arXiv subjects

Antonio R. G. Garcia

Publications and source records attributed to Antonio R. G. Garcia.

4 recordsLinked to original sources

arXiv2026

Generalized junction conditions for discontinuous metrics

In this work, the Darmois-Israel junction formalism is extended to the case of discontinuous metrics within the framework of Colombeau algebras of generalized functions. This formulation provides a mathematically consistent treatment of nonlinear operations involving singular quantities, such as products and derivatives of distributions. By relaxing the usual continuity condition on the metric, the generalized junction conditions naturally include higher-order singular terms in the curvature and in the surface energy-momentum tensor. These additional contributions represent new geometric degrees of freedom associated with genuine discontinuities in the space-time geometry. The resulting formalism recovers the traditional Darmois-Israel conditions as a limiting case, while offering a coherent extension applicable to geometric boundaries and abrupt transitions in space-time.

gr-qc↗

arXiv2023

An Approach to the Primordial Universe Using Colombeau's Simplified Algebra

The proposal "no boundary" of physicists Hartle and Hawking seeks to build a satisfactory model of the early Universe, in a way that avoids the singularity "Big Bang" of the beginning of the Universe. As a consequence of this proposal, the concept of metric signature change arises, which is approached in different ways in the literature. Here, we reinterpret the Mansouri-Nozari approach, which modifies the FLRW metric, and uses the formalism of Colombeau's Algebras, to develop its equations. In addition, we write the function that changes the sign in terms of redshift. Finally, we developed Friedmann's equations, of the modified metric, as well as the equation of state and other relevant equations in Cosmology.

gr-qc↗

arXiv2017

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized pointvalues, Colombeau's differential algebra, holomorphic and analytic functions. We show that the Embedding Theorem and the Open Mapping Theorem hold in this framework. Moreover, we study some applications in differential equations.

math.CA↗

arXiv2017

Dissipative property for a class of non local evolution equations

In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(βK(f\circ u)(x,t)+βh), ~x \inΩ, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminusΩ, ~t\in[0,\infty[;\\ u(x,0)=u_0(x),~x\in\mathbb{R}^N, \end{cases} \] where $Ω$ is a smooth bounded domain in $\mathbb{R}^N, ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying certain growing condition and $K$ is an integral operator with symmetric kernel, $ Kv(x)=\int_{\mathbb{R}^{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.

math.DS↗