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

By M. Mitrea, Michael Taylor. 2000

150 citations recorded by the source corpus.

Abstract 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 Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C 1+ 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/d740f0da0566cdfebbeb91c34ab7653d832dc496

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In Applied Mathematics

Citations in the corpus

Differential operators and boundary value problems on hypersurfaces (2006, methodology): “For starters, the treatment of the classical Dirichlet and Neumann boundary problems for the Laplace-Beltrami operator in Lipschitz subdomains of Riemannian manifolds from [ MT ] translate into the well-posedness results about 8 <”
Boundary value problems for the Laplacian in convex and semiconvex domains (2010, methodology): “The work in [41] has been extended in [28,66] to allow Neumann boundary condition and variable coefficient operators, and further, in [50,51], to allow data from B α , F p,q α , 0<p,q <∞, α ∈ R for an optimal range of indices.”
Non-self-adjoint operators, infinite determinants, and some applications (2005, methodology): “Indeed, with H−s+3/2(Ω) replaced by H−s+3/2(Ω ∩B) for a sufficiently large ball B ⊂ R, this is proved in [35] and the behavior at infinity is easily taken care of by employing the exponential decay of E.”
Boundary Value Problems, Weyl Functions, and Differential Operators (2020, methodology): “In this context we also mention the papers [585, 586, 587, 588] by Mitrea and Taylor for general layer potential methods in Lipschitz domains on Riemannian manifolds.”
Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schr\ (2008, unknown): “Its mappings properties on the scale of Sobolev spaces have been analyzed in [74] and we note here that D−1 : H (∂Ω) → H(Ω), 0 ≤ s ≤ 1, (2.”
The Stationary Navier-Stokes System in Nonsmooth Manifolds: The Poisson Problem in Lipschitz and C1 Domains (2004, background): “For a more detailed exposition, the interested reader is referred to [3], [33], [43], [63].”
The regularity of the Stokes operator and the Fujita–Kato approach to the Navier–Stokes initial value problem in Lipschitz domains (2008, background): “116) have been proved in [40] and [38].”
Navier-Stokes equations on Lipschitz domains in Riemannian manifolds (2001, background): “Carrying out this extension requires further work, but this has led to many results on Dirichlet, Neumann, and other natural boundary problems for the Laplace operator and the Hodge Laplacian in [12], [13], and [11].”