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Regularity Results for Elliptic Equations in Lipschitz Domains

✍ Scribed by Giuseppe Savaré

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
487 KB
Volume
152
Category
Article
ISSN
0022-1236

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✦ Synopsis

We develop a simple variational argument based on the usual Nirenberg difference quotient technique to deal with the regularity of the solutions of Dirichlet and Neumann problems for some linear and quasilinear elliptic equation in Lipschitz domains. We obtain optimal regularity results in the natural family of Sobolev spaces associated with the variational structure of the equations. In the linear case, we obtain in a completely different way some of the results of D. Jerison and C. E. Kenig about the Laplace equation.

1998 Academic Press then u belongs to H 2 loc (0) and this regularity holds up to the boundary, i.e., u # H 2 (0), if 0 is of class C 1, 1 or 0 is convex (see, e.g.

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✍ Martin Costabel 📂 Article 📅 1990 🏛 John Wiley and Sons 🌐 English ⚖ 202 KB

## Abstract Let u be a vector field on a bounded Lipschitz domain in ℝ^3^, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space __H__^1/2^ on the domain