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Zbl 1150.35051
Cavalheiro, Albo Carlos
Regularity of weak solutions to certain degenerate elliptic equations.
(English)
[J] Commentat. Math. Univ. Carol. 47, No. 4, 681-693 (2006). ISSN 0010-2628

Denote by $L$ an elliptic operator $Lu=-\sum D_j(a_{ij}(x) D_iu)(x)-\sum b_i(x)D_i u(x)$ whose coefficients $a_{ij}$ and $b_i$ are measurable, real-valued functions, and whose coefficient matrix $A=(a_{ij})$ is symmetric and satisfies the degenerate ellipticity condition $\lambda \omega (x)\vert \psi \vert ^2 \le \sum a_{ij}\psi _i \psi _j \le \Lambda \omega (x)\vert \psi \vert ^2$ in a bounded open set $\Omega$. Here $\omega$ is a weight function, $\lambda$ and $\Lambda$ are positive constants. Let $u$ be a weak solution in the weighted Sobolev space $W^{1,2}(\Omega , \omega )$ of the equation $Lu=g$. Suppose that $g/\omega \in L ^2 (\Omega , \omega )$, $\omega$ is uniformly in the Muckenhoupt class $A_2$, $\vert b(x)\vert \le C\omega (x)$ in $\Omega$. Denote by $\Delta _k^h u(x)$ the difference quotient of $u$ at $x$ in the direction $x_k$. Suppose $\vert \Delta _k^h a_{ij} (x)\vert \le C_1 \omega (x)$, $x\in \Omega ' \subset \Omega$, $0<\vert h \vert < \text {dist}\,(\Omega ', \partial \Omega )$, with a constant $C_1$ independent of $\Omega '$ and $h$. It is shown that $u$ is locally in $W^{2,2}(\Omega , \omega )$.
[Dagmar Medková (Praha)]
MSC 2000:
*35J70 Elliptic equations of degenerate type
35D10 Regularity of generalized solutions of PDE

Keywords: degenerate elliptic equation; weighted Sobolev spaces

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