arXiv subjects

Greta Marino

Publications and source records attributed to Greta Marino.

17 recordsLinked to original sources

arXiv2026

Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms

We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to possible degeneracies and prove, as our first main result, its global-in-time existence. The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired nonlocal weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. Finally, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts.

math.AP↗

arXiv2025

A free boundary model for transport induced neurite growth

We introduce a free boundary model to example the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length depending on the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

math.AP↗

arXiv2024

Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system

We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.

math.AP↗

arXiv2023

Lipschitz regularity for solutions of a general class of elliptic equations

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(Ω)\ni v\mapsto \int_ΩF(Dv)\, dx \] where $Ω\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado, i.e. a convex $C^1$-function such that the ratio between the maximum and minimum eigenvalues of $D^2F$ is essentially bounded. This class of integrands inculdes the standard singular/degenerate functions $F(z)=|z|^p$ for any $p>1$ and arises naturally as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.

math.AP↗

arXiv2022

Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

math.AP↗

arXiv2021

Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

math.AP↗

arXiv2021

Uncertainty Analysis for Drift-Diffusion Equations

We study evolution equations of drift-diffusion type when various parameters are random. Motivated by applications in pedestrian dynamics, we focus on the case when the total mass is, due to boundary or reaction terms, not conserved. After providing existence and stability for the deterministic problem, we consider uncertainty in the data. Instead of a sensitivity analysis we propose to measure functionals of the solution, so-called quantities of interest (QoI), by involving scalarizing statistics. For these summarizing statistics we provide probabilistic continuity results.

math.PR↗

arXiv2021

Existence results for double phase problems depending on Robin and Steklov eigenvalues for the $p$-Laplacian

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.

math.AP↗

arXiv2020

Existence and uniqueness of elliptic systems with double phase operators and convection terms

In this paper we study quasilinear elliptic systems driven by so-called double phase operators and nonlinear right-hand sides depending on the gradients of the solutions. Based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems. Furthermore, under some additional conditions on the data, the uniqueness of weak solutions is shown.

math.AP↗

arXiv2020

Implicit equations involving the $p$-Laplace operator

In this work we study the existence of solutions $u \in W^{1,p}_0(Ω)$ to the implicit elliptic problem $ f(x, u, \nabla u, Δ_p u)= 0$ in $ Ω$, where $ Ω$ is a bounded domain in $ \mathbb R^N $, $ N \ge 2 $, with smooth boundary $ \partial Ω$, $ 1< p< +\infty $, and $ f\colon Ω\times \mathbb R \times \mathbb R^N \times \R \to \R $. We choose the particular case when the function $ f $ can be expressed in the form $ f(x, z, w, y)= φ(x, z, w)- ψ(y) $, where the function $ ψ$ depends only on the $p$-Laplacian $ Δ_p u $. We also present some applications of our results.

math.AP↗

arXiv2020

Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment.

math.AP↗

arXiv2020

$L^\infty$-bounds for general singular elliptic equations with convection term

In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $Ω$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where the growth condition for the function $B\colon Ω\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.

math.AP↗

arXiv2020

Existence and $L^{\infty}$-estimates for elliptic equations involving convolution

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $Ω\subset \mathbb{R}^N$ whose boundary $\partialΩ$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to $u$ the convolution $ρ\ast E(u)$ of $ρ$ with $E(u)$, where $ρ$ is an integrable function on $\mathbb{R}^N$ and $E$ is an extension operator related to $Ω$. Under verifiable conditions, we prove the existence of a (weak) solution to our problem by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to our problem is bounded.

math.AP↗

arXiv2019

Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to $L^\infty(\close)\times L^\infty(\close)$. The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.

math.AP↗

arXiv2018

On a Dirichlet problem with $(p,q)$-Laplacian and parametric concave-convex nonlinearity

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $λ>0$ varies, is proven. Since for every admissible $λ$ the problem has a smallest positive solution $\bar u_λ$, both monotonicity and continuity of the map $ λ\mapsto \bar u_λ$ are studied.

math.AP↗

arXiv2018

Moser iteration applied to elliptic equations with critical growth on the boundary

This paper deals with boundedness results for weak solutions of an elliptic equation where the functions are Carathéodory functions satisfying certain $p$-structure conditions that have critical growth even on the boundary. Based on a modified version of the Moser iteration we are able to prove that every weak solution of our problem is bounded up to the boundary. Under some additional assumptions this leads directly to $C^{1,α}$-regularity for weak solutions of the problem.

math.AP↗

arXiv2016

Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to $u""+qu"+f(u)=0$, under various hypotheses on the parameter $q$ and on the potential $F(t)=\int_0^tf(s)\, ds$, generally assumed to be bounded from below. We prove a non-existence result in the case $q\le 0$ and an existence result of periodic solution for: 1) almost every suitably small (depending on $F$), positive values of $q$; 2) all suitably large (depending on $F$) values of $q$. Finally, we describe some conditions on $F$ which ensure that some (or all) solutions $u_q$ to the equation satisfy $\|u_q\|_\infty\to 0$, as $q\downarrow 0$.

math.CA↗