Searcharxiv⌕ Search

arXiv subjects

Quoc-Hung Nguyen

Publications and source records attributed to Quoc-Hung Nguyen.

69 records · Page 4Linked to original sources
arXiv2018

Quasilinear and Hessian Lane-Emden type systems with measure data

We study nonlinear systems of the form $-Δ\_pu=v^{q\_1}+μ,\;-Δ\_pv=u^{q\_2}+η$ and $F\_k[-u]=v^{s\_1}+μ,\;F\_k[-v]=u^{s\_2}+η$ in a bounded domain $Ω$ or in $\mathbb{R}^N$ where $μ$ and $η$ are nonnegative Radon measures, $Δ\_p$ and $F\_k$ are respectively the $p$-Laplacian and the $k$-Hessian operators and $q\_1$, $q\_2$, $s\_1$ and $s\_2$ positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.

math.AP↗
arXiv2018

A maximal function characterization of absolutely continuous measures and Sobolev functions

In this note, we give a new characterisation of Sobolev $W^{1,1}$ functions among $BV$ functions via Hardy-Littlewood maximal function. Exploiting some ideas coming from the proof of this result, we are also able to give a new characterisation of absolutely continuous measures via a weakened version of Hardy-Littlewood maximal function. Finally, we show that the approach adopted in [Crippa and De Lellis, J. Reine Angew. Math. (2008)] and [Jabin, J. Math. Pures Appl. (2010)] to establish existence and uniqueness of regular Lagrangian flows associated to Sobolev vector fields cannot be further extended to the case of $BV$ vector fields.

math.CA↗
arXiv2018

Good-$λ$ and Muckenhoupt-Wheeden type bounds in quasilinear measure datum problems, with applications

Weighted good-$λ$ type inequalities and Muckenhoupt-Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in $\mathbb{R}^n$ in the sense of Reifenberg. The principal operator here is modeled after the $p$-Laplacian, where for the first time singular case $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$ is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

math.AP↗
arXiv2017

Porous medium equation with nonlocal pressure in a bounded domain

We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space domain. We prove existence of weak solutions and suitable a priori bounds and regularity estimates.

math.AP↗
arXiv2017

Gradient estimates for singular quasilinear elliptic equations with measure data

In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=μ,$$ in a bounded domain $Ω\subset\mathbb{R}^{N}$, where $A(x,\nabla u)\nabla u \asymp |\nabla u|^p$, $p\in (1,2-\frac{1}{n}]$ and $μ$ is a Radon measure in $Ω$

math.AP↗
arXiv2017

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to the linear parabolic equations with $\mathbf{A}_q$-weights $$u_t-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and $Ω$ is a Lipchistz domain with a small Lipschitz constant.

math.AP↗
arXiv2015

Wiener criteria for existence of large solutions of nonlinear parabolic equations with absorption in a non-cylindrical domain

We obtain a necessary and a sufficient condition expressed in terms of Wiener type tests involving the parabolic $W\_{q'}^{2,1}$- capacity, where $q'=\frac{q}{q-1}$, for the existence of large solutions to equation $\prt\_tu-Δu+u^q=0$ in non-cylindrical domain, where $q\textgreater{}1$. Also, we provide a sufficient condition associated with equation $\prt\_tu-Δu+e^u-1=0$ . Besides, we apply our results to equation: $\prt\_tu-Δu+a|\nabla u|^p+bu^{q}=0$ for $a,b\textgreater{}0$, $11$.

math.AP↗
arXiv2015

Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+μ,$$ where $q>1$ and $μ$ is a bounded Radon measure.

math.AP↗
arXiv2015

An elliptic semilinear equation with source term and boundary measure data: the supercritical case

We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-Δu = u^q ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*}where $Ω$ is a either a bounded smooth domain or $\mathbb{R}\_+^{N}$, $q\textgreater{}1$ and $σ\in \mathfrak{M}^+(\partialΩ)$ is a nonnegative Radon measure on $\partialΩ$. One of the criteria we obtain is expressed in terms of some Bessel capacities on $\partialΩ$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed terms. \begin{align*} -Δu = |u|^{q\_1-1}u|\nabla u|^{q\_2} ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*} where $q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2$, $σ\in \mathfrak{M}(\partialΩ)$ is a Radon measure on $\partialΩ$.

math.AP↗
arXiv2014

Stability properties for quasilinear parabolic equations with measure data

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where $p>1$, $μ\in\mathcal{M}_{b}(Q)$ and $u_{0}\in L^{1}(Ω).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u\longmapsto\mathcal{A}(u)=$div$(A(x,t,\nabla u))$\textit{. }

math.AP↗
arXiv2014

Evolution equations of p-Laplace type with absorption or source terms and measure data

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We consider problems\textit{ }of the type % \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u\pm\mathcal{G}(u)=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where ${Δ_{p}}$ is the $p$-Laplacian, $μ$ is a bounded Radon measure, $u_{0}\in L^{1}(Ω),$ and $\pm\mathcal{G}(u)$ is an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We prove the existence of renormalized solutions for any measure $μ$ in the subcritical case, and give sufficient conditions for existence in the general case, when $μ$ is good in time and satisfies suitable capacitary conditions.

math.AP↗
arXiv2014

Quasilinear and Hessian type equations with exponential reaction and measure data

We prove existence results concerning equations of the type $-Δ_pu=P(u)+μ$ for $p>1$ and $F_k[-u]=P(u)+μ$ with $1\leq k<\frac{N}{2}$ in a bounded domain $Ω$ or the whole $\mathbb{R}^N$, where $μ$ is a positive Radon measure and $P(u)\sim e^{au^β}$ with $a>0$ and $β\geq 1$. Sufficient conditions for existence are expressed in terms of the fractional maximal potential of $μ$. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of $μ$. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form $u={\bf W}_{α,p}^R[P(u)]+f$ in $\mathbb{R}^N$, where $0

math.AP↗