A large scaling property of level sets for degenerate $p$-Laplacian equations with logarithmic BMO matrix weights

TL;DR

This paper extends regularity and gradient estimates for solutions to degenerate p-Laplace equations with logarithmic BMO matrix weights, using advanced covering techniques and fractional maximal operators.

Contribution

It introduces new gradient estimates for level sets of solutions to degenerate p-Laplace equations under logarithmic BMO conditions, broadening the scope of regularity results.

Findings

Established gradient estimates for level sets of solutions.

Extended Calderón-Zygmund estimates to more subtle function spaces.

Developed a covering method for super-level sets using fractional maximal operators.

Abstract

In this study, we deal with generalized regularity properties for solutions to $p$-Laplace equations with degenerate matrix weights. It has already been observed in previous interesting works [A. Kh. Balci, L. Diening, R. Giova, A. Passarelli di Napoli, SIAM J. Math. Anal. 54(2022), 2373-2412] and [A. Kh. Balci, S.-S. Byun, L. Diening, H.-S. Lee, J. Math. Pures Appl. (9) 177(2023), 484-530] that gaining Calder\'on-Zygmund estimates for nonlinear equations with degenerate weights under the so-called $\log$-$\mathrm{BMO}$ condition and minimal regularity assumption on the boundary. In this paper, we also follow this direction and extend general gradient estimates for level sets of the gradient of solutions up to more subtle function spaces. In particular, we construct a covering of the super-level sets of the spatial gradient $|\nabla u|$ with respect to a large scaling parameter via fractional maximal operators.