C1,α domains and unique continuation at the boundary

✍ Scribed by Vilhelm Adolfsson; Luis Escauriaza

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 286 KB
• Volume: 50
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199710)50:10<935::aid-cpa1>3.0.co;2-h

✦ Synopsis

It is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset V contained in the boundary of a Dini domain or whose normal derivative vanishes on an open subset V in the boundary of a C 1,1 domain in R d satisfies the doubling property with respect to balls centered at points Q ∈ V . Under any of the above conditions, the module of the gradient of u is a B2(dσ)-weight when restricted to V , and the Hausdorff dimension of the set of points {Q ∈ V : ∇u(Q) = 0} is less than or equal to d-2. These results are generalized to solutions to elliptic operators with Lipschitz second-order coefficients and bounded coefficients in the lower-order terms.