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Miltiadis Paschalis

Publications and source records attributed to Miltiadis Paschalis.

6 recordsLinked to original sources
arXiv2026

A topology-changing variational framework for the Einstein-Hilbert functional

Motivated by recent developments in the theory of gravitation, we revisit the idea of topological variations, originally introduced by Wheeler and Hawking, from a rigorous perspective. Starting from a localized version of the Einstein-Hilbert variational principle, we encode the key aspects of the variational procedure in the form of a topology on a suitable space of Sobolev variational configurations, which is the final topology generated by the admissible variational maps. This framework naturally lends itself to generalization, and we rigorously introduce two distinct types of topological variations, corresponding to the infinitesimal addition of disconnected components and to infinitesimal surgeries, both motivated by related physical concepts. Using tools from the theory of Sobolev spaces and precise asymptotics, we establish dimensional obstructions for the continuity and differentiability of the Einstein-Hilbert action with respect to these variations, and show that in the extended variational framework the action does not admit critical points in dimension $n=4$, while higher dimensions are free of this problem. We also discuss the deeper geometric issue of scalar curvature blow-up of degenerating metrics within the context of our framework, and finally demonstrate the non-trivial effect of added higher order curvature terms on the critical dimension.

math.DG↗
arXiv2024

Hardy inequalities and uncertainty principles in the presence of a black hole

In this paper we establish Hardy and Heisenberg uncertainty-type inequalities for the exterior of a Schwarzschild black hole. The weights that appear in both inequalities are tailored to fit the geometry, and can both be compared to the related Riemannian distance from the event horizon to yield inequalities for that distance. Moreover, in both cases the classic Euclidean inequalities with a point singularity can be recovered in the limit where one stands "far enough" from the black hole, as expected from the asymptotic flatness of the metric.

math.AP↗
arXiv2024

Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities

Let $Ω\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $Σ\subset\partialΩ$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $Σ$ by $d_Σ$. In this paper, we study positive solutions of the equation $(*)\, -Δu -μu/d_Σ^2 = g(u,|\nabla u|)$ in $Ω$, where $μ\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g:\mathbb{R}\times\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data $u=ν$ in two main cases, provided that the total mass of $ν$ is small. In the first case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity $g(u,|\nabla u|) = |u|^p|\nabla u|^q$, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that $ν$ is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of $Σ$, or under other equivalent conditions.

math.AP↗
arXiv2021

Geometric Hardy inequalities via integration on flows

We introduce a geometric approach of integral curves for functional inequalities involving directional derivatives in the general context of differentiable manifolds that are equipped with a volume form. We focus on Hardy-type inequalities and the explicit optimal Hardy potentials that are induced by this method. We then apply the method to retrieve some known inequalities and establish some new ones.

math.AP↗
arXiv2021

Finsler-Rellich inequalities involving the distance to the boundary

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of half-spaces we obtain the sharp constant while for a general convex domain we obtain estimates that are better than those obtained by comparison with the polyharmonic operator.

math.AP↗