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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
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.
Received: 18.12.2018
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
Linking options:
https://www.mathnet.ru/eng/aa1699 https://www.mathnet.ru/eng/aa/v32/i3/p39
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Abstract page: | 169 | Full-text PDF : | 16 | References: | 20 | First page: | 15 |
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