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C 1,α domains and unique continuation at the boundary
Author(s) -
Adolfsson Vilhelm,
Escauriaza Luis
Publication year - 1997
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199710)50:10<935::aid-cpa1>3.0.co;2-h
Subject(s) - mathematics , lipschitz continuity , bounded function , boundary (topology) , domain (mathematical analysis) , open set , mathematical analysis , hausdorff dimension , order (exchange) , harmonic function , dimension (graph theory) , directional derivative , function (biology) , pure mathematics , finance , evolutionary biology , economics , biology
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 B 2 ( d σ)‐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. © 1997 John Wiley & Sons, Inc.