Higher Order Boundary Harnack Principle via Degenerate Equations

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Susanna Terracini, Giorgio Tortone, Stefano Vita
{"title":"Higher Order Boundary Harnack Principle via Degenerate Equations","authors":"Susanna Terracini,&nbsp;Giorgio Tortone,&nbsp;Stefano Vita","doi":"10.1007/s00205-024-01973-1","DOIUrl":null,"url":null,"abstract":"<div><p>As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type </p><div><div><span>$$\\begin{aligned} -\\textrm{div}\\left( \\rho ^aA\\nabla w\\right) =\\rho ^af+\\textrm{div}\\left( \\rho ^aF\\right) \\quad \\text {in}\\; \\Omega \\end{aligned}$$</span></div></div><p>for exponents <span>\\(a&gt;-1\\)</span>, where the weight <span>\\(\\rho \\)</span> vanishes with non zero gradient on a regular hypersurface <span>\\(\\Gamma \\)</span>, which can be either a part of the boundary of <span>\\(\\Omega \\)</span> or mostly contained in its interior. As an application, we extend such estimates to the ratio <i>v</i>/<i>u</i> of two solutions to a second order elliptic equation in divergence form when the zero set of <i>v</i> includes the zero set of <i>u</i> which is not singular in the domain (in this case <span>\\(\\rho =u\\)</span>, <span>\\(a=2\\)</span> and <span>\\(w=v/u\\)</span>). We prove first the <span>\\(C^{k,\\alpha }\\)</span>-regularity of the ratio from one side of the regular part of the nodal set of <i>u</i> in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension <span>\\(n=2\\)</span>, we provide local gradient estimates for the ratio, which hold also across the singular set.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-01973-1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

Abstract

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type

$$\begin{aligned} -\textrm{div}\left( \rho ^aA\nabla w\right) =\rho ^af+\textrm{div}\left( \rho ^aF\right) \quad \text {in}\; \Omega \end{aligned}$$

for exponents \(a>-1\), where the weight \(\rho \) vanishes with non zero gradient on a regular hypersurface \(\Gamma \), which can be either a part of the boundary of \(\Omega \) or mostly contained in its interior. As an application, we extend such estimates to the ratio v/u of two solutions to a second order elliptic equation in divergence form when the zero set of v includes the zero set of u which is not singular in the domain (in this case \(\rho =u\), \(a=2\) and \(w=v/u\)). We prove first the \(C^{k,\alpha }\)-regularity of the ratio from one side of the regular part of the nodal set of u in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension \(n=2\), we provide local gradient estimates for the ratio, which hold also across the singular set.

Abstract Image

通过退化方程的高阶边界哈纳克原理
作为第一个结果,我们证明了奇异/退化椭圆方程的解的高阶 Schauder 估计值,其类型为 $$\begin{aligned} -\textrm{div}\left(\rho ^aA\nabla w\right) =\rho ^af+\textrm{div}\left(\rho ^aF\right) \quad \text {in};\对于指数 \(a>-1\),权重 \(\rho\)在规则超曲面 \(\Gamma\)上以非零梯度消失,这个超曲面可以是 \(\Omega\)边界的一部分,也可以大部分包含在它的内部。作为应用,我们把这种估计扩展到发散形式的二阶椭圆方程的两个解的v/u之比,当v的零集包括u的零集,而u在域中不是奇异的时候(在这种情况下,\(\rho =u\),\(a=2\)和\(w=v/u\))。我们首先根据萨文(Discrete Contin Dyn Syst 35-12:6155-6163, 2015)中高阶边界哈纳克原理的精神,证明了来自u的结点集正则部分一边的比率的(C^{k,\alpha }\ )正则性。然后,根据胶合定理,估计值扩展到节点集的正则部分。最后,利用维度(n=2)的保形映射,我们提供了比率的局部梯度估计,该估计在奇点集中也成立。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信