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Quelques inégalités pour les potentiels sur un domaine lipschitzien. (Some inequalities for potentials on a Lipschitz domain). (French) Zbl 0604.31002
Let \(\Omega\) be a domain in \({\mathbb{R}}^ 2\) delimited by the graph \(\Gamma\) of a Lipschitzian function, \(V^ 1(\Omega)\) be the pre- hilbertian space of finite energy functions on \(\Omega\) and \(V^{1/2}(\Gamma)\) is the space of ”Lipschitz-square-integrable functions on \(\Gamma\) ”. It is first proved a Dirichlet type theorem, that is there exists a continuous map \(\gamma\) from \(V^ 1(\Omega)\) onto \(V^{1/2}(\Gamma)\) admitting a continuous harmonic lifting.
Using appropriate definitions of normal and tangential derivatives in \(V^ 1(\Omega)\) this leads to a Green formula and moreover to the solution of a Neumann problem.
If F(f) is the Cauchy integral of a function f on \(\Gamma\), the singular Cauchy integral of f is defined by \(I(f)=2\gamma F(f)-f\) and continuity properties of this map are then investigated.
Reviewer: J.Lacroix
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)