Elliptic equations with VMO a, b$\in L_d$, and c$\in L_{d/2}$

Abstract

We consider elliptic equations with operators $L=a^{ij}{D}_{ij}+b^i{D}_i-c$ with $a$ being almost in VMO, $b\in L_d$ and $c\in L_q$, $c\ge 0$, $d>q\ge d/2$. We prove the solvability of $Lu=f\in L_p$ in bounded $C^{1,1}$-domains, $1<p\le q$, and of $\lambda u-Lu=f$ in the whole space for any $\lambda >0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.

19 p, two references added, few glitches are corrected and numerous comments by Hongjie Dong are taken into account, this version is accepted by the Trans. AMS

References in corpus (2)

• On stochastic equations with drift in $L_d$
• Dirichlet boundary value problems for elliptic operators with measure data: a probabilistic approach

Cited by in corpus (1)

• Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients