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Explicit solutions of the equation –div z = f under the constraint ∥ z ∥ ∞ = 1
Author(s) -
Milbers Zoja
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700733
Subject(s) - mathematics , differentiable function , eigenvalues and eigenvectors , domain (mathematical analysis) , laplace's equation , constraint (computer aided design) , work (physics) , mathematical analysis , euler's formula , laplace operator , pure mathematics , partial differential equation , physics , geometry , quantum mechanics , thermodynamics
We consider the eigenvalue problem for the 1‐Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler‐Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2‐dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler‐Lagrange equation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)