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Nodal Sets for "Broken" Quasilinear PDEs
Seoul Natl Univ, Dept Math Sci, Seoul 08826, South Korea..
Seoul Natl Univ, Dept Math Sci, Seoul 08826, South Korea.;Korea Inst Adv Study, Seoul 02455, South Korea..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2019 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 68, no 4, p. 1113-1148Article in journal (Refereed) Published
Abstract [en]

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, div(A(s) (x, u)del u) = div (f) over bar (x), where A(s) (x, u) has "broken" derivatives of orders s >= 0, such as A(s)(x, u) = a(x) + b(x) (u(+))(s), with (u(+))(0) being understood as the characteristic function on {u > 0}. The vector (f) over bar (x) is assumed to be C-alpha in case s = 0, and C-1,C-alpha (or higher) in case s > 0. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More precisely, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

Place, publisher, year, edition, pages
INDIANA UNIV MATH JOURNAL , 2019. Vol. 68, no 4, p. 1113-1148
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Mathematics
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URN: urn:nbn:se:kth:diva-261064DOI: 10.1512/iumj.2019.68.7711ISI: 000484366600003OAI: oai:DiVA.org:kth-261064DiVA, id: diva2:1356326
Note

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approved

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Shahgholian, Henrik

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