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

Minh-Phuong TRAN; The-Quang TRAN; Thanh-Nhan NGUYEN

doi:10.1007/s10473-024-0412-8

Acta mathematica scientia, Series B >

2024 , Vol. 44 >Issue 4: 1394 - 1414

DOI: https://doi.org/10.1007/s10473-024-0412-8

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

Minh-Phuong TRAN ,
The-Quang TRAN ,
Thanh-Nhan NGUYEN

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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

E-mail: tranminhphuong@tdtu.edu.vn; quangtranthe@gmail.com

Received date: 2023-01-27

Revised date: 2023-10-20

Online published: 2024-08-30

Supported by

The research was supported by Ministry of Education and Training (Vietnam), under grant number B2023-SPS-01.

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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

Cite this article

Minh-Phuong TRAN , The-Quang TRAN , Thanh-Nhan NGUYEN . GLOBAL BOUND ON THE GRADIENT OF SOLUTIONS TO ${p}$-LAPLACE TYPE EQUATIONS WITH MIXED DATA[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1394 -1414 . DOI: 10.1007/s10473-024-0412-8

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