Global bound on the gradient of solutions to p-Laplace type equations with mixed data

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: \( - {\rm{div}}\left( {{{({s^2}|\nabla u{|^2})}^{{{p - 2} \over 2}}}\nabla u} \right) = - {\rm{div}}\,\,(|{\bf{f}}{|^{p - 2}}{\bf{f}}) + {\rm{g}}\,\,\,\,{\rm{in}}\,\,\Omega ,\,\,\,\,u = {\rm{h}}\,\,\,{\rm{in}}\,\,\partial \Omega ,\) $$-\text{div}\left({(s^2|\nabla u{|}^2)}^{\frac{p-2}{2}}\nabla u\right)=-\text{div}(|f{|}^{p-2}f)+\text{g in}\Omega ,u=\text{h in}\partial \Omega ,$$ with the (sub-elliptic) degeneracy condition s ∈ [0,1] and with mixed data \({\bf{f}} \in {L^p}(\Omega ;{\mathbb{R}^n})\) $$f\in L^p(\Omega ;{ℝ}^n)$$ , \(g \in {L^{{p \over {p - 1}}}}(\Omega ;{\mathbb{R}^n})\) $$g\in L^{\frac{p}{p-1}}(\Omega ;{ℝ}^n)$$ for p ∈ (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 \({{\cal M}_\alpha }\) $${ℳ}_{\alpha }$$ 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 α. Our approach therefore has its own interest.