The Helmholtz decomposition of weightedL q-spaces for Muckenhoupt weights
Copyright policy )The Helmholtz decomposition of weightedL q-spaces for Muckenhoupt weights
The aim of this article is to prove the existence of a Helmholtz decomposition for Muckenhoupt weights. A compact embedding and a Poincaré inequality are proved in weighted Sobolev spaces for a bounded Lipschitz domain in \(\mathbb{R}^n\). These technical tools permit the author to solve the weak Neumann problem for the Laplace equation in weighted spaces on \(\mathbb{R}^n\) on \(\mathbb{R}^n_+\), on bounded domains and on exterior domains with boundary of class \(C^1\). This yields the desired Helmholtz decomposition by adapting a method of Simader and Sohr. The present research extends previous work by Farwig and Sohr.
- TU Darmstadt Germany
bounded Lipschitz domain, weak Neumann problem, Maximal functions, Littlewood-Paley theory, Poincaré inequality, Sobolev spaces, compact embedding, Laplace equation, Navier-Stokes equations, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, weighted Sobolev spaces, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
bounded Lipschitz domain, weak Neumann problem, Maximal functions, Littlewood-Paley theory, Poincaré inequality, Sobolev spaces, compact embedding, Laplace equation, Navier-Stokes equations, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, weighted Sobolev spaces, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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