Linear and fully nonlinear elliptic equations with L_d-drift

Abstract

In subdomains of ${\mathbb{R}}^d,$ we consider uniformly elliptic equations $H(v(x),Dv(x),D^{2}v(x),x)=0$ with the growth of H with respect to $\left|Dv\right|$ controlled by the product of a function from L_d and $\left|Dv\right|.$ The dependence of H on x is assumed to be of BMO type. Among other things we prove that there exists $d_0\in (d/2,d)$ such that for any $p\in (d_0,d)$ the equation with prescribed continuous boundary data has a solution in class $W_{p,\text{loc}}^2.$ Our results are new even if H is linear.

Keywords:

• Fully nonlinear equations
• interior estimates
• solvability
• unbounded coefficients

2010 Mathematics Subject Classification:

• 35J60
• 35J15

Acknowledgments

The author is very grateful to Hongjie Dong, A. I. Nazarov, and M. Safonov, who read the first draft of the paper and pointed out several glitches in it. The referee’s comments were very helpful in what concerns the references, the author brings his many thanks to him for that.