Full and partial regularity for a class of nonlinear free boundary problems

IF 1.8 1区数学 Q1 MATHEMATICS, APPLIED

Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-07-01 DOI:10.1016/j.anihpc.2020.09.008

Aram Karakhanyan

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引用次数: 2

Abstract

In this paper we classify the nonnegative global minimizers of the functional $J_{F} (u) = \int_{Ω} F (| \nabla u |^{2}) + λ^{2} χ_{{u > 0}},$ where F satisfies some structural conditions and $χ_{D}$ is the characteristic function of a set $D \subset R^{n}$ . We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial {u > 0}$ can be seen as a rectifiable $n - 1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if $n = 3$ and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

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一类非线性自由边界问题的全正则性和部分正则性

本文对泛函jf (u)=∫ΩF(|∇u|2)+λ2χ{u>0}的非负全局极小值进行了分类，其中F满足某些结构条件，χD是集合D∧Rn的特征函数。我们计算了能量的二阶变分，并研究了稳定算子的性质。自由边界∂{u>0}可以看作是一个可校正的n - 1变量。如果自由边界是一个Lipschitz多重图，那么我们证明了这个变分的第一个变分是有界的。因此可以用Allard的单调性公式来证明切锥模于一组小Hausdorff维数的存在性。特别地，我们证明了如果n=3且由F生成的拟线性椭圆算子的椭圆常数接近于1，则圆锥自由边界一定是平坦的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annales De L Institut Henri Poincare-Analyse Non Lineaire 数学-应用数学

CiteScore

4.10

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.