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Moshe Marcus

Publications and source records attributed to Moshe Marcus.

31 records · Page 2Linked to original sources
arXiv2012

Fading absorption in non-linear elliptic equations

We study the equation $-Δu+h(x)|u|^{q-1}u=0$, $q>1$, in $R^N_+=R^{N-1}\ti R_+$ where $h\in C(\bar{R^N_+})$, $h\geq 0$. Let $(x_1,..., x_N)$ be a coordinate system such that $R^N_+=[x_N>0]$ and denote a point $x\in \RN$ by $(x',x_N)$. Assume that $h(x', x_N)>0$ when $x'\neq 0$ but $h(x',x_N)\to 0$ as $|x'|\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.

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

Complete classification of the positive solutions of $-Δu + u^q=0$

We study the equation $-Δu+u^q=0$, $q>1$, in a bounded $C^2$ domain $Ω\subset R^N$. A positive solution of the equation is moderate if it is dominated by a harmonic function and $σ$-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the sub-critical case, $1 2$. In this paper we prove that, for all $q\ge q_c$, every positive solution is $σ$-moderate. We use purely analytic techniques which apply to the full super-critical range. The main tools come from linear and non-linear potential theory. Combined with previous results, this establishes a 1-1 correspondence between positive solutions and their boundary traces in the sense of [35].

math.AP↗
arXiv2011

Remarks on nonlinear equations with measures

We study the Dirichlet boundary value problem for equations with absorption of the form $-Δu+g\circ u=μ$ in a bounded domain $Ω\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $μ$ is a finite measure. We also discuss the notion of solution when the measure $μ$ is positive and blows up on a compact subset of $\Gw$.

math.CA↗
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↗
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

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

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

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↗
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↗