Interior estimates

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In this chapter we establish interior Hölder (C0,α) estimates, W1,p estimates, and Lipschitz (C0,1) estimates, that are uniform in ε > 0, for solutions of Lε (uε) = F, where Lε = -div(A(x/ε)∇). As a result, we obtain uniform size estimates of Γe(x, y)∇xΓ ε (x, y)∇yΓ ε (x, y), and ∇xyΓ ε (x, y), where Γ ε (x, y) denotes the matrix of fundamental solutions for Lε in ℝd. This in turn allows us to derive asymptotic expansions, [Formula presented.]. Thus no uniform regularity beyond Lipschitz estimates should be expected (unless div(A) = 0, which would imply Xβ j = 0).

Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
Number of pages33
StatePublished - 2018

Publication series

NameOperator Theory: Advances and Applications
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Bibliographical note

Publisher Copyright:
© Springer Nature Switzerland AG 2018.

ASJC Scopus subject areas

  • Analysis


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