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Laurent Veron

Publications and source records attributed to Laurent Veron.

At least 55 records · Page 3Linked to original sources
arXiv2012

Capacitary estimates of solutions of semilinear parabolic equations

We prove that any positive solution of $ \prt_tu-Δu+u^q=0$ ($q>1$) in $\BBR^N\ti(0,\infty)$ with initial trace $(F,0)$, where $F$ is a closed subset of $\BBR^N$ can be estimated from above and below and up to two universal multiplicative constants, by a series involving the Bessel capacity $C_{2/q,q'}$. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of $u(x,t)$ when $t\downarrow 0$, by using the "density" of $F$ expressed in terms of the $C_{2/q,q'}$-capacity.

math.AP↗
arXiv2012

Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations

We study the boundary value problem with measures for (E1) $-\Gd u+g(|\nabla u|)=0$ in a bounded domain $\Gw$ in $\BBR^N$, satisfying (E2) $ u=\gm$ on $\prt\Gw$ and prove that if $g\in L^1(1,\infty;t^{-(2N+1)/N}dt)$ is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When $g(r)\geq r^q$ with $q>1$ we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1

math.AP↗
arXiv2011

Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case

We study the generalized boundary value problem for nonnegative solutions of of $-Δu+g(u)=0$ in a bounded Lipschitz domain $Ω$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $Ω$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $Ω$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $Ω$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.

math.AP↗
arXiv2011

A note on the equation $-Δu+e^{u}-1=0$

If $Ω$ is a bounded domain in $\mathbb R^N$, we study conditions on a Radon measure $μ$ on $\partialΩ$ for solving the equation $-Δu+e^{u}-1=0$ in $Ω$ with $u=μ$ on $\partialΩ$. The conditions are expressed in terms of nonlinear capacities.

math.AP↗
arXiv2011

Separable solutions of quasilinear Lane-Emden equations

For $0 < p-1 < q$ and $\ge=\pm 1$, we prove the existence of solutions of $-\Gd_pu=\ge u^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\prt C_S$, under the form $u(x)=|x|^\gb\gw(\frac{x}{|x|})$. The problem reduces to a quasilinear elliptic equation on $S$ and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.

math.AP↗
arXiv2011

Propagation of Singularities of Nonlinear Heat Flow in Fissured Media

In this paper we investigate the propagation of singularities in a nonlinear parabolic equation with strong absorption when the absorption potential is strongly degenerate following some curve in the $(x,t)$ space. As a very simplified model, we assume that the heat conduction is constant but the absorption of the media depends stronly of the characteristic of the media. More precisely we suppose that the temperature $u$ is governed by the following equation \label{I-1} \partial_{t}u-Δu+h(x,t)u^p=0\quad \text{in}Q_{T}:=R^N\times (0,T) where $p>1$ and $h\in C(\bar Q_{T})$. We suppose that $h(x,t)>0$ except when $(x,t)$ belongs to some space-time curve.

math.AP↗
arXiv2011

A note on maximal solutions of nonlinear parabolic equations with absorption

If $Ω$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-Δu+f(u)=0$ in $Q_\infty^Ω:=Ω\times (0,\infty)$, $u=\infty$ on the parabolic boundary $\partial_{p}Q$. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in $Ω$.

math.AP↗
arXiv2011

Initial trace of positive solutions of a class of degenerate heat equation with absorption

We study the initial value problem with unbounded nonnegative functions or measures for the equation $ \prt_tu-\Gd_p u+f(u)=0$ in $\BBR^N\ti(0,\infty)$ where $p>1$, $\Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u)$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $k\gd_0$ and we study their limit when $k\to\infty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)=\int_0^s f(\gs)d\gs$. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case $f(u)=u^\ga \ln^\gb(u+1)$, where $\ga>0$ and $\gb\geq 0$.

math.AP↗
arXiv2010

Local and global properties of solutions of heat equation with superlinear absorption

We study the limit, when $k\to\infty$ of solutions of $u_t-Δu+f(u)=0$ in $R^N\times(0,\infty)$ with initial data $k\gd$, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible behaviour according $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We emphasize the case where f(u)=u((\ln u+1))^α. We use these results for giving a general result on the existence of the initial trace and some non-uniqueness results for regular solutions with unbounded initial data.

math.AP↗
arXiv2010

Boundary value problems with measures for elliptic equations with singular potentials

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-Δu+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure $\gm$ on $\prt\Gw$ so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In the appendix A. Ancona solves a question raised by M. Marcus and L. Véron concerning the vanishing set of the Poisson kernel of $L_V$ for an important class of potentials $V$.

math.AP↗
arXiv2010

Boundary Value Problems with Measures for Elliptic Equations with Singular Potentials

We study the boundary value problem with Radon measures for nonnegative solutions of $-Δu+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure $\gm$ on $\prt\Gw$ so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions.

math.AP↗
arXiv2009

Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains

We study the generalized boundary value problem for nonnegative solutions of $-Δu+g(u)=0$ in a bounded Lipschitz domain $\Gw$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Gw$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Gw$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Gw$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace. We obtain sharp results involving Besov spaces with negative index on k-dimensional edges and apply our results to the characterization of removable sets and good measures on the boundary of a polyhedron.

math.AP↗
arXiv2009

Isolated Boundary Singularities of Semilinear Elliptic Equations

Given a smooth domain $Ω\subset\RR^N$ such that $0 \in \partialΩ$ and given a nonnegative smooth function $ζ$ on $\partialΩ$, we study the behavior near 0 of positive solutions of $-Δu=u^q$ in $Ω$ such that $u = ζ$ on $\partialΩ\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.

math.AP↗
arXiv2008

Maximal solutions for $-Δu+u^q=0$ in open or finely open sets

We study the existence and uniqueness of new classes of solutions of the superlinear equation $-Δu+u^q=0$ (q>1) in a domain of R^N or in a finely open set for the topology associated to the Bessel capacity C_{2,q'}. Condition of existence or uniqueness of solutions with boundary blow-up are obtained generalizing the results of Dhersin-Le Gall and of Labutin.

math.AP↗
arXiv2008

Solutions of some nonlinear parabolic equations with initial blow-up

We study the existence and uniqueness of solutions of $\partial_tu-Δu+u^q=0$ ($q>1$) in $Ω\times (0,\infty)$ where $Ω\subset\mathbb R^N$ is a domain with a compact boundary, subject to the conditions $u=f\geq 0$ on $\partialΩ\times (0,\infty)$ and the initial condition $\lim_{t\to 0}u(x,t)=\infty$. By means of Brezis' theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when $f=0$, whatever is the regularity of the boundary of the domain. When $\partialΩ$ satisfies the parabolic Wiener criterion and $f$ is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at $t=0$.

math.AP↗