Second-order elliptic equations with variably partially VMO coefficients

TL;DR

This paper proves the solvability of second-order elliptic equations with coefficients that are measurable in one direction and VMO in orthogonal directions, extending previous results to more general coefficient conditions.

Contribution

It introduces a new solvability result for elliptic equations with coefficients that vary measurably in one direction and have VMO regularity in others, depending on the location.

Findings

Solvability in $W^{2}_{p}(R^{d})$ spaces established for these equations.

Generalizes previous results from continuous or VMO coefficients to more variable coefficients.

Provides a framework for equations with direction-dependent coefficient regularity.

Abstract

The solvability in $W^{2}_{p}(\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients.