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Algebra i Analiz, 2020, Volume 32, Issue 3, Pages 39–64 (Mi aa1699)  

Research Papers

Almost everywhere regularity for the free boundary of the $p$-harmonic obstacle problem $p>2$

J. Andersson

Institutionen för matematik KTH, 100 44 Stockholm, Sweden
References:
Abstract: Let $u$ be a solution to the normalized $p$-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and
$$ \mathrm{div}\,( |\nabla u|^{p-2}\nabla u)=\chi_{\{u>0\}}\textrm{ in }B_1(0) $$
where $u(x)\ge 0$ and $\chi_A$ is the characteristic function of the set $A$. The main result is that for almost every free boundary point with respect to the $(n-1)$-Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,\beta}$-graph. That is, for $\mathcal{H}^{n-1}$-a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta}$.
Keywords: $p$-Laplace operator, blow-up, Carleson measure Hausdorff measure.
Funding agency Grant number
Swedish Research Council 2016-03639
This research was supported by the Swedish research council grant 2016-03639.
Received: 18.12.2018
English version:
St. Petersburg Mathematical Journal, 2021, Volume 32, Issue 3, Pages 415–433
DOI: https://doi.org/10.1090/spmj/1654
Document Type: Article
Language: English
Citation: J. Andersson, “Almost everywhere regularity for the free boundary of the $p$-harmonic obstacle problem $p>2$”, Algebra i Analiz, 32:3 (2020), 39–64; St. Petersburg Math. J., 32:3 (2021), 415–433
Citation in format AMSBIB
\Bibitem{And20}
\by J.~Andersson
\paper Almost everywhere regularity for the free boundary of the $p$-harmonic obstacle problem $p>2$
\jour Algebra i Analiz
\yr 2020
\vol 32
\issue 3
\pages 39--64
\mathnet{http://mi.mathnet.ru/aa1699}
\transl
\jour St. Petersburg Math. J.
\yr 2021
\vol 32
\issue 3
\pages 415--433
\crossref{https://doi.org/10.1090/spmj/1654}
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