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Laurent Veron

Publications and source records attributed to Laurent Veron.

At least 73 records · Page 4Linked to original sources
arXiv2008

On uniqueness of large solutions of nonlinear parabolic equations in nonsmooth domains

We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-Δu+u^q=0$ ($q>1$) in $Ω\times (0,\infty)$, $u=\infty$ on $\partialΩ\times (0,\infty)$ and $u(.,0)\in L^1(Ω)$, when $Ω$ is a bounded domain in $\mathbb R^N$. We construct a maximal solution, prove that this maximal solution is a large solution whenever $q<N/(N-2)$ and it is unique if $\partialΩ=\partial\barΩ^c$. If $\partialΩ$ has the local graph property, we prove that there exists at most one solution to problem (P)

math.AP↗
arXiv2008

Reduced measures associated to parabolic problems

We study the existence and the properties of the reduced measures for the parabolic equations $\partial_tu-Δu+g(u)=0$ in $Ω\times (0,\infty)$ subject to the conditions ($P$): $u=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=μ$ and ($P'$): $u=μ'$ on $\partialΩ\times (0,\infty)$, $u(x,0)=0$ where $μ$ and $μ'$ are positive Radon measures and $g$ a continuous nondecreasing function

math.AP↗
arXiv2008

Maximal solutions of equation u = uq in arbitrary domains

We prove bilateral capacitary estimates for the maximal solution $U_F$ of $-Δu+u^q=0$ in the complement of an arbitrary closed set $F\subset\mathbb R^N$, involving the Bessel capacity $C_{2,q'}$, for $q$ in the supercritical range $q\geq q_{c}:=N/(N-2)$. We derive a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that $U_F(x)\to\infty$ as $x\to y$ for given $y\in\prt F$. Finally we prove a general uniqueness result for large solutions.

math.AP↗
arXiv2008

The balance between diffusion and absorption in semilinear parabolic equations

Let $h:[0,\infty)\mapsto [0,\infty)$ be continuous and nondecreasing, $h(t)>0$ if $t>0$, and $m,q$ be positive real numbers. We investigate the behavior when $k\to\infty$ of the fundamental solutions $u=u_{k}$ of $\prt_{t} u-Δu^m+h(t)u^q=0$ in $Ω\ti (0,T)$ satisfying $u_{k}(x,0)=kδ_0$. The main question is wether the limit is still a solution of the above equation with an isolated singularity at $(0,0)$, or a solution of the associated ordinary differential equation $ u'+h(t)u^q=0$ which blows-up at $t=0$.

math.AP↗
arXiv2008

Diffusion versus absorption in semilinear parabolic equations

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-Δu+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=kδ_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to $\int_{0}^1\sqrt{\gw(t)}t^{-1}dt<\infty$, the limit function $u_{\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\gw(t)\equiv 1$, $u_{\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\prt_{t}u-\Gd u+ h(t)e^{u}=0$.

math.AP↗
arXiv2008

Capacitary representations of positive solutions of semilinear parabolic equations

We give a global bilateral estimate on the maximal solution $\bar u_F$ of $ \prt_tu-Δu+u^q=0$ in $\BBR^N\times (0,\infty)$, $q>1$, $N\geq 1$, which vanishes at $t=0 $ on the complement of a closed subset $F\subset \BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q'}$. We deduce from this estimate that $\bar {u}_F$ is $σ$-moderate in Dynkin's sense.

math.AP↗
arXiv2008

Boundary singularities of solutions of N-harmonic equations with absorption

We study the boundary behaviour of solutions $u$ of $-Δ_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $Ω\subset\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if $q\geq 2N-1$ such a $u$ is identically zero, while, if $N-1<q<2N-1$, $u$ inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.

math.AP↗
arXiv2008

Maximal Solutions of Semilinear Elliptic Equations with Locally Integrable Forcing Term

We study the existence of a maximal solution of $-\Gd u+g(u)=f(x)$ in a domain $\Gw\subset \BBR^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$ and that $g$ is nondecreasing, $g(0)\geq 0$ and $g$ satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classical $C_{1,2}$ Wiener criterion then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition we discuss the question of uniqueness of large solutions.

math.AP↗
arXiv2008

Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations

If $h$ is a nondecreasing real valued function and $0\leq q\leq 2$, we analyse the boundary behaviour of the gradient of any solution $u$ of $-Δu+h(u)+\abs {\nabla u}^q=f$ in a smooth N-dimensional domain $Ω$ with the condition that $u$ tends to infinity when $x$ tends to $\partialΩ$. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to $\partialΩ$. Motivated by the blow--up argument in our proof, we also give in Appendix a symmetry result for some related problems in the half space.

math.AP↗
arXiv2007

Boundary singularities of positive solutions of some nonlinear elliptic equations

We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and existence or non existence and uniqueness results for solutions of some nonlinear elliptic equations on the half unit sphere.

math.AP↗
arXiv2006

The precise boundary trace of positive solutions of the equation $Δu=u^q$ in the supercritical case

We construct the precise boundary trace of positive solutions of $Δu=u^q$ in a smooth bounded domain in $R^N$, for $q$ in the super-critical range $q\geq (N+1)/(N-1)$. The construction is performed in the framework of the fine topology associated with the Bessel capacity $C_{2/q,q'}$ on the boundary of the domain. We prove that the boundary trace is a Borel measure (in general unbounded), which is outer regular relative to this capacity. We provide a necessary and sufficient condition for such measures to be the boundary trace of a positive solution and prove that the corresponding generalized boundary value problem is well-posed in the class of $σ$-moderate solutions.

math.AP↗