arXiv subjects

Michele Miranda Jr

Publications and source records attributed to Michele Miranda Jr.

10 recordsLinked to original sources

arXiv2025

Heat content asymptotics for sets with positive reach

In this paper we study the heat content for sets with positive reach. In details, we investigate the asymptotic behavior of the heat content of bounded subsets of the Euclidean space with positive reach. The concept of positive reach was introduced by Federer in \cite{fed_1959} and widely developed in the following years (see for instance the recent book by Rataj and Zh{ä}le \cite{rat_zah_2019}). It extends the class of sets with smooth boundaries to include certain non-smooth and singular sets while still admitting a well-defined normal geometry. For such sets $E\subseteq\Rn$, we analyze the short-time asymptotics of the heat content $\|T_t\mathbbm{1}_E\|_2$, where $T_t\mathbbm{1}_E$ is the soluzion of the heat equation in $\Rn$ with initial condition $\mathbbm{1}_E$. The present paper is in the spirit of Angiuli, Massari and Miranda Jr.\cite{ang_mas_mir_2013}, but the technique's used here are completely different and also the final result is slightly different.

math.AP↗

arXiv2025

A Schrödinger operator with confining potential having quadratic growth

We study the spectral properties of a Schrödinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.

math.AP↗

arXiv2024

Gradient contractivity of a rescaled resolvent on domains in Wiener spaces

Given an abstract Wiener space $(X,γ,H)$, we consider an open set $O\subseteq X$ which satisfies certain smoothness and mean-curvature conditions. We prove that the rescaled resolvent operator associated to the Ornstein-Uhlenbeck operator with homogeneous Dirichlet boundary conditions on $O$ is gradient contractive in $L^p(X,γ)$ for every $p\in(1,\infty)$. This is the Gaussian counterpart of an analogous result for the rescaled resolvent operator associated to the Laplace operator $Δ$ in $L^p$ with respect to the Lebesgue measure, $p\in[1,\infty)$, with homogeneous Dirichlet boundary conditions on a bounded convex open set $O\subseteq \mathbb R^n$.

math.AP↗

arXiv2022

Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

In this paper we consider an abstract Wiener space $(X,γ,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,γ)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,γ)$ can be characterized as the space of functions in $W^{1,p}(O,γ)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,γ)$.

math.FA↗

arXiv2021

On $BV$ functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation ($BV$) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,μ)$ equipped with a non-negative Radon measure $μ$ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$ for any $p\in[1,\infty]$ and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss-Green formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$ vector field. This differential machinery is also the natural framework to specialize our analysis for ${\mathsf{RCD}(K,\infty)}$ spaces, where we exploit the underlying geometry to determine the Leibniz rules for $\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on the Gauss-Green formulas.

math.DG↗

arXiv2021

Rough traces of $BV$ functions in metric measure spaces

Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation ($BV$) in the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such a function. We then compare our analysis with the discussion done in a recent work by P. Lahti and N. Shanmugalingam, where traces of $BV$ functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.

math.MG↗

arXiv2019

Characterization of BV functions on open domains: the Gaussian case and the general case

We provide three different characterizations of the space $BV(O,γ)$ of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure $ γ$ on open domains $O$ in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition for belonging to $BV(O,γ)$ by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our technique to Fomin differentiable probability measures $ν$ on a Hilbert space $X$, inferring a characterization of the space $BV(O,ν)$ of the functions of bounded variation with respect to $ν$ on open domains $O\subseteq X$.

math.FA↗

arXiv2013

Local higher integrability for parabolic quasiminimizers in metric spaces

Using variational methods, we prove local higher integrability for the minimal p-weak upper gradients of parabolic quasiminimizers in metric measure spaces. We assume the measure to be doubling and the underlying space to be such that a weak Poincaré inequality is supported. We give proofs to density results concerning the space of test functions used when proving estimates for parabolic quasiminimizers.

math.AP↗

arXiv2012

Newtonian Lorentz Metric Spaces

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space carries a doubling measure and a weak Poincare inequality) and some restrictions on q, we show that the Lipschitz functions are dense in those spaces. Moreover, in the same setting we show that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provide a counterexample to the density result of Lipschitz functions in the Euclidean setting when q is infinite.

math.MG↗

arXiv2011

Perimeter of sublevel sets in infinite dimensional spaces

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains.

math.FA↗