Abstract:
Partial regularity of a generalized solution $u\colon\Omega\subset\mathbb R^n\to\mathbb R^N$, $n>2$, $N>1$, of a quasilinear elliptic system is proved under a nonsmooth condition on the conormal derivative. The singular set
$\Sigma\subset\overline\Omega$ is described; it is proved that for some $p>2$ the Hausdorff dimension of $\Sigma$ is equal to $n-p$. In the proof essential use is made of a theorem proved earlier by the author on reverse inequalities with surface integrals.
Citation:
A. A. Arkhipova, “Partial regularity of solutions of quasilinear elliptic systems with nonsmooth condition on the conormal derivative”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 215–230
\Bibitem{Ark93}
\by A.~A.~Arkhipova
\paper Partial regularity of solutions of quasilinear elliptic systems with nonsmooth condition on the~conormal derivative
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 78
\issue 1
\pages 215--230
\mathnet{http://mi.mathnet.ru/eng/sm966}
\crossref{https://doi.org/10.1070/SM1994v078n01ABEH003466}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1214946}
\zmath{https://zbmath.org/?q=an:0814.35010}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1994NR97600014}
Linking options:
https://www.mathnet.ru/eng/sm966
https://doi.org/10.1070/SM1994v078n01ABEH003466
https://www.mathnet.ru/eng/sm/v184/i2/p87
This publication is cited in the following 4 articles:
A. Arkhipova, “Boundary regularity of unbounded weak solutions of the oblique derivative problem for a class of strongly nonlinear elliptic systems”, St. Petersburg Math. J., 2025
A. A. Arkhipova, “Boundary regularity of unbounded weak solutions of the oblique derivative problem for a class of strongly nonlinear elliptic systems”, Algebra i analiz, 36:1 (2024), 60–94
Kim D., “Global Regularity of Solutions to Quasilinear Conormal Derivative Problem with Controlled Growth”, J. Korean. Math. Soc., 49:6 (2012), 1273–1299
A. A. Arkhipova, “On the regularity of the solutions of the Neumann problem for quasilinear parabolic systems”, Russian Acad. Sci. Izv. Math., 45:2 (1995), 231–253
Citation in format AMSBIB
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