On the Solvability of Elliptic Equations with Coefficients in $$\boldsymbol{BMO}$$: An Alternative Approach to Elliptic Equations with Irregular Coefficients via Functional Analysis

Abstract: This paper investigates partial differential equations with irregular coefficients, focusing on the Leray–Lions operator. We present existence and uniqueness results for solutions to the Dirichlet problem when the coefficients belong to the space of functions with bounded mean oscillation. The main reason to deal with BMO coefficients instead of the standard bounded coefficients is that BMO coefficients allow for more irregular behavior while still maintaining essential solvability and regularity properties. Moreover, BMO functions can have unbounded oscillations locally, but their mean oscillation remains controlled. Our main contribution is an alternative proof for the existence and uniqueness of solutions to linear equations with -coefficients using the celebrated Banach–Nečas–Babuška theorem. Our approach employs the Hodge decomposition in both standard and weighted Sobolev spaces.