a variational formulation, where we ask the derivative of the solutions to be in Lp. We show that an exponential decay at infinity for the second member implies exponential decay for the derivative of the solutions. We also give an application of this result to the study of boundary layers problems.     

The porous medium equation with nonlinear absorption and moving boundaries

We consider a nonlinear parabolic equation containing the porous medium operator and a nonlinear absorption term, which causes the appearance of a moving boundary. Basic behavior and regularity results are obtained for the solution and the moving boundary under two different boundary conditions. Also, the behavior of the solution and the moving boundary as time goes to infinity is investigated. Supported in part by NSF grant number MCS 8104220.     

N-soliton solution of a lattice equation related to the discrete MKdV equation

We present an N-soliton solution of a lattice equation related to the discrete MKdV equation under an arbitrary boundary value at infinity.     

A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton–Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton–Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.