arXiv subjects

Luis J. Alías

Publications and source records attributed to Luis J. Alías.

4 recordsLinked to original sources

arXiv2026

On isoperimetric local-Bollobás-Thomason inequalities

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $σ\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical vectors $\{e_i:i\inτ\cup([n]\setminusσ)\}$, for every $τ\subseteqσ$, bounds from below the volume of the corresponding projections of $K$.

math.MG↗

arXiv2025

On local Liakopoulos-Meyer type inequalities and their functional counterparts

We provide a functional Rogers-Shephard type inequality for log-concave functions on $\mathbb R^n$ and any $1$-reducible $s$-cover of $[n]$. As a consequence, we derive a sharp local Liakopoulos-Meyer type inequality for $n$-dimensional convex bodies and $1$-reducible $s$-covers of any $σ\subset[n]$, solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Gutiérrez, Bernués, Brazitikos, Carbery in [3].

math.MG↗

arXiv2024

Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs

In this paper we establish maximum principles for weakly 1-coercive operators $L$ on complete, non-compact Riemannian manifolds $M$. In particular, we search for conditions under which one can guarantee that solutions $u$ of differential equations of the form $L(u)\geq f(u)$ satisfy $f(u)\leq 0$ on $M$. The case of weakly $p$-coercive operators with $p>1$, including the $p$-Laplacian and in particular the Laplace-Beltrami operator for $p=2$, has been considered in a recent paper of ours. As a consequence of the main results we infer comparison principles for that kind of operators. Furthermore we apply them to geometric situations dealing with the mean curvature operator, which is a typical weakly 1-coercive operator. We first consider the case of $\mathcal C^1$ operators $L$ acting on functions $u$ of class $\mathcal C^2$ and, in the last section of the paper, we show how our results can be extended to the case of less regular operators $L$ acting on functions $u$ which are just continuous and locally $W^{1,1}$ regular.

math.AP↗

arXiv2013

On the first stability eigenvalue of constant mean curvature surfaces into homogeneous 3-manifolds

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we derive some consequences for strongly stable surfaces in such ambient spaces. Moreover, we also get a characterization of Hopf tori in certain Berger spheres.

math.DG↗