Special Session 89: Partial Differential Equations: Diverse Applications and Connections

Global boundedness of solutions to fully anisotropic variational problems

Angela Alberico Italian National Research Council - Institute for Applied Calculus (Napoli) Italy Co-Author(s):

Abstract: A class of nonlinear, fully anisotropic variational problems whose anisotropy is governed by a general $n$-dimensional convex function of the gradient is analyzed. Global boundedness for both solutions to fully anisotropic nonlinear elliptic Dirichlet problems, and minima of fully anisotropic variational integrals, is established under general growth conditions, not necessarily of power type. The results unify and extend several existing contributions in the literature, yielding new regularity conclusions across a variety of anisotropic frameworks.

Finite energy solutions for a class of Dirichlet problem with very singular data

Giuseppa Rita Cirmi University of Catania Italy Co-Author(s):    Lucio Boccardo, Gustavo Madeira

Abstract: We study the homogeneous Dirichlet problem for a class of nonlinear elliptic equations of the form: \begin{equation*} -\text{div} \big( [a(x) + |u|^\theta] |Du|^{p-2} Du \big) = f \quad \text{in } \Omega, \end{equation*} where $\Omega$ is a bounded open set in $\mathbb{R}^N$ ($N > 2$), $1 < p < N$, and $\theta > 0$. The coefficient matrix $a(x)$ is assumed to be elliptic with bounded entries, while the source term $f$ satisfies minimal summability assumptions. We prove the existence of finite energy solutions, highlighting how the results depend on both the summability of $f$ and the value of the parameter $\theta$. Furthermore, we discuss the existence of $W_0^{1,1}(\Omega)$ solutions for small values of $\theta$ and for $p$ close to $1$.

On the existence of $W^{1,1}_0$ distributional solutions to nonlinear elliptic equations with lower-order terms.

Salvatore D`Asero University of Catania Italy Co-Author(s):

Abstract: We study the existence of distributional solutions in $W^{1,1}_0$ for nonlinear elliptic equations with lower-order terms and possibly singular right-hand side terms.

On the Obstacle Problem for Nonlinear Materials via the Monotonicity Principle

Luisa Faella University of Cassino and Southern Lazio Italy Co-Author(s):    Antonio Corbo Esposito, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash, Antonello Tamburrino

Abstract: In this talk, we discuss the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponents. This setting is particularly relevant for applications, as it enables the use of fast imaging methods based on the MP in problems involving multiple materials, at least one of which exhibits nonlinear behavior. The proposed framework is quite general and can accommodate a variety of practical configurations, including Superconducting (SC), Perfect Electric Conducting (PEC), and Perfect Electric Insulating (PEI) materials. A central role is played by the average Dirichlet-to-Neumann operator. We show how this approach can be extended to the more general setting.

Boundedness of Weak Solutions for a Degenerate Elliptic Equation Modelling Spider Orb Webs

Maria Stella Fanciullo Universit\`a degli Studi di Catania Italy Co-Author(s):    Giuseppe Di Fazio, Antonino Morassi, Pietro Zamboni

Abstract: We present a boundedness result for weak solutions to a class of two-weight degenerate elliptic equations. The motivating example is the equation governing the static equilibrium of a spider orb web, whose continuous membrane model was proposed by Morassi, Soler and Zaera in 2017. The concentration of radial threads near the center induces a singularity at the origin. This is reflected in the degenerate character of the equation of the model through two weights whose ratios are neither bounded away from zero nor bounded from above. Our approach combines a refined two-weight Sobolev-Poincar\`e inequality with the assumption that lower-order coefficients belong to a generalized Stummel-Kato class. These tools, together with a generalized Moser iteration scheme, yield an explicit $L^\infty$ bound for weak solutions in terms of the data and the weights.

Asymptotic Behavior and Blow-up Analysis for Parabolic Obstacle Problems with Drifts

Fernando Farroni Universit\\\\grave{a} degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni R. Caccioppoli Italy Co-Author(s):

Abstract: We study the evolution for a class of parabolic obstacle problems. The aim of this talk is two-fold. For global-in-time solutions, we describe the large-time behavior. For solutions defined on bounded time horizons, we perform a blow-up analysis.

Noncoercive Parabolic Systems

Luigi Greco Universit\\`a di Napoli Italy Co-Author(s):

Abstract: We prove existence results for the Cauchy--Dirichlet problem for convection--diffusion parabolic systems with singular coefficients in the convective term. The principal part of our operator is of $p$-Laplacian type and standard monotonicity and growth conditions are assumed. However, the coefficients of the lower order terms are time dependent and belong to borderline mixed Lebesgue--Marcinkiewicz spaces, and this causes lack of coercivity. We are not assuming any additional structure condition, such as for example the well-known Landes condition, introduced to deal with systems. We also show optimality of our results in the linear case.

BV functions, hyperbolic conservation laws and fractal sets

Emanuel Guariglia Kean University Italy Co-Author(s):

Abstract: In this talk, we review some important results on the uniqueness of solutions to hyperbolic conservation laws. More precisely, we show that every weak solution taking values in the domain of the semigroup, and such that shocks satisfy the Liu conditions, coincides with a semigroup trajectory. Finally, we give an application in fractal geometry.

Isoperimetric Inequalities for a Capillarity Problem

Giulio Pascale Universit\\'{a} degli Studi di Napoli Federico II Italy Co-Author(s):

Abstract: The classical capillarity problem consists of minimizing, among sets satisfying a volume constraint within a given container, a suitably weighted perimeter. The contribution of the interface that touches the boundary of the container is weighted by a fixed constant representing the relative adhesion coefficient between the liquid drop and the solid walls of the container. When the container is a half-space, the isoperimetric sets for this problem are suitably truncated balls lying on the boundary of the half-space. The aim of this talk is to present some quantitative isoperimetric inequalities for the capillarity problem in a half-space, which estimate different notions of asymmetry of a competitor with respect to the optimal bubble in terms of the corresponding isoperimetric deficit. I will also mention some applications in which nonlocal interactions are taken into account.

Behavior of the solutions to nonlinear parabolic equations with absorption terms

Maria Michaela MM Porzio Sapienza University of Rome Italy Co-Author(s):

Abstract: In this talk we discuss the influence of an absorption term of power type on the regularity and time behavior of the solutions to a class of nonlinear parabolic problems. We will show that important and unexpected changes occur. For example, in absence of a forcing term it can produces an immediately boundedness in cases when it is well known that, in absence of such a lower order term, the solutions remain unbounded. Moreover, we prove that regularization phenomena appear also in presence of forcing terms.

Higher differentiability for solutions to a class of elliptic systems with lower order terms

Teresa Radice University of Naples Federico II Italy Co-Author(s):    Fernando Farroni

Abstract: This talk concerns the higher differentiability of the solution $u$ to the Dirichlet problem: \begin{equation*} \begin{cases} \textrm{div} (A(x, Du)) + B(x,u)=f & \hbox{in $\Omega$} \cr \cr u=0 &\hbox{on $\partial \Omega$} \end{cases} \end{equation*} on a bounded lipschitz domain $\Omega$ in $\mathbb R^n$. The lower order term $B(x,u)$ is controlled with respect to spatial variable by a function $b(x)$ belonging to the Marcinkiewicz space $L^{n,\infty}$. This work is in collaboration with Fernando Farroni.

Sobolev embeddings and divergence operator

Roberta Schiattarella Dipartimento di Matematica e applicazioni, Universita` degli Studi di Napoli Federico II Italy Co-Author(s):    Gianluigi Manzo

Abstract: We investigate the relationship between Sobolev-type embeddings and the solvability of the divergence equation $\mathrm{div}\,u = f$ in the setting of Banach function spaces. Given two spaces $X, Y \subset L^{1}(Q_{0})$ where $Q_0=[0,1]^n$, we denote by $\mathbf{Y}$ the space of vector fields whose components belong to $Y$, and by $X`$, $Y`$ their associate spaces. We show that the existence of solutions $u \in \mathbf{Y}$ to $\mathrm{div}\,u = f$ for every $f \in X$ is equivalent to a dual Sobolev embedding of the form \[ W^{1}_{0}Y` \hookrightarrow X`. \] This establishes a general duality principle linking divergence-type equations and Sobolev embeddings beyond the classical Lebesgue framework. We also present examples highlighting borderline phenomena, where standard regularity and gradient representation fail. This work is based on a joint project with Gianluigi Manzo.

Existence and Qualitative Properties of Solutions for a Class of Nonlinear Fokker-Planck Equations with Superlinear drift

Gabriella Zecca Dipartimento di Matematica e Applicazioni `R.Caccioppoli` - Università degli Studi di Napoli Federico II Italy Co-Author(s):    Stefano Buccheri, Fernando Farroni

Abstract: In this talk, I will deal with a class of nonlinear Fokker-Planck equations with the following structure: \begin{equation} \partial_t u - \text{div}\big(M\nabla u+ E h(u)\big)=0, \end{equation} where $M$ is a bounded elliptic matrix, $E$ is a vector field in a suitable Lebesgue space, and $h(u)$ exhibits a superlinear growth. I will provide existence results for $C([0,T),L^1)$ distributional solutions to initial-boundary value problems related to the above equation under general assumptions on the coefficients. The approach followed is entirely non-variational, does not require specific structural assumptions on the vector field $E$ (such as being a gradient), and avoids the use of representation formulas. Additionally, some qualitative properties of the solutions will be discussed. These results have been obtained in collaboration with Stefano Buccheri and Fernando Farroni.