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On properness and related properties of quasilinear systems on unbounded domains
Author(s) -
Krömer Stefan,
Lilli Markus
Publication year - 2011
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.200810145
Subject(s) - compact space , mathematics , sobolev space , lemma (botany) , nonlinear system , divergence (linguistics) , pure mathematics , sequence (biology) , class (philosophy) , measure (data warehouse) , domain (mathematical analysis) , set (abstract data type) , operator (biology) , mathematical analysis , computer science , philosophy , genetics , biology , physics , poaceae , gene , repressor , database , artificial intelligence , ecology , linguistics , chemistry , biochemistry , quantum mechanics , transcription factor , programming language
The purpose of this paper is to provide tools for analyzing the compactness properties of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general class of nonlinear operators arising in quasilinear systems of partial differential equations of second order, in divergence form. Our approach, based on a suitable decomposition lemma, admits the discussion of problems with some inherent loss of compactness, for example due to a domain with infinite measure or a lower order term with critical growth. As an application, we obtain a characterization of properness which is considerably easier to verify than the definition. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim