Domains In Manifolds Research Articles

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor g j k d x j ⊗ d x k g_{jk} dx_j\otimes dx_k has low regularity. Under the assumption that \[ | g j k ( x ) − g j k ( y ) | ≤ C ω ( | x − y | ) , |g_{jk}(x)-g_{jk}(y)|\leq C\,\omega (|x-y|), \] where the modulus of continuity ω \omega satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with L p L^p boundary data, for sharp ranges of p p ’s and with optimal nontangential maximal function estimates.

A smoothly bounded domain in a complex surface with a compact quotient

We study the classification of smoothly bounded domains in complex manifolds that cover compact sets. We prove that a smoothly bounded domain in a hyperbolic complex surface that covers a compact set is either biholomorphic to the ball or covered by the bidisc.

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.

Plemelj Projection Operators over Domain Manifolds

Plemelj Projection Operators over Domain Manifolds

Plemelj Projection Operators over Domain Manifolds

Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in R^n, and are consequently called domain manifolds. The key ingredients used here are techniques from both real and complex Clifford analysis. Analogues of the Kerzman-Stein kernel and Szego projection operators ar introduced, and their conformal covariance is described.

Plemelj Projection Operators over Domain Manifolds

Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in R^n, and are consequently called domain manifolds. The key ingredients used here are techniques from both real and complex Clifford analysis. Analogues of the Kerzman-Stein kernel and Szego projection operators ar introduced, and their conformal covariance is described.

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C1+γ, for some γ>0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig.

Potential theory on lipschitz domains in riemannian manifolds: hölder continuous metric tensors

In our papers [MT], [MT2], [MT3], and [MMT], we have developed the method of layer potentials as a tool to treat boundary problems for the Laplace operator and related operators on Lipschitz domains in Riemannian manifolds, extending work done on Lipschitz domains in Euclidean space with its standard flat metric, beginning with the papers of [FJR], [Ve], and [DK]. We worked under the hypothesis that the metric tensor was at least Lipschitz. Here our goal is to relax this regularity hypothesis and work with metric tensors that are merely Holder continuous.

Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds

We extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the Laplace equation and certain other constant coefficient elliptic partial differential equations on Lipschitz domains in Euclidean space. We treat the Laplace operator on Lipschitz domains in a manifold withC1metric tensor and study the Dirichlet, Neumann, and oblique derivative boundary problems.

Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds

Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds

Gradient Estimates for Harmonic Functions on Regular Domains in Riemannian Manifolds

Derivative formulae for heat semigroups are used to give gradient estimates for harmonic functions on regular domains in Riemannian manifolds. This probabilistic method provides an alternative to coupling techniques, as introduced by Cranston, and allows us to improve some known estimates. We discuss two slightly different ways to exploit derivative formulae where each one should be interesting by itself.

On the differentiation of heat semigroups and poisson integrals

We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain differentiation formulae for Poisson integrals in the same way as for heat semigroups involving boundary conditions. In particular, our results yield Bismut type representations for the logarithmic derivative of the Poisson kernel on regular domains in Riemannian manifolds corresponding to elliptic PDOs of Hormander type. Such formulae provide a direct approach to gradient estimates for harmonic functions on Riemannian manifolds

Homotopy formulas and $$\bar \partial $$ -equation on localq-convex domains in Stein manifolds-equation on localq-convex domains in Stein manifolds

The homotopy formulas of (r, s) differential forms and the solution of\(\bar \partial \)-equation of type (r, s) on localq-convex domains in Stein manifolds are obtained. The homotopy formulas on localq-convex domains have important applications in uniform estimates of\(\bar \partial \)-equation and holomorphic extension of CR-manifolds.

Symmetry properties of positive solutions of elliptic equations on symmetric domains

An improved version of the maximum principle is used to generalize Alexandrov's moving-planes method. Symmetry and monotonicity properties are obtained for positive solutions of second order semilinear elliptic equations in symmetric domains in ℝn as well as in domains in Riemannian manifolds.

Stability Domain and Invariant Manifolds of 2d Area-Preserving Diffeomorphisms

We study the stability domain of generic 2D area-preserving polynomialdiffeomorphisms. The starting point of our analysis is the study of thedistribution of stable and unstable fixed points. We show that the locationof fixed points and their stability type are linked to the degree of thepolynomial map. These results are based on a classification Theorem forplane automorphisms by Friedland and Milnor. Then we discuss the problem ofdetermining the domain in phase space where stable motion occurs. We showthat the boundary of the stability domain is given by the invariantmanifolds emanating from the outermost unstable fixed point of low period(one or two). This fact extends previous results obtained for reversiblearea-preserving polynomial maps of the plane. This analysis is based onanalytical arguments and is supported by the results of numericalsimulations.

Domains in Riemannian manifolds and inverse spectral geometry

Domains in Riemannian manifolds and inverse spectral geometry

Line bundle convexity of pseudoconvex domains in complex manifolds

Line bundle convexity of pseudoconvex domains in complex manifolds

Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature

Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature

Spectrum of domains in Riemannian manifolds

Soit M n une variete de Riemann complete, de dimension n, avec une courbure de Ricci bornee inferieurement par −(n-1)c, C≥0. Soit Ω un domaine a frontiere lisse contenu dans M. On considere le laplacien Δ=d*d agissant sur L 2 Ω avec des conditions aux limites de Dirichlet. On etudie le spectre de Δ sur differents domaines Ω