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The Jump of the Laplacian on a Submanifold
Author(s) -
Dudek Ewa,
Holly Konstanty
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971880106
Subject(s) - submanifold , mathematics , fubini's theorem , jump , codimension , laplace operator , pure mathematics , projection (relational algebra) , mathematical analysis , function (biology) , combinatorics , physics , algorithm , quantum mechanics , evolutionary biology , biology
Assume that a submanifold M ⊂ ℝ n of an arbitrary codimension k ϵ {1, …, n } is closed in some open set O →ℝ n . With a given function u ϵ C 2 ( O \ M ) we may associate its trivial extension u : O →ℝ such that u| O \M = u and u| m ≡ 0. The jump of the Laplacian of the function u on the submanifold M is defined by the distribution Δ u — Δ u . By applying some general version of the Fubini theorem to the nonlinear projection onto M we obtain the formula for the jump of the Laplacian (Theorem 2.2).