arXiv subjects

Renzo Bruera

Publications and source records attributed to Renzo Bruera.

5 recordsLinked to original sources

arXiv2026

Topological entropy of Turing complete dynamics

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

math.DS↗

arXiv2026

Regularity of stable solutions to the MEMS problem up to the optimal dimension 6

In this article we address the regularity of stable solutions to semilinear elliptic equations $-Δu = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $Ω\subset \mathbb{R}^n$ and $f:[0,1)\to (0,+\infty)$ blowing up at $u=1$ and nonintegrable near 1. In this context, a solution $u$ is regular if $u<1$ in all $Ω$ or, equivalently, if $-Δu = f(u)<+\infty$ in $Ω$. This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that $u$ may satisfy. Our results hold up to the optimal dimension $n=6$ (there are counterexamples for $n\geq 7$) but require a Crandall-Rabinowitz type assumption on the nonlinearity $f$. Our main estimate controls the $L^\infty$ norm of $F(u)$ in a ball, where $F$ is a primitive of $f$, by only the $L^1$ norm of $u$ in a larger ball. Under the same assumptions, we also give global estimates in dimensions $n\leq 6$ for the Dirichlet problem with vanishing boundary condition, improving previously known results. For $n\leq 2$, we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1.

math.AP↗

arXiv2026

On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$

In this paper, we study planar nonlocal Delaunay sets. That is, open sets in $\mathbb{R}^2$ with constant nonlocal mean curvature that are periodic in $x_1$, and even in $x_1$ and in $x_2$. Using bifurcation analysis and fine explicit computations, we prove that every sufficiently $C^{1,β}$-flat nonlocal Delaunay set in $\mathbb{R}^2$ that is not a straight band is unstable with respect to volume-preserving periodic variations. Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands.

math.AP↗

arXiv2026

Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature

In this article we prove existence and symmetry properties of periodic surfaces of revolution with constant anisotropic nonlocal mean curvature, generalizing a classical result of Delaunay to the anisotropic nonlocal setting. First, by studying the corresponding periodic isoperimetric problem, under natural assumptions on the kernel, we use rearrangement inequalities to extend a periodic version of the Wulff inequality to the nonlocal setting. This leads to the existence and symmetry properties of minimizers for every given volume in each period, thus generalizing the results of Cabré, Csató, and Mas to the anisotropic case. Second, under the same hypotheses on the kernel, we prove the existence of a one-parameter family of Delaunay near-cylinders in $\mathbb{R}^2$ bifurcating from a straight cylinder and having each constant anisotropic mean curvature. This extends the results of Cabré, Fall, Solà-Morales, and Weth to the anisotropic case. The stability of these near-cylinders will be studied in a forthcoming paper.

math.AP↗

arXiv2024

Mixing and Geometry in the North Atlantic Meridional Overturning Circulation

Vertical motions across the ocean are central to processes, like CO$_2$ fixation, heat removal or pollutant transport, which are essential to the Earth's climate. This work explores 3D conveyor routes {associated with} the Atlantic Meridional Overturning Circulation (AMOC). Our findings show the geometry of mixing structures in the upper and deep ocean layers by means of Lagrangian Coherent Structures. This tool identifies among others, zones linked to vertical transport and characterizes vertical transport time scales. We focus the study in two regions. The first one is the Flemish Cap region, a zone of interaction between the major AMOC components, where our analysis identifies a domain of deep waters that ascend very rapidly to the ocean surface. The second one is the Irminger Sea, where our analysis confirms the existence of a downwelling zone, and reveals a previously unreported upwelling connection between very deep waters and the ocean surface.

physics.ao-ph↗