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An elliptic free boundary arising from the jump of conductivity
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2017 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 161, p. 1-29Article in journal (Refereed) Published
Abstract [en]

In this paper we consider a quasilinear elliptic PDE, div(A(x,u)∇u)=0, where the underlying physical problem gives rise to a jump for the conductivity A(x,u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

Place, publisher, year, edition, pages
Elsevier Ltd , 2017. Vol. 161, p. 1-29
Keywords [en]
Conductivity jump, Free boundary problem, Quasilinear elliptic equation, Mathematical techniques, Nonlinear analysis, Free boundary, Free-boundary problems, Lipschitz regularity, Matrix-valued, Monotonicity, Quasi-linear elliptic, Quasilinear elliptic equations, Linear equations
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-216189DOI: 10.1016/j.na.2017.05.010ISI: 000407182400001Scopus ID: 2-s2.0-85020887578OAI: oai:DiVA.org:kth-216189DiVA, id: diva2:1168192
Note

QC 20171220

Available from: 2017-12-20 Created: 2017-12-20 Last updated: 2017-12-20Bibliographically approved

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Shahgholian, Henrik

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