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Giovanni Leoni

Publications and source records attributed to Giovanni Leoni.

At least 19 recordsLinked to original sources
arXiv2025

Second-Order $\Gamma$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, II

This paper continues the study of the asymptotic development of order 2 by $\Gamma$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.

math.AP↗
arXiv2024

Motion of Islands of Elastic Thin Films in the Dewetting Regime

This paper addresses a two-dimensional sharp interface variational model for solid-state dewetting of thin films with surface energies, introduced by Wang, Jiang, Bao, and Srolovitz in \cite{jiang2016solid}. Using the $H^{-1}$-gradient flow structure of the evolution law, short-time existence for a surface diffusion evolution equation with curvature regularization is established in the context of epitaxially strained two-dimensional films. The main novelty, as compared to the study of the wetting regime, is the presence of moving contact lines.

math.AP↗
arXiv2024

On the Trace of $\dot{W}_{a}^{m+1,1}(\mathbb{R}_{+}^{n+1})$

In this paper we prove extension results for functions in Besov spaces. Our results are new in the homogeneous setting, while our technique applies equally in the inhomogeneous setting to obtain new proofs of classical results. While our results include $p>1$, of principle interest is the case $p=1$, where we show that \begin{equation*} \int_{\mathbb{R}_{+}^{n+1}}t^{a}|\nabla^{m+1}u(x,t)|\;dtdx\lesssim\left\vert f\right\vert _{B^{m-a,1}(\mathbb{R}^{n})} \end{equation*} for all $f \in \dot{B}^{m-a,1}(\mathbb{R}^{n})$ (the homogeneous Besov space) where $u$ is a suitably scaled heat extension of $f$.

math.AP↗
arXiv2024

Existence of an equilibrium with limited stock market participation and power utilities

For constants $γ\in (0,1)$ and $A\in (1,\infty)$, we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE \begin{align*} \begin{cases} h'(y) = \frac{1+γ}{y}\big( γ- h(y)\big)+h(y)\frac{γ+ \big((A-γ)e^{\int_y^1 \frac{1-h(q)}{1-q}dq}-A\big)h(y)}{1-y},\quad y\in(0,1), h(0) = γ, \quad h(1) = 1. \end{cases} \end{align*} As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogenous power-utility investors in the limited participation model from Basak and Cuoco (1998).

q-fin.MF↗
arXiv2023

A First Course in Fractional Sobolev Spaces

This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagliardo-Nirenberg type interpolation inequalities, and trace theory. The third part explores some applications: interior regularity for the Poisson problem with the right-hand side in a fractional Sobolev space and some basic properties of the fractional Laplacian. The first part of the book is accessible to advanced undergraduates with a strong background in integration theory; the second part to graduate students having familiarity with measure and integration and some functional analysis. Basic knowledge of Sobolev spaces would help, but is not necessary. The book can also serve as a reference for mathematicians working in the calculus of variations and partial differential equations as well as for researchers in other disciplines with a solid mathematics background. It contains several exercises and is self-contained.

math.AP↗
arXiv2021

Traveling wave solutions to the free boundary incompressible Navier-Stokes equations

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n \ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e. time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n \ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an over-determined Stokes problem and its under-determined adjoint, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions.

math.AP↗
arXiv2018

Higher order Gamma-limits for singularly perturbed Dirichlet-Neumann problems

A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.

math.AP↗
arXiv2018

Traces for homogeneous Sobolev spaces in infinite strip-like domains

In this paper we construct a trace operator for homogeneous Sobolev spaces defined on infinite strip-like domains. We identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse. The intrinsic seminorm involves two features not encountered in the trace theory of bounded Lipschitz domains or half-spaces. First, due to the strip-like structure of the domain, the boundary splits into two infinite disconnected components. The traces onto each component are not completely independent, and the intrinsic seminorm contains a term that measures the difference between the two traces. Second, as in the usual trace theory, there is a term in the seminorm measuring the fractional Sobolev regularity of the trace functions with a difference quotient integral. However, the finite width of the strip-like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation. The screened homogeneous fractional Sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest, and we study their basic properties. We conclude the paper with applications of the trace theory to partial differential equations.

math.AP↗
arXiv2017

A second order minimality condition for a free-boundary problem

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_Ω\big(|\nabla v|^{2}+χ_{\{v>0\}}Q^{2} \big)\,dx, \] introduced in a classical paper of Alt and Caffarelli. For a special choice of $Q$ this includes water waves. The second variation is obtained by computing the second derivative of the functional along suitable variations of the free boundary. It is proved that the strict positivity of the second variation gives a sufficient condition for local minimality. Also, it is shown that smooth critical points are local minimizers in a small tubular neighborhood of the free-boundary.

math.AP↗
arXiv2017

A Note Regarding Second-Order $Γ$-limits for the Cahn--Hilliard Functional

This note completely resolves the asymptotic development of order $2$ by $Γ$-convergence of the mass-constrained Cahn--Hilliard functional, by showing that one of the critical assumptions of the authors' previous work (Leoni, Murray, Second-order $Γ$-limit for the Cahn--Hilliard functional, Arch. Ration. Mech. Anal. 219, 3, 2016) is unnecessary.

math.AP↗
arXiv2016

Asymptotic Analysis of Second Order Nonlocal Cahn-Hilliard-Type Functionals

In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.

math.AP↗
arXiv2016

Domain formation in membranes near the onset of instability

The formation of microdomains, also called rafts, in biomembranes can be attributed to the surface tension of the membrane. In order to model this phenomenon, a model involving a coupling between the local composition and the local curvature was proposed by Seul and Andelman in 1995. In addition to the familiar Cahn-Hilliard/Modica-Mortola energy, there are additional `forces' that prevent large domains of homogeneous concentration. This is taken into account by the bending energy of the membrane, which is coupled to the value of the order parameter, and reflects the notion that surface tension associated with a slightly curved membrane influences the localization of phases as the geometry of the lipids has an effect on the preferred placement on the membrane. The main result of the paper is the study of the $Γ$-convergence of this family of energy functionals, involving nonlocal as well as negative terms. Since the minimizers of the limiting energy have minimal interfaces, the physical interpretation is that, within a sufficiently strong interspecies surface tension and a large enough sample size, raft microdomains are not formed.

math.AP↗
arXiv2015

Regularity in Time for Weak Solutions of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

The evolution equation derived by Xiang (SIAM J. Appl. Math. 63:241--258, 2002) to describe vicinal surfaces in heteroepitaxial growth is $$ h_t=-\left[ H(h_x)+\left(h_x^{-1}+h_x \right) h_{xx}\right]_{xx}, $$ where $h$ denotes the surface height of the film, and $H$ is the Hilbert transform. Existence of solutions was obtained by Dal Maso, Fonseca and Leoni (Arch. Rational Mech. Anal. 212: 1037--1064, 2014). The regularity in time was left unresolved. The aim of this paper is to prove existence, uniqueness, and Lipschitz regularity in time for weak solutions, under suitable assumptions on the initial datum.

math.AP↗
arXiv2015

Second-Order $Γ$-limit for the Cahn-Hilliard Functional

The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order $2$ by $Γ$-convergence of the mass-constrained Cahn-Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.

math.AP↗