Noncompact Boundary Research Articles

Tilted spacetime positive mass theorem with arbitrary ends

In this paper, we prove the spacetime positive mass theorem for asymptotically flat spin initial data sets with arbitrary ends and a non-compact boundary. Moreover, we demonstrate a quantitative shielding theorem, subject to the tilted boundary dominant energy condition. Our results are established by solving a mixed boundary value problem for the Dirac-Witten operator with a Callias potential.

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Asymptotic spatial behavior for the heat equation on noncompact regions

AbstractWe consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flow measure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates for mean‐square cross‐sectional measures of the time‐weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time‐weighted temperature. In particular, when the base heat flow measure is positive, the time‐weighted temperature becomes pointwise unbounded at large spatial distance.

Mapping class groups of surfaces with noncompact boundary components

Mapping class groups of surfaces with noncompact boundary components

The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

Qualitative Properties of Solutions of Degenerate Parabolic Equations via Energy Approaches

We consider several problems for degenerate parabolic equations exhibiting nonlinearities of various kinds. For example the equations may contain superlinear sources, causing blow up of the solutions, or damping terms; the principal part of the operator is also nonlinear. We mention as unifying features the fact that the spatial domains have non-compact boundary, and the technical approach which is based on energy methods and a priori estimates. The issues investigated include existence under optimal assumptions on the data, asymptotic behavior of solutions, existence or non-existence of global in time solutions.

𝒵–compactifiable manifolds which are not pseudocollarable

It is shown that there exist $\mathcal{Z}$-compactifiable manifolds with noncompact boundary which fail to be pseudo-collarable.

Asymptotic Behavior of the Solution of Doubly Degenerate Parabolic Equations with Inhomogeneous Density

In this paper we study the large time behaviour for solutions to the Cauchyproblem for degenerate parabolic equations with inhomogeneous density.Under the suitable assumptions on the data of the problem and on the behaviour of the densityat infinity we establish new sharp bound of solutions for a large time.One of the main tool of the proof is new weighted embedding result whichis of independent interest. In addition, the proof of uniform estimatesof the solution is carried out by modified version of the classical methodof De-Giorgi--Ladyzhenskaya--Uraltseva--DiBenedetto. Similar resultsin the case of power-like density was obtained by one of the author~[10].The approach of this work can be applied for example when studying the qualitativeproperties of solutions to the Neumann problem for a doubly nonlinear parabolic equationwith inhomogeneous density in domains with non-compact boundaries.

Evaluation of the Mass of an Asymptotically Hyperbolic Manifold

We show that the mass functional of an asymptotically hyperbolic manifold with a noncompact boundary can be evaluated via the Ricci tensor and the second fundamental form by using purely coordinates. The method extends Miao-Tam’s approach to the asymptotically hyperbolic manifold. We also calculate a graphical example.

Decay properties for the incompressible Navier-Stokes flows in a half space

In this article, we give a comprehensive characterization of$L^1$-summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that$L^q-L^r$estimates for the Stokes flow don't work in this end-point case:$q=r=1$; the projection operator$P: L^1\longrightarrow L^1_\sigma$is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted$L^1$-estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection$P$by solving an elliptic problem with homogeneous Neumann boundary condition.

Lattice regularisation of a non-compact boundary conformal field theory

Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions.The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable {D}_2^2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.

An APS index theorem for even-dimensional manifolds with non-compact boundary

An APS index theorem for even-dimensional manifolds with non-compact boundary

The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary

We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application, we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

AbstractIn this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

We study the index of the APS boundary value problem for a strongly Callias-type operator $${\mathcal {D}}$$ on a complete Riemannian manifold M. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds $$N_0$$ and $$N_1$$ . If the dimension of M is odd we show that the latter index depends only on the restrictions $${\mathcal {A}}_0$$ and $${\mathcal {A}}_1$$ of $${\mathcal {D}}$$ to $$N_0$$ and $$N_1$$ and thus is an invariant of the boundary. We use this invariant to define the relative $$\eta $$ -invariant $$\eta ({\mathcal {A}}_1,{\mathcal {A}}_0)$$ . We show that even though in our situation the $$\eta $$ -invariants of $${\mathcal {A}}_1$$ and $${\mathcal {A}}_0$$ are not defined, the relative $$\eta $$ -invariant behaves as if it were the difference $$\eta ({\mathcal {A}}_1)-\eta ({\mathcal {A}}_0)$$ .

Characterization of Pseudo-Collarable Manifolds with Boundary

In this paper we obtain a complete characterization of pseudo-collarable n -manifolds for n ≥ 6 . This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann’s dissertation that can be applied to manifolds with nonstable fundamental group at infinity, our main theorem can also be viewed as an extension of the recent Gu–Guilbault characterization of completable n -manifolds in a manner that is applicable to manifolds whose fundamental group at infinity is not peripherally stable.

Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expression.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS

We prove two existence results for the nonlinear elliptic boundary value system involving $p$-Laplacian over an unbounded domain in $R^N$ with noncompact boundary. The proofs are based on variational methods applied to weighted spaces.

On the Mixed Steklov—Neumann and Steklov-type Biharmonic Problems in Unbounded Domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Steklov–Neumann and Steklov-type biharmonic problems in unbounded domains with a compact and non-compact boundaries (in particular, the exterior of a compact set, half-space, the domain with conical points) under the assumption that generalized solutions of these problems has a bounded Dirichlet (energy) integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

The mass in terms of Einstein and Newton

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second fundamental form of the boundary. This extends to this setting previous results by several authors in the boundaryless case. The method outlined below, which is based on a coordinate-free approach due to Herzlich, also applies to asymptotically hyperbolic manifolds, again with a noncompact boundary, for which a similar notion of mass has been recently considered by Almaraz and the first named author, and both cases will be discussed here.