Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (4): 1394-1414.doi: 10.1007/s10473-024-0412-8

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GLOBAL BOUND ON THE GRADIENT OF SOLUTIONS TO ${p}$-LAPLACE TYPE EQUATIONS WITH MIXED DATA

Minh-Phuong Tran1, The-Quang Tran2, Thanh-Nhan Nguyen3,*   

  1. 1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
    2. Nguyen Huu Huan High School, Thu Duc City, Vietnam;
    3. Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Vietnam
  • Received:2023-01-27 Revised:2023-10-20 Online:2024-08-25 Published:2024-08-30
  • Contact: *E-mail: nhannt@hcmue.edu.vn
  • About author:E-mail: tranminhphuong@tdtu.edu.vn; quangtranthe@gmail.com
  • Supported by:
    The research was supported by Ministry of Education and Training (Vietnam), under grant number B2023-SPS-01.

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

CLC Number: 

  • 35J62
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