Semilinear Elliptic Equations and Fixed Points

In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with C\sp{1,1} boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.Comment: 4 page

A note on asymptotically monotone basic sequences and well-separated sets

We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm \vertiii{\cdot} on $X$ such that the unit sphere (S_{(X, \vertiii{\cdot})}) contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.Comment: 9 pages, comments are more than welcome. Corrected versio

On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.Comment: A subtle mistake in the proof of main result makes obsolete the pape

The fixed point property for a class of nonexpansive maps in L\sp\infty(\Omega,\Sigma,\mu)

For a finite and positive measure space $(\Omega,\Sigma,\mu)$ and any weakly compact convex subset of L\sp\infty(\Omega,\Sigma,mu), a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space $C(\Omega)$. An illustrative example is given.Comment: 4 page

Measures of Weak Compactness and Fixed Point Theory

In this paper, we study a class of Banach spaces, called \phi-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.Comment: 9 page

Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds

We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.Comment: 7 Pages, written in Latex. Revised version of the paper "Curvature free lower bounds for the first eigenvalue of normal geodesic balls

A Topological and Geometric Approach to Fixed Points Results for Sum of Operators and Applications

In the present paper we establish a fixed point result of Krasnoselskii type for the sum $A+B$, where $A$ and $B$ are continuous maps acting on locally convex spaces. Our results extend previous ones. We apply such results to obtain strong solutions for some quasi-linear elliptic equations with lack of compactness. We also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a Banach space. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in L\sp p(E).Comment: 24 page

A note on the first eigenvalue of spherically symmetric manifolds

We give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifolds. These lower and upper bounds are $C^{0}$-dependent on the metric coefficients. It gives better lower bounds for the first eigenvalue of spherical caps than those from Betz-Camera-Gzyl.Comment: 6 pages. We apply Barta's Theorem to give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifold

Boundary Conditions and Vacuum Fluctuations in AdS4\mathrm{AdS}_4

Initial conditions given on a spacelike, static slice of a non-globally hyperbolic spacetime may not define the fates of classical and quantum fields uniquely. Such lack of global hyperbolicity is a well-known property of the anti-de Sitter solution and led many authors to question how is it possible to develop a quantum field theory on this spacetime. Wald and Ishibashi took a step towards the healing of that causal issue when considering the propagation of scalar fields on AdS. They proposed a systematic procedure to obtain a physically consistent dynamical evolution. Their prescription relies on determining the self-adjoint extensions of the spatial component of the differential wave operator. Such a requirement leads to the imposition of a specific set of boundary conditions at infinity. We employ their scheme in the particular case of the four-dimensional AdS spacetime and compute the expectation values of the field squared and the energy-momentum tensor, which will then bear the effects of those boundary conditions. We are not aware of any laws of nature constraining us to prescribe the same boundary conditions to all modes of the wave equation. Thus, we formulate a physical setup in which one of those modes satisfy a Robin boundary condition, while all others satisfy the Dirichlet condition. Due to our unusual settings, the resulting contributions to the fluctuations of the expectation values will not respect AdS invariance. As a consequence, a back-reaction procedure would yield a non-maximally symmetric spacetime. Furthermore, we verify the violation of weak energy condition as a direct consequence of our prescription for dynamics.Comment: 23 pages, 4 figure

Vacuum Fluctuations and Boundary Conditions in a Global Monopole

We study the vacuum fluctuations of a massless scalar field $\hat{\Psi}$ on the background of a global monopole. Due to the nontrivial topology of the global monopole spacetime, characterized by a solid deficit angle parametrized by $\eta^2$, we expect that \left_{\text{ren}} and \left_{\text{ren}} are nonzero and proportional to $\eta^2$, so that they annul in the Minkowski limit $\eta\to0$. However, due to the naked singularity at the monopole core, the evolution of the scalar field is not unique. In fact, they are in one to one correspondence with the boundary conditions which turn into self-adjoint the spatial part of the wave operator. We show that only Dirichlet boundary condition corresponds to our expectations and gives zero contribution to the vacuum fluctuations in Minkowski limit. All other boundary conditions give nonzero contributions in this limit due to the nontrivial interaction between the field and the singularity.Comment: 7 pages, submitted to PR