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Phuoc-Tai Nguyen

Publications and source records attributed to Phuoc-Tai Nguyen.

35 records · Page 2Linked to original sources
arXiv2020

Boundary value problems in Euclidean space for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.

math-ph↗
arXiv2020

Green kernel and Martin kernel of Schrödinger operators with singular potential and application to the B.V.P. for linear elliptic equations

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $K \subset Ω$ be a compact, $C^2$ submanifold in $\mathbb{R}^N$ without boundary, of dimension $k$ with $0\leq k < N-2$. We consider the Schrödinger operator $L_μ= Δ+ μd_K^{-2}$ in $Ω\setminus K$, where $d_K(x) = \text{dist}(x,K)$. The optimal Hardy constant $H=(N-k-2)/2$ is deeply involved in the study of $-L_μ$. When $μ\leq H^2$, we establish sharp, two-sided estimates for Green kernel and Martin kernel of $-L_μ$. We use these estimates to prove the existence, uniqueness and a priori estimates of the solution to the boundary value problem with measures for linear equations associated to $-L_μ$

math.AP↗
arXiv2019

Energy conservation for inhomogeneous incompressible and compressible Euler equations

Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By exploiting a suitable test function, the spatial regularity for the density is only required to be of order $2/3$ in the incompressible case, and of order $1/3$ in the compressible case. When the density is constant, we recover the existing results for classical incompressible Euler equation.

math.AP↗
arXiv2019

Energy equalities for compressible Navier-Stokes equations

The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hold. The method of proof is suitable for the case of periodic as well as homogeneous Dirichlet boundary conditions. In particular, by a careful analysis using the homogeneous Dirichlet boundary condition, no boundary layer assumptions are required when dealing with bounded domains with boundary.

math.AP↗
arXiv2019

Semilinear elliptic equations with Hardy potential and gradient nonlinearity

Let $Ω\subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $δ$ be the distance to $\partial Ω$. We study positive solutions of equation (E) $-L_μu+ g(|\nabla u|) = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2} $, $μ\in (0,\frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ν$. When $g(t)=t^q$ with $q \in (1,2)$, we show that equation (E) admits a critical exponent $q_μ$ (depending only on $N$ and $μ$). In the subcritical case, namely $1<q<q_μ$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $\partial Ω$. In the supercritical case, i.e. $q_μ\leq q<2$, we demonstrate a removability result in terms of Bessel capacities.

math.AP↗
arXiv2018

On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-Δ)^s u &= v^p+μ\quad &&\text{in } Ω\\ (-Δ)^s v &= u^q+ν\quad &&\text{in } Ω\\ u = v &= 0 \quad &&\text{in } Ω^c={\mathbb R}^N \setminus Ω, \end{aligned} \right. \end{equation*} where $Ω$ is a $C^2$ bounded domains in ${\mathbb R}^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $μ,\, ν$ are positive measures in $Ω$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $μ$ and $ν$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0 \leq μ,\, ν\in L^r(Ω)$ for some $r>\frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.

math.AP↗
arXiv2018

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $δ(x)=\text{dist}(x,\partial Ω)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$ -Δu- \fracμ{δ^2}u= u^p +τ\quad \text{in } Ω, \quad \quad u=ν\quad\text{on } \partial Ω, $$ where $μ\in \mathbb{R}$, $p>0$, $τ$ and $ν$ are measures on $Ω$ and $\partial Ω$ respectively. We then establish existence results for the system $$ \left\{ \begin{aligned} &-Δu- \fracμ{δ^2}u = ε\, v^p +τ\quad \text{in } Ω, \\ &-Δv- \fracμ{δ^2}v = ε\, u^{\tilde p}+\tilde τ\quad \text{in } Ω, \\ &u=ν, \quad v= \tilde ν\quad \text{on } \partial Ω, \end{aligned} \right. $$ where $ε=\pm 1$, $p>0$, $\tilde p>0$, $τ$ and $\tilde τ$ are measures on $Ω$, $ν$ and $\tilde ν$ are measures on $\partial Ω$. We also deal with elliptic systems where the nonlinearities are more general.

math.AP↗
arXiv2018

Schrödinger equations with singular potentials: linear and nonlinear boundary value problems

Let $Ω\subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $F \subset \partial Ω$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$. Put $δ_F(x)=dist(x,F)$, $V=δ_F^{-2}$ in $Ω$ and $L_{γV}=Δ+ γV$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $Ω$. We study positive solutions of equations (LE) $-L_{γV} u = 0$ and (NE) $-L_{γV} u+ f(u) = 0$ in $Ω$ when $γ< C_H(V)$ and $f \in C({\mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=\partial Ω$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $Ω$ and establish existence and stability results when the data is concentrated on the set of subcritical points.

math.AP↗
arXiv2018

Boundary singularities of solutions to semilinear fractional equations

We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.

math.AP↗
arXiv2017

On the generalized principal eigenvalue of quasilinear operators

The notions of generalized principal eigenvalue for linear second order elliptic operators in general domains introduced by Berestycki et al. \cite{BNV,BR0,BR3} have become a very useful and important tool in analysis of partial differential equations. In this paper, we extend these notions for quasilinear operator of the form $$\CK_V[u]:=-Δ_p u +Vu^{p-1},\quad\quad u \geq0.$$ This operator is a natural generalization of self-adjoint linear operators. If $Ø$ is a smooth bounded domain, we already proved in \cite{NV} that the generalized principal eigenvalue coincides with the (classical) first eigenvalue of $\CK_V$. Here we investigate the relation between three types of the generalized principal eigenvalue for quasilinear operator on general smooth domain (possibly unbounded), which plays an important role in the investigation of their asymptotic properties. These results form the basis for the study of the simplicity of the generalized principal eigenvalues, the maximum principle and the spectrum of $\CK_V$. We further discuss applications of the notions by providing some examples.

math.AP↗
arXiv2015

Semilinear elliptic equations with Hardy potential and subcritical source term

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^N$ and $δ(x)=\text{dist}\,(x,\partial Ω)$. Assume $μ>0$, $ν$ is a nonnegative finite measure on $\partial Ω$ and $g \in C(Ω\times \mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -Δu - \fracμ{δ^2} u = g(x,u) \text{ in } Ω, \qquad \text{tr}^*(u)=ν. $$ Here $\text{tr}^*(u)$ denotes the normalized boundary trace of $u$ which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<μ< C_H(Ω)$ (the Hardy constant for $Ω$) and provide some qualitative properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we prove that there is a critical value $q^*$ (depending only on $N$, $μ$) for (P) in the sense that if $1<q<q^*$ then (P) admits a solution under a smallness assumption on $ν$, but if $q \geq q^*$ this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where $g$ is subcritical. We also investigate the case where the $g$ is linear or sublinear and give some existence results for (P).

math.AP↗
arXiv2015

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

We study the following quasilinear elliptic equation $$ -Δ_p u + (βΦ(x)-a(x)) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad (P_β) $$ where $p>1$, $Φ\in L^\infty_{loc}(\mathbb{R}^N)$, $a,b \in L^\infty(\mathbb{R}^N)$, $β,b,g \geq 0$. We provide a sharp criterion in terms of generalized principal eigenvalue for existence/nonexistence of positive solutions of ($P_β$) in suitable classes of functions. We derive the uniqueness result of ($P_β$) in those classes. Under additional conditions on $Φ$, we further show that : i) either for every $β\geq 0$ nonexistence phenomenon occurs, ii) or there exists a threshold value $β^*>0$ in the sense that for every $β\in [0,β^*)$ existence and uniqueness phenomenon occurs and for every $β\geq β^*$ nonexistence phenomenon occurs. In the latter case, we study the limits, as $β\to 0$ and $β\toβ^*$, of the sequence of positive solutions of $(P_β)$. Our results are new even in the case $p=2$.

math.AP↗
arXiv2014

Elliptic equations with nonlinear absorption depending on the solution and its gradient

We study positive solutions of equation (E1) $-Δu + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-Δu + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $Ω\subset \mathbb{R}^N$. We obtain a sharp condition on $p$ and $q$ under which, for every positive, finite Borel measure $μ$ on $\partial Ω$, there exists a solution such that $u=μ$ on $\partial Ω$. Furthermore, if the condition mentioned above fails then any isolated point singularity on $\partial Ω$ is removable, namely there is no positive solution that vanishes on $\partial Ω$ everywhere except at one point. With respect to (E2) we also prove uniqueness and discuss solutions that blow-up on a compact subset of $\partial Ω$. In both cases we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when $p=0$ but not the general case.

math.AP↗
arXiv2014

Moderate solutions of semilinear elliptic equations with Hardy potential

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_μu+ u^q = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2}$, $0<μ$, $q>1$ and $δ(x)=\mathrm{dist}\,(x,\partialΩ)$. A positive solution of (E) is moderate if it is dominated by an $L_μ$-harmonic function. If $μ<C_H(Ω)$ (the Hardy constant for $Ω$) every positive $L_μ$- harmonic functions can be represented in terms of a finite measure on $\partialΩ$ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, $1<q<q_{μ,c}$. (The critical value depends only on $N$ and $μ$.) For $q\geq q_{μ,c}$ there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator $L_μ$. These results form the basis for the study of the nonlinear problem.

math.AP↗
arXiv2013

Parabolic equations with exponential nonlinearity and measure data

Let $Ω$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - Δu \pm g(u) &= μ\quad &\text{in } Q_T:=Ω\times (0,T) \\ \phantom{------,} u&=0 &\text{on } \partial Ω\times (0,T)\\ \phantom{----,} u(.,0) &=ω&\text{in } Ω. \end{array} \right. \end{equation} where $μ$ and $ω$ are bounded Radon measures in $Q_T$ and $Ω$ respectively and $g(u) \sim e^{a |u|^q} $ with $a>0$ and $q \geq 1$. We provide a sufficient condition in terms of fractional maximal potentials of $μ$ and $ω$ for solving (P).

math.AP↗
arXiv2013

Positive solutions of quasilinear elliptic equations with subquadratic growth in the gradient

We study positive solutions of equation (E) $-Δu + u^p|\nabla u|^q= 0$ ($0 1$) and other related equations in a smooth bounded domain $Ω\subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for every positive, finite Borel measure $μ$ on $\partial Ω$, there exists a solution of (E) such that $u=μ$ on $\partial Ω$. Furthermore, if $N(p+q-1)\geq p+1$ then an isolated point singularity on $\partial Ω$ is removable. In particular there is no solution with boundary data $δ_y$ (=Dirac measure at a point $y\in \partial Ω$). Finally we obtain a classification of positive solutions with an isolated boundary singularity.

math.AP↗