{"title":"Higher Order Boundary Harnack Principle via Degenerate Equations","authors":"Susanna Terracini, Giorgio Tortone, 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>-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
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.
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.