Abstract:

In this paper, the study of gradient regularity for solutions of a class of elliptic problems of $p$-Laplace type is offered. In particular, we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:

$\begin{align*}-\mathrm{div}\left((s^2+|\nabla u|^2)^{\frac{p-2}{2}}\nabla u\right) &= \ -\mathrm{div}\left(|\mathbf{f}|^{p-2}\mathbf{f}\right) + \mathsf{g} \quad \text{in} \ \Omega, \quad u = \mathsf{h} \quad \text{in} \ \partial\Omega,\end{align*}$

with the (sub-elliptic) degeneracy condition $s\in [0,1]$ and with mixed data $\mathbf{f} \in L^p(\Omega;\mathbb{R}^n)$, $\mathsf{g} \in L^{\frac{p}{p-1}}(\Omega;\mathbb{R}^n)$ for $p \in (1,n)$. This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory, electro-rheology, radiation of heat, plastic moulding and many others. Building on the idea of level-set inequality on fractional maximal distribution functions, it enables us to carry out a global regularity result of the solution via fractional maximal operators. Due to the significance of $\mathcal{M}_\alpha$ and its relation with Riesz potential, estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of order $\alpha$. Our approach therefore has its own interest.

Key words: gradient estimates, $p$-Laplace, quasilinear elliptic equation, fractional maximal operators, Lorentz-Morrey spaces