Very weak solutions of linear elliptic PDEs with singular data and irregular coefficients

J. Merker
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引用次数: 1

Abstract

In this article it is shown that linear elliptic PDEs admit very weak solutions for rather singular data – like non-integrable right hand sides or singular Neumann boundary conditions – not only in case of continuous coefficients, but even for general bounded measurable coefficients. This is rather astonishing, as under such weak assumptions on the coefficients generally strong solutions do not exist, thus the duality between very weak solutions and strong solutions seems to indicate that very weak solutions do not exist either. We circumvent this problem by using an appropriate functional analytic setting and particularly Hölder continuity of weak solutions established by de Giorgi Nash Moser to obtain existence of very weak solutions to singular data for irregular coefficients.
具有奇异数据和不规则系数的线性椭圆偏微分方程的极弱解
本文证明了线性椭圆偏微分方程不仅在连续系数的情况下,而且在一般有界可测系数的情况下,对于相当奇异的数据,如不可积的右侧或奇异的诺伊曼边界条件,都有非常弱的解。这是相当惊人的,因为在对系数的这种弱假设下,一般强解不存在,因此极弱解和强解之间的对偶性似乎表明极弱解也不存在。我们利用适当的泛函解析设置,特别是利用de Giorgi Nash Moser建立的Hölder弱解的连续性,得到了不规则系数奇异数据的极弱解的存在性,从而规避了这一问题。
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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