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This paper establishes the existence of a SOLA for quasilinear elliptic Dirichlet problems featuring finite Radon measure data and a degenerate matrix-valued weight. It then proves a local Calderón-Zygmund estimate for the gradient of the solution, specifically for $|\mathbb{M}|^{\frac{1}{p-1}}\mathbb{M} Du$, under a small log-BMO condition on the weight matrix $\mathbb{M}$. This result quantifies the regularity of solutions in the presence of significant degeneracy and measure data.
Degenerate elliptic equations with measure data exhibit surprising gradient regularity, bounded by fractional maximal functions of the data and weight.
This paper is concerned with the existence and local gradient regularity of solutions to a quasilinear elliptic Dirichlet problem with finite Radon measure and a degenerate matrix-valued weight. Under a small log-BMO condition on $\mathbb M$, we first prove the existence of a SOLA (solution obtained by a limit of approximations) and then establish a local Calder\'on-Zygmund estimate for $|\mathbb{M}|^{\frac{1}{p-1}}\mathbb{M} Du$ in terms of the fractional maximal function $\mathcal{M}_1(\mu)^{\frac{1}{p-1}}$.