Critical Perturbations for Second Order Elliptic Operators—Part II: Non-tangential Maximal Function Estimates

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Simon Bortz, Steve Hofmann, José Luis Luna Garcia, Svitlana Mayboroda, Bruno Poggi
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引用次数: 0

Abstract

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators \(-\textrm{div}A \nabla \) by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the \(L^2\) well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi–Nash–Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-\(L^p\)\(N<S\)” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full \(L^2\) bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the \(L^p\)-solvability of boundary value problems for the magnetic Schrödinger operator \(-(\nabla -i\textbf{a})^2+V\) when the magnetic potential \(\textbf{a}\) and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.

二阶椭圆算子的临界扰动--第二部分:非切向最大函数估计值
这是我们通过层势方法研究发散形式二阶椭圆算子(-\textrm{div}A \nabla \)的一阶和零阶项的扰动的系列论文的最后一部分,其复数系数位于临界空间。特别是,我们证明了上半空间中复赫米特、块形式或常系数发散形式椭圆算子的 Dirichlet、Neumann 和正则性问题的 \(L^2\) 拟合性(具有自然非切线最大函数估计值)在这种扰动下都是稳定的。由于在我们的设置中缺乏经典的 De Giorgi-Nash-Moser 理论,我们证明非切线最大函数估计的方法依赖于一个全新的论证:我们得到了某个弱的(L^p\)"(N<S\)"估计值,并最终将其与平方函数边界、加权外推法理论和引导论证结合起来,从而恢复了完整的(L^2\)边界。最后,我们在一个相对广泛的类别中证明了解的存在性和唯一性。作为推论,当磁势\(textbf{a}\)和电势V在尺度不变的勒贝格空间的规范中相应较小时,我们声称在无界域中关于磁薛定谔算子边界值问题的\(L^p\)-可解性的第一个结果。
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
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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