REGULARITY FOR ENTROPY SOLUTIONS TO DEGENERATE ELLIPTIC EQUATIONS WITH A CONVECTION TERM

Abstract

We deal with entropy solutions to degenerate elliptic equations of the form

where the Carathéodory function $\mathcal{𝒜}:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to {ℝ}^n$ satisfies degenerate coercivity condition

$$\mathcal{𝒜}(x,s,\xi )\cdot \xi \ge \alpha \frac{|\xi {|}^p}{{(1+|s|)}^{\tau }}$$

and controllable growth condition

$$|\mathcal{𝒜}(x,s,\xi )|\le \beta |\xi {|}^{p-1}$$

for almost all $x\in \mathrm{\Omega }$ and all $(s,\xi )\in ℝ\times {ℝ}^n$. We let , , , we let $f$ and $E$ belong to some Marcinkiewicz spaces, and we give some regularity properties for entropy solutions. We derive a generalized version of Stampacchia’s lemma in order to prove the main theorem.