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Regularity for entropy solutions to degenerate elliptic equations. (English) Zbl 1447.35090

Summary: We derive estimates for entropy solutions to degenerate elliptic equations of the form \[ \begin{cases} - \operatorname{div} \mathcal{A} (x, u(x), \nabla u(x)) = f (x) , \quad & x \in \Omega, \\ u (x) = 0, & x \in \partial \Omega, \end{cases} \] where the Carathéodory function \(\mathcal{A} : \Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n\) satisfies degenerate coercivity condition \[\mathcal{A}(x, s, \xi) \cdot \xi \geq \alpha \frac{ | \xi |^p}{ (1 + | s |)^\theta}\] and controllable growth condition \[| \mathcal{A}(x, s, \xi) | \leq \beta | \xi |^{p - 1}\] for \(1 < p < n\), \(0 \leq \theta < p - 1\) and \(0 < \alpha \leq \beta < \infty \), and \(f\) lies in Marcinkiewicz spaces. We use generalized Stampacchia Lemma to prove the main theorem. Counterexample shows that one of the results in this paper is optimal.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J62 Quasilinear elliptic equations
35J70 Degenerate elliptic equations
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