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Quasi–concave envelope of a function and convexity of level sets of solutions to elliptic equations

✍ Scribed by Andrea Colesanti; Paolo Salani

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
170 KB
Volume
258
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Given a C^2^ function u, we consider its quasi–convex envelope u* and we investigate the relationship between D^2^u and D^2^u* (the latter intended in viscosity sense); we obtain two inequalities between the tangential Laplacian of u and u* and the normal second derivative of u and u* (the words tangential and normal are referred to a level set of the involved functions). Then we apply the result to prove convexity of level sets of solutions of elliptic equations in convex rings. Our results can be applied to a class of elliptic operator which can be naturally decomposed in a tangential and a normal part, such as Laplacian, p–Laplacian or the Mean Curvature operator. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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