On the existence of bounded solutions for a nonlinear elliptic system

This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain \Omega\subset\RR^n, where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. We prove that, under appropriate conditions on the parameters ($a,b,p,q,m,n$), any non-negative solution $(u,v)$ of the system is bounded by a constant independent of $(u,v)$. Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case $m=1$ was considered by Souplet in \cite{PS}. Our paper generalize to $m\ge 1$ the results of that paper

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n+2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially selfadjoint, but it does satisfy a weaker property that we call here quasi essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.Comment: 36 pages. Final version to appear in Classical and Quantum Gravit

Proyector de Calderón asociado a un operador elíptico con coeficientes Lipschitz

El proyector sobre los datos de Cauchy, llamado habitualmente proyector de Calderón, es una de las herramientas más importante para tratar problemas elípticos de borde. Es un operador pseudodiferencial de orden 0 [1], [10], si los coeficientes del operador elíptico a tratar son C∞ y el borde de la región en la que está definido es también C∞. Permite definir problemas elípticos con condiciones de borde más generales que las definidas por las condiciones de Lopatinsky. Para analizar sus propiedades cuando el borde de las regiones es menos regular (Lipschitz), resultó necesario avanzar en el estudio de la integral de Cauchy en curvas Lipschitz [2], [4]. Los resultados obtenidos permitieron demostrar la continuidad de los operadores involucrados en la definición del proyector de Calderón para el Laplaciano, [7], [20], [6], entre otros. El objetivo de este trabajo es construir y analizar este proyector para operadores elípticos del tipo L = - div (A∇), cuando los coeficientes de la matriz A son Lipschitz, y en este caso se obtienen las mismas propiedades que en el caso del Laplaciano.Tesis digitalizada en SEDICI gracias a la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP).Facultad de Ciencias Exacta

Proyector de Calderón asociado a un operador elíptico con coeficientes Lipschitz

El proyector sobre los datos de Cauchy, llamado habitualmente proyector de Calderón, es una de las herramientas más importante para tratar problemas elípticos de borde. Es un operador pseudodiferencial de orden 0 [1], [10], si los coeficientes del operador elíptico a tratar son C∞ y el borde de la región en la que está definido es también C∞. Permite definir problemas elípticos con condiciones de borde más generales que las definidas por las condiciones de Lopatinsky. Para analizar sus propiedades cuando el borde de las regiones es menos regular (Lipschitz), resultó necesario avanzar en el estudio de la integral de Cauchy en curvas Lipschitz [2], [4]. Los resultados obtenidos permitieron demostrar la continuidad de los operadores involucrados en la definición del proyector de Calderón para el Laplaciano, [7], [20], [6], entre otros. El objetivo de este trabajo es construir y analizar este proyector para operadores elípticos del tipo L = - div (A∇), cuando los coeficientes de la matriz A son Lipschitz, y en este caso se obtienen las mismas propiedades que en el caso del Laplaciano.Tesis digitalizada en SEDICI gracias a la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP).Facultad de Ciencias Exacta

On the existence of bounded solutions for a nonlinear elliptic system

This work deals with the system (−Δ)mu = a(x) vp, (−Δ)mv = b(x) uq with Dirichlet boundary condition in a domain Ω⊂Rn , where Ω is a ball if n ≥ 3 or a smooth perturbation of a ball when n = 2. We prove that, under appropriate conditions on the parameters (a, b, p, q, m, n), any nonnegative solution (u, v) of the system is bounded by a constant independent of (u, v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m = 1 was considered by Souplet (Nonlinear Partial Differ Equ Appl 20:464–479, 2004). Our paper generalize to m ≥ 1 the results of that paper.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

On the existence of bounded solutions for a nonlinear elliptic system

This work deals with the system (−Δ)mu = a(x) vp, (−Δ)mv = b(x) uq with Dirichlet boundary condition in a domain Ω⊂Rn , where Ω is a ball if n ≥ 3 or a smooth perturbation of a ball when n = 2. We prove that, under appropriate conditions on the parameters (a, b, p, q, m, n), any nonnegative solution (u, v) of the system is bounded by a constant independent of (u, v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m = 1 was considered by Souplet (Nonlinear Partial Differ Equ Appl 20:464–479, 2004). Our paper generalize to m ≥ 1 the results of that paper.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

A discussion on the natural domain of Radial Toeplitz operators in Segal-Bargmann space

We discuss the domain of Toeplitz operators with radial symbols in the Segal-Bargmann space: we point out and correct missleading staments in previous works, establishing the conditions under which a given Toeplitz operator is unitarily equivalent to a diagonal operator in the space l^2(C) of square summable complex sequences.Facultad de Ciencias ExactasInstituto de Física La Plat

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n + 2)-dimensional static and spherically symmetric spacetimes. They are related to the properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially self-adjoint, but it does satisfy a weaker property that we call here quasi-essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.Instituto de Física La Plat