arXiv subjects

Minh-Phuong Tran

Publications and source records attributed to Minh-Phuong Tran.

24 records · Page 2Linked to original sources

arXiv2020

Global Lorentz estimates for non-uniformly nonlinear elliptic equations via fractional maximal operators

This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows \begin{align*} -\mathrm{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u) = - \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F} + a(x)|\mathbf{F}|^{q-2}\mathbf{F}), \end{align*} that arises from double phase functional problems. In particular, the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators. This work extends that of \cite{CoMin2016,Byun2017Cava} by dealing with the global estimates in Lorentz spaces. This work also extends our recent result in \cite{PNJDE}, which is devoted to the new estimates of divergence elliptic equations using cut-off fractional maximal operators. For future research, the approach developed in this paper allows to attain global estimates of distributional solutions to non-uniformly nonlinear elliptic equations in the framework of other spaces.

math.AP↗

arXiv2019

An application of global gradient estimates in Lorentz-Morrey spaces: The existence of stationary solution to degenerate diffusive Hamilton-Jacobi equations

In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes, is also known by the different name as the quasilinear Riccati type equation. The existence of solutions to this type of equation under some assumptions and requirements, still remains an interesting open problem at the moment. In our previous studies \cite{MP2018, MPT2019}, we obtained the global bounds and gradient estimates for quasilinear elliptic equations with measure data. There have been many applications are discussed related to these works, and main goal of this paper is to obtain the existence of a renormalized solution to the quasilinear stationary solution to the degenerate diffusive Hamilton-Jacobi equation with the finite measure data in Lorentz-Morrey spaces.

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arXiv2019

A remark on the interpolation inequality between Sobolev spaces and Morrey spaces

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic and/or parabolic equations. Since many of researchers who are working on this domain, the main motivation of our work here is to provide an important observation of $L^p$-boundedness property with respect to the interpolation inequalities. In this paper, from an idea of integral approximation theory and Sobolev, Morrey embeddings, we construct a nontrivial counterexample for interpolation inequalities in the connection of results between Sobolev and Morrey spaces. Our proofs rely on the integral representation and the theory of maximal and sharp maximal functions.

math.CA↗

arXiv2019

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} div(A(x,\nabla u)) &= \ div(|F|^{p-2}F) \quad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ σ\qquad \qquad \qquad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), the nonlinearity $A$ is a monotone Carathéodory vector valued function defined on $W^{1,p}_0(Ω)$ for $p>1$ and the $p$-capacity uniform thickness condition is imposed on the complement of our bounded domain $Ω$. Moreover, for given data $F \in L^p(Ω;\mathbb{R}^n)$, the problem is set up with general Dirichlet boundary data $σ\in W^{1-1/p,p}(\partialΩ)$. In this paper, the optimal good-$λ$ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works \cite{55QH4, MPT2018, MPT2019} and references therein.

math.AP↗

arXiv2019

Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ u &=0 \quad \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz-Morrey spaces, where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), $μ$ is a finite Radon measure, $A$ is a monotone Carathéodory vector valued function defined on $W^{1,p}_0(Ω)$ and the $p$-capacity uniform thickness condition is imposed on the complement of our domain $Ω$. It is remarkable that the local gradient estimates has been proved firstly by G. Mingione in \cite{Mi3} at least for the case $2 \le p \le n$, where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz-Morrey and Morrey regularities were obtained by N.C.Phuc in \cite{55Ph1} for regular case $p>2 - \frac{1}{n}$. Here in this study, we particularly restrict ourselves to the singular case $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$. The results are central to generalize our technique of good-$λ$ type bounds in previous work \cite{MP2018}, where the local gradient estimates of solution to this type of equation was obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.

math.AP↗

arXiv2018

Good-$λ$ type bounds of quasilinear elliptic equations for the singular case

In this paper, we study the good-$λ$ type bounds for renormalized solutions to nonlinear elliptic problem: \begin{align*} \begin{cases} -÷(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ u &=0 \quad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$, $μ$ is a finite Radon measure and $A$ is a monotone Carathédory vector valued function defined on $W^{1,p}_0(Ω)$. The operator $A$ satisfies growth and monotonicity conditions, and the $p$-capacity uniform thickness condition is imposed on $\mathbb{R}^n \setminus Ω$, for the singular case $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$. In fact, the same good-$λ$ type estimates were also studied by Quoc-Hung Nguyen and Nguyen Cong Phuc. For instance, in \cite{55QH4,55QH5}, authors' method was also confined to the case of $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$ but under the assumption of $Ω$ is the Reifenberg flat domain and the coefficients of $A$ have small BMO (bounded mean oscillation) semi-norms. Otherwise, the same problem was considered in \cite{55Ph0} in the regular case of $p>2-\frac{1}{n}$. In this paper, we extend their results, taking into account the case $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$ and without the hypothesis of Reifenberg flat domain on $Ω$ and small BMO semi-norms of $A$. Moreover, in rest of this paper, we also give the proof of the boundedness property of maximal function on Lorentz spaces and also the global gradient estimates of solution.

math.AP↗