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Leonardo Marazzi

Publications and source records attributed to Leonardo Marazzi.

12 recordsLinked to original sources
arXiv2022

A Strong Minimum principle and Large Time Asymptotics for viscosity solutions to a class of doubly nonlinear possibly degenerate parabolic equations

We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $Ω\times [0,T)$},$$ where $Ω\subset \mathbb{R}^n$ is a bounded domain and $0<T\leq \infty$. The spatial operator $H$ is homogeneous with power $k$.

math.AP↗
arXiv2017

On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations

In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+χ(t)|Du|^Γ-f(u)u_t=$ in $Ω\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions, $Γ>0$ and depends on the degree of homogeneity of $H$, $f>0$, increasing and meets a concavity condition. We also consider the case $f\equiv 1$ and prove existence of solutions without sign restrictions.

math.AP↗
arXiv2016

On the viscosity solutions to some nonlinear elliptic equations

We consider viscosity solutions of a class of nonlinear degenerate elliptic equations on bounded domains. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of a first eigenvalue and a first positive eigenfunction.

math.AP↗
arXiv2015

On the viscosity solutions to Trudinger's equation

We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains $Ω\times[0, T)$, where $Ω\subset \mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We show existence for general domains $Ω,$ when $n<p<\infty$. For $2\leq p\leq n$, we prove existence for domains $Ω$ that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron's method.

math.AP↗
arXiv2013

An Eigenvalue problem for the Infinity-Laplacian

We study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The work also contains existence results when the parameter, in the equation, is less than the first eigenvalue. A comparison principle applicable to these problems is also proven. Some additional results are shown, in particular, that on star- shaped domains and on C^2 domains higher eigenfunctions change sign. When the domain is a ball, we prove that the first eigenfunction has one sign, radial principal eigenfunction exist and are unique up to scalar multiplication, and that there are infinitely many eigenvalues.

math.AP↗
arXiv2009

Asymptotically hyperbolic manifolds with polyhomogeneous metric

We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds.

math.AP↗
arXiv2009

Inverse scattering on conformally compact manifolds

We study inverse scattering for $Δ_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(λ_1,$ $λ_2)$ in a suitable subset of $\mc$, determines $\alf,$ and the Taylor series of both the potential and the metric at the boundary.

math.AP↗