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C 1, 1 Regularity in semilinear elliptic problems
Author(s) -
Shahgholian Henrik
Publication year - 2003
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/cpa.10059
Subject(s) - mathematics , bounded function , corollary , simple (philosophy) , lipschitz continuity , unit sphere , monotonic function , mathematical proof , pure mathematics , ball (mathematics) , discrete mathematics , mathematical analysis , geometry , philosophy , epistemology
In this paper we give an astonishingly simple proof of C 1, 1 regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optical results. The setting we'll consider is the following. Let u be a solution to$$\Delta u = f(x, u)\,\,\,\, \rm{in} \,\it{B},$$where B , is the unit ball in ℝ n , f ( x , t ) is a bounded Lipschitz function in x , and f t ′ is bounded from below. Then we prove that u ⊇ C 1, 1 ( B 1/2 ). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig. © 2002 Wiley Periodicals, Inc.