Universität zu Köln

Kawohl, Bernd ORCID: 0000-0003-2918-7318 and Horak, Jiri (2017). ON THE GEOMETRY OF THE p-LAPLACIAN OPERATOR. Discret. Contin. Dyn. Syst.-Ser. S, 10 (4). S. 799 - 814. SPRINGFIELD: AMER INST MATHEMATICAL SCIENCES-AIMS. ISSN 1937-1179

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Abstract

The p-Laplacian perator Delta(p)u = div (vertical bar del u vertical bar(p-2)del u) is not uniformly elliptic for any p is an element of (1, 2) boolean OR ( 2, infinity) and degenerates even more when p -> infinity or p -> 1. In those two cases the Dirichlet and eigenvalue problems associated with the p-Laplacian lead to intriguing geometric questions, because their limits for p -> infinity or p -> 1 can be characterized by the geometry of Omega. In this little survey we recall some well- known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general p is an element of [1, infinity]. We report also on results concerning the normalized or game-theoretic p-Laplacian Delta(N)(p) u := 1/p vertical bar del u vertical bar(2-p)Delta(p)u = 1/p Delta(N)(1)u+p-1/p Delta(N)(infinity)u and its parabolic counterpart u(t) - Delta(N)(p)u = 0 These equations are homogeneous of degree 1 and Delta(N)(p) p is uniformly elliptic for any p is an element of(1, infinity). In this respect it is more benign than the p-Laplacian, but it is not of divergence type.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Kawohl, Bernd UNSPECIFIED orcid.org/0000-0003-2918-7318 UNSPECIFIED Horak, Jiri UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-224578
DOI:	10.3934/dcdss.2017040
Journal or Publication Title:	Discret. Contin. Dyn. Syst.-Ser. S
Volume:	10
Number:	4
Page Range:	S. 799 - 814
Date:	2017
Publisher:	AMER INST MATHEMATICAL SCIENCES-AIMS
Place of Publication:	SPRINGFIELD
ISSN:	1937-1179
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language VISCOSITY SOLUTIONS; EIGENVALUE PROBLEM; INFINITY; EQUATION Multiple languages Mathematics, Applied Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/22457