Critical Sets of Elliptic Equations
Critical Sets of Elliptic Equations
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DOI：
10.1002/cpa.21518
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发表时间：
2012-07
影响因子：
3
通讯作者：
J. Cheeger;A. Naber;Daniele Valtorta
中科院分区：
文献类型：
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作者：
J. Cheeger;A. Naber;Daniele Valtorta
给定具有Lipschitz系数的线性齐次二阶椭圆方程的解u，我们引入用于给出临界集𝒷（u）u {x：|δu|（x）= 0}，以及对有效临界集r（u）的第一次估计𝒷，它大致由点x组成，使得u的梯度在Br（x）上的某处与u的不恒定性相比是小的。这些结果甚至对调和函数都是新的。给定这样的u，u的标准一阶分层{lk}基于u-u（x）的首阶多项式的对称度来分离点x。在本文中，我们给出了u的定量分层{ ln，rk }，它基于u的近似首阶多项式在各种尺度下的几乎对称的数量来分离点。我们证明了每个{ ln，rk }的管状邻域的体积的有效估计，这直接导致u的临界集的（n-2 + n）-Minkowski型估计。通过对方程系数的一些额外的正则性假设，我们改进了估计，给出了u的临界集和奇异集上的一致（n-2）-Hausdorff测度估计的新证明。
Given a solution u to a linear, homogeneous, second‐order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set 𝒷(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set 𝒷r(u), which roughly consists of points x such that the gradient of u is small somewhere on Br(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ℝn. Given such a u, the standard first‐order stratification {lk} of u separates points x based on the degrees of symmetry of the leading‐order polynomial of u‐u(x). In this paper we give a quantitative stratification { ln,rk } of u, which separates points based on the number of almost symmetries of approximate leading‐order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each { ln,rk } , which lead directly to (n‐2 + ɛ)‐Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n‐2)‐Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc.