𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

✍ Scribed by Eugene Fabes; Osvaldo Mendez; Marius Mitrea

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
656 KB
Volume
159
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

✦ Synopsis

We study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev Besov spaces. As such, this is a natural continuation of work in [Jerison and Kenig, J. Funct. Anal. (1995), 16 219] where the inhomogeneous Dirichlet problem is treated via harmonic measure techniques. The novelty of our approach resides in the systematic use of boundary integral methods. In this regard, the key results are establishing the invertibility of the classical layer potential operators on scales of Sobolev Besov spaces on Lipschitz boundaries for optimal ranges of indices. Applications to L p -based Helmholtz type decompositions of vector fields in Lipschitz domains are also presented.

📜 SIMILAR VOLUMES

Potential Theory on Lipschitz Domains in
Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem
✍ Marius Mitrea; Michael Taylor 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 450 KB

We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on L p 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 L