Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds

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.

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

In this paper we use a layer potential analysis to show the existence of solutions for a Dirichlet‐transmission problem for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in compact boundaryless Riemannian manifolds. Compactness and invertibility properties of corresponding layer potential operators on Lp, Sobolev, or Besov scales are also obtained.

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.

In the previous work, the author and M. Mitrea presented a method of solving the stationary Navier-Stokes equation on Lipschitz domains in Riemannian manifolds via the boundary integral technique, where only the vanishing Dirichlet boundary condition was considered. In this paper, more sophisticated estimates are developed, which allows us to consider arbitrary large (dim M ≤ 4) Dirichlet boundary data for this equation.KeywordsRiemannian ManifoldStokes EquationLipschitz DomainStokes ProblemStokes SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

R n, n≥ 3, and with L2 boundary data. Shen [47] studied the L p Dirichlet problem for the Stokes system in Lipschitz domains in Rn, n≥ 3, and obtained optimal estimates when n= 3, as well as a weak estimate of solutions for certain range of p when n≥ 4 (see also [48]). Mitrea and Taylor [40–45] developed the potential theory for elliptic operators on Lipschitz domains in Riemannian manifolds and used this theory to treat the

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to domains with corners.

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.

Potential theory on Lipschitz domains in Riemannian manifolds: $L^p$ Hardy, and Holder space results

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

We examine solutions u = PIf to Δu − Vu = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M, satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L p -Sobolev space results of the form and variants, when the metric tensor on M has limited regularity, described by a Hölder or a Dini-type modulus of continuity. We also discuss related estimates for solutions to the Neumann problem.

Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem $Lu-F(x,u)=0$ in $\Omega$. We improve our previous results by studying more general nonlinear terms $F(x,u)$ with polynomial (and in some cases exponential) growth in the variable $u$. We also study the case of nonnegative solutions.

Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds

Gradient Estimates for Harmonic Functions on Regular Domains in Riemannian Manifolds

In this paper we use the method of boundary integral equations to treat some transmission problems for Brinkman-type operators on Lipschitz and C1 domains in Riemannian manifolds.