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A semilinear PDE with free boundary
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-1316-7913
2017 (English)In: Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X, Vol. 151, p. 145-163Article in journal (Refereed) Published
Abstract [en]

We study the semilinear problem Δu=λ+(x)(u+)q−1−λ−(x)(u−)q−1inB1, from a regularity point of view for solutions and the free boundary ∂{±u>0}. Here B1 is the unit ball, 1<q<2 and λ± are Lipschitz. Our main results concern local regularity analysis of solutions and their free boundaries. One of the main difficulties encountered in studying this equation is classification of global solutions. In dimension two we are able to present a fairly good analysis of global homogeneous solutions, and hence a better understanding of the behavior of the free boundary. In higher dimensions the problem becomes quite complicated, but we are still able to state partial results; e.g. we prove that if a solution is close to one-dimensional solution in a small ball, then in an even smaller ball the free boundary can be represented locally as two C1-regular graphs Γ+=∂{u>0} and Γ−=∂{u<0}, tangential to each other. It is noteworthy that the above problem (in contrast to the case q=1) introduces interesting and quite challenging features, that are not encountered in the case q=1. E.g. one obtains homogeneous global solutions that are not one-dimensional. This complicates the analysis of the free boundary substantially.

Place, publisher, year, edition, pages
Elsevier, 2017. Vol. 151, p. 145-163
Keywords [en]
Free boundary, Regularity, Semilinear elliptic equation
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-200888DOI: 10.1016/j.na.2016.11.019ISI: 000393013300009Scopus ID: 2-s2.0-85007448553OAI: oai:DiVA.org:kth-200888DiVA, id: diva2:1071102
Note

QC 20170203

Available from: 2017-02-03 Created: 2017-02-03 Last updated: 2017-03-03Bibliographically approved

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf