GRADIENT ESTIMATES FOR ENTROPY SOLUTIONS TO ELLIPTIC EQUATIONS WITH DEGENERATE COERCIVITY

Abstract

We derive some a priori estimates and then prove the existence and regularity of distributional 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

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

for $1<p<n$, $0\le 𝜃<p-1$ and $0<\alpha \le \beta <\infty$, with $f$ a Marcinkiewicz function. We obtain a gradient estimate for entropy solutions, and show that the result is optimal by a counterexample. We also consider degenerate elliptic equations with a lower order term,

We show that the presence of the lower order term has regularizing effects on our result.