An higher integrability result for the second derivatives of the solutions to a class of elliptic PDE’s
| Autor: | Claudia Capone |
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| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
a priori estimate General Mathematics 010102 general mathematics approximation Function (mathematics) Algebraic geometry 01 natural sciences Omega Combinatorics Elliptic curve Number theory Embedding theorem 0103 physical sciences reverse inequality 010307 mathematical physics 0101 mathematics Linear growth Second derivative Mathematics |
| Zdroj: | Manuscripta mathematica 164 (2021): 375–393. doi:10.1007/s00229-020-01186-2 info:cnr-pdr/source/autori:Capone C./titolo:An higher integrability result for the second derivatives of the solutions to a class of elliptic PDE's/doi:10.1007%2Fs00229-020-01186-2/rivista:Manuscripta mathematica/anno:2021/pagina_da:375/pagina_a:393/intervallo_pagine:375–393/volume:164 |
| ISSN: | 1432-1785 0025-2611 |
| Popis: | In this paper we establish an higher integrability result for second derivatives of the local solution of elliptic equation 0.1 $$\begin{aligned} \text {div} (A(x, Du)) =0 \,\,\,\,\,\, \hbox {in } \Omega \end{aligned}$$ where $$\Omega \subseteq \mathbb {R}^n$$ , $$n\ge 2$$ and $$A(x,\xi )$$ has linear growth with respect to $$\xi $$ variable. Concerning the dependence on the x-variable, we shall assume that, for the map $$x \rightarrow A(x,\xi )$$ , there exists a non negative function k(x), such that 0.2 $$\begin{aligned} |D_x A(x, \xi )| \leqslant k(x)\,(1+ |\xi |) \end{aligned}$$ for every $$\xi \in \mathbb {R}^n$$ and a.e. $$x \in \Omega $$ . It is well known that there exists a relationship between this condition and the regularity of the solutions of the equation. Our pourpose is to establish an higher integrability result for second derivatives of the local solution, by assuming k(x) in a suitable Zygmund class. |
| Databáze: | OpenAIRE |
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