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Nguyen Cong Phuc

Publications and source records attributed to Nguyen Cong Phuc.

At least 19 recordsLinked to original sources
arXiv2025

Some capacitary strong type inequalities and related function spaces

We verify a conjecture of D. R. Adams on a capacitary strong type inequality that generalizes the classical capacitary strong type inequality of V. G. Maz'ya. As a result, we characterize related function spaces as Köthe duals to a class of Sobolev multiplier type spaces. Moreover, using tools from nonlinear potential theory, weighted norm inequalities, and Banach function space theory, we show that these spaces are also isomorphic to more concrete spaces that are easy to use and fit in well with the modern theory of function spaces of harmonic analysis.

math.CA↗
arXiv2022

Uniqueness of entire solutions to quasilinear equations of p-Laplace type

We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ is a nonnegative locally finite measure in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$) on $\mathcal{A}(x, ξ)$ ($x, ξ\in \mathbb{R}^n$); the model case $\mathcal{A}(x, ξ)=ξ| ξ|^{p-2}$ corresponds to the $p$-Laplace operator $Δ_p$ on $\mathbb{R}^n$. Our main results establish uniqueness of solutions to a similar problem, \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σu^q +μ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} in the sub-natural growth case $0<q<p-1$, where $μ, σ$ are nonnegative locally finite measures in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and $\mathcal{A}(x, ξ)$ satisfies an additional homogeneity condition, which holds in particular for the $p$-Laplace operator.

math.AP↗
arXiv2022

Universal potential estimates for $1<p\leq 2-\frac{1}{n}$

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla u))=μ \end{equation*} in a bounded open subset $Ω$ of $\mathbb{R}^n$, $n\geq 2$, with a finite signed measure $μ$ in $Ω$. The operator $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian $Δ_p u:= {\rm div}\, (|\nabla u|^{p-2}\nabla u)$, where the nonlinearity $A(x, ξ)$ ($x, ξ\in \mathbb{R}^n$) is assumed to satisfy natural growth and monotonicity conditions of order $p$, as well as certain additional regularity conditions in the $x$-variable.

math.AP↗
arXiv2022

A comparison estimate for singular $p$-Laplace equations and its consequences

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincaré type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

math.AP↗
arXiv2021

BMO solutions to quasilinear equations of $p$-Laplace type

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-Δ_{p} u = μ$ in $\mathbb{R}^n$, $u\ge 0$, where $μ$ is a locally finite Radon measure, and $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). We also characterize BMO solutions to equations $-Δ_{p} u = σu^{q} + μ$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $μ$ and $σ$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in place of $Δ_{p}$.

math.AP↗
arXiv2021

On a capacitary strong type inequality and related capacitary estimates

We establish a Maz'ya type capacitary inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy-Littlewood maximal function in a sublinear setting.

math.CA↗
arXiv2020

Characterizations of predual spaces to a class of Sobolev multiplier type spaces

We characterize preduals and Köthe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. We make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results.

math.AP↗
arXiv2020

Quasilinear Riccati type equations with oscillatory and singular data

We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + σ\quad \text{in} ~Ω, \\ u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} with a distributional or measure datum $σ$. Here ${\rm div}\,\mathcal{A}(x, \nabla u)$ is a quasilinear elliptic operator modeled after the $p$-Laplacian ($p>1$), and $Ω$ is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that $p>1$ and $q>p$. For measure data, we assume that they are compactly supported in $Ω$, $p>\frac{3n-2}{2n-1}$, and $q$ is in the sub-linear range $p-1<q<1$. We also assume more regularity conditions on $\mathcal{A}$ and on $\partialΩ$ in this case.

math.AP↗
arXiv2020

Existence and regularity estimates for quasilinear equations with measure data: the case $1<p\leq \frac{3n-2}{2n-1}$

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= δ\, |\nabla u|^q +μ$ in a bounded main $\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $δ=0$ or $δ=1$, $μ$ is a finite signed Radon measure in $Ω$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $δ=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.

math.AP↗
arXiv2019

Pointwise gradient estimates for a class of singular quasilinear equation with measure data

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=μ$ in a bounded and possibly nonsmooth domain $Ω$ in $\mathbb{R}^n$. Here $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian. Our results extend earlier known results to the singular case in which $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$.

math.AP↗
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↗
arXiv2018

Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $Ω$. The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz-Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $W^{1,p}_{0}(Ω)$ for some $p>1$ but not necessarily in $W^{1,2}_{0}(Ω)$) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

math.AP↗
arXiv2018

Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

We obtain necessary and sufficient conditions with sharp constants on the distribution $σ$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form $-Δ_p u = |\nabla u|^p + σ$ in a bounded open set $Ω\subset \mathbb{R}^n$. Here $Δ_p$, $p>1$, is the standard $p$-Laplacian operator defined by $Δ_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(Ω)$ such that $e^{μ u}\in W^{1,p}_0(Ω)$ for some $μ>0$ and the inequality \begin{equation*} \int_Ω |φ|^p |\nabla u|^p dx \leq A \int_Ω|\nabla φ|^p dx \end{equation*} holds for all $φ\in C_c^\infty(Ω)$ with some constant $A>0$. This is a natural class of solutions at least when the distribution $σ$ is nonnegative. The study of $-Δ_p u = |\nabla u|^p + σ$ is applied to show the existence of globally finite energy solutions to the quasilinear equation of Schrödinger type $-Δ_p v = σ\, v^{p-1}$, $v\geq 0$ in $Ω$, and $v=1$ on $\partialΩ$, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$.

math.AP↗
arXiv2018

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form $-Δ_p u = |\nabla u|^p + σ$ in a bounded domain $Ω\subset \mathbb{R}^n$. Here $Δ_p$, $p>1$, is the standard $p$-Laplacian operator defined by $Δ_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$, and the datum $σ$ is a signed distribution in $Ω$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(Ω)$ such that $|\nabla u|\in M(W^{1,p}(Ω)\rightarrow L^p(Ω))$, a space pointwise Sobolev multipliers consisting of functions $f\in L^{p}(Ω)$ such that \begin{equation*} \int_Ω |f|^{p} |φ|^p dx \leq C \int_Ω (|\nabla φ|^p + |φ|^p) dx \quad \forall φ\in C^\infty(Ω), \end{equation*} for some $C>0$. This is a natural class of solutions at least when the distribution $σ$ is nonnegative and compactly supported in $Ω$. We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write $σ={\rm div}\, F$ for a vector field $F$ such that $|F|^{\frac{1}{p-1}}\in M(W^{1,p}(Ω)\rightarrow L^p(Ω))$. As an important application, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$, we obtain an existence result for the quasilinear equation of Schrödinger type $-Δ_p v = σ\, v^{p-1}$, $v\geq 0$ in $Ω$, and $v=1$ on $\partialΩ$, which is interesting in its own right.

math.AP↗
arXiv2017

Local energy bounds and $ε$-regularity criteria for the 3D Navier-Stokes system

The system of three dimensional Navier-Stokes equations is considered. We obtain some new local energy bounds that enable us to improve several $ε$-regularity criteria. They key idea here is to view the `head pressure' as a signed distribution belonging to certain fractional Sobolev space of negative order. This allows us to capture the oscillation of the pressure in our criteria.

math.AP↗
arXiv2015

Characterizations of signed measures in the dual of $BV$ and related isometric isomorphisms

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded open set $Ω$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(Ω)^*$. One of the goals of this paper is to make precise the definition of $BV_0(Ω)$, which is the space of functions of bounded variation with zero trace on the boundary of $Ω$. We show that the measures in $BV_0(Ω)^*$ coincide with the measures in $W^{1,1}_0(Ω)^*$. Finally, the class of finite measures in $BV(Ω)^*$ is also characterized.

math.AP↗