Special Session 88: Diffusion problems with non-standard growth conditions

Regularity results for a degenerate double phase type operator with irregular data

Fessel Achhoud University of Messina Italy Co-Author(s):    Rita CIRMI and Salvatore D`ASERO

Abstract: Our talk deals with the following nonlinear Dirichlet problem associated to the model equation $$ -\operatorname{div}\left(a(x) \vert\nabla v\vert^{p-2}\nabla v\right)-\operatorname{div}\left(\vert v\vert^{q(r-1)+1} \vert\nabla v\vert ^{q-2}\nabla v\right)=f\quad \text{in}\; \Omega, $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N, N>2,$ $ 1 < q < p \frac{1}{q'}$, $f$ is a function with poor summability and the function $a(x)$ is a measurable function such that $$ \alpha \leq a(x) \leq \beta, \quad \text{a.e. } x \in \Omega $$ with $ \alpha, \beta>0.$ Previous result showed that, even for $f \in L^1\log L^1(\Omega)$, the presence of the degenerate term $-div(\vert v\vert^{q(r-1)+1}|\nabla v|^{q-2}\nabla v)$ yields a strong regularizing effect, ensuring existence and improved regularity of distributional solutions. Building on that, we extended these results to the case where the datum $f \in L^m(\Omega)$, with $m>1$, including the regime $ 1 < m < (p^*)'$. Furthermore, we addressed a new borderline case, obtaining existence result in the space $W^{1,1}_0(\Omega)$. These findings highlight the crucial role played by degenerate gradient terms in overcoming the lack of regularity of the data.

Existence and regularity results for a class of singular parabolic problems with $L^1$ data

Ida de Bonis Sapienza University of Rome Italy Co-Author(s):

Abstract: We prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in $L^1(\Omega)$, there exists a solution that ``instantly'' becomes bounded. The results are obtained in collaboration with M.M. Porzio.

Nonlinear elliptic problems with quadratic gradient growth and singular Robin eigenvalues

Francesco Della Pietra Universita degli studi di Napoli Federico II Italy Co-Author(s):    Giuseppina di Blasio, Giuseppe Riey

Abstract: In the talk I will describe an existence result for Robin boundary value problems modeled on \[ \begin{cases} \Delta u + |\nabla u|^2 + \lambda f(x) = 0 & \text{in } \Omega \ \frac{\partial u}{\partial \nu} + \beta u = 0 & \text{on } \partial\Omega \end{cases} \] where $\Omega$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$\beta>0$}, under a smallness assumption on the datum $\lambda$. In order to study such problem, I will show several properties of the weighted, singular Robin eigenvalue problem \[ \lambda_{1,f,\gamma}(\Omega)= \inf_{\psi\in H^{1},\;\int_{\Omega}f\psi^{2}=1}\left\{\int_{\Omega}|\nabla \psi|^{2}dx+\gamma\int_{\partial\Omega}\psi^{2}\right\}. \]

Recent developments on noncoercive anisotropic operators with singular lower order terms

Giuseppina di Blasio University of Campania "L. Vanvitelli" Italy Co-Author(s):

Abstract: The aim of this talk is to provide an overview of several results obtained in recent years concerning a class of noncoercive anisotropic elliptic equations. We focus on problems where the presence of a convection term characterized by coefficients belonging to suitable Marcinkiewicz spaces, leads to a loss of coercivity in the associated differential operator. In this talk, starting from the classical results, I will illustrate the most recent developments achieved in collaboration with F. Feo and G. Zecca, trying to provide an overview of the current state of the art on this topic.

Schiffer type problems on the plane and the 2-sphere.

Antonio J. Fern\`andez Universidad Aut\`onoma de Madrid Spain Co-Author(s):

Abstract: The so-called Schiffer conjecture can be stated as follows: If a nonconstant Neumann eigenfunction of the Laplacian on a smooth bounded domain is constant on the boundary, then the domain is a ball. In this talk we will consider two slightly relaxed versions of this conjecture, and provide suitable counterexamples to them. Specifically, we will consider annular type domains on the plane, and spherical cap type domains on the 2-sphere. The proofs rely on a local bifurcation argument. We combine the use of anisotropic H\older spaces for the functional setting with computer-assisted techniques to check the bifurcation conditions. The talk is based on joint works with A. Enciso, D. Ruiz and P. Sicbaldi (planar case), and with G. Cao-Labora (2-sphere case).

Quasilinear elliptic systems with critical growth

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

Abstract: We study the Diriclet problem for nonlinear elliptic systems of convection-diffusion type with singular coefficients in the convective term. Our operators are monotone, with principal part of $p$-Laplacian type, satisfy appropriate ellipticity conditions and have ctirical growth. This may produce lack of coercivity and compactness. We extend to systems results already proven for the scalar case. To treat the vectorial case, we need an additional structural condition. We discuss some of these conditions, which are inspired by the well-known Landes` condition. Our results are contained in a joint work with Gabriella Zecca.

Periodic solutions to the Lorentz force equation

Salvador L\`{o}pez Mart\`{i}nez Autonomous University of Madrid Spain Co-Author(s):    Manuel Garz\`{o}n

Abstract: The motion of a charged particle in an electromagnetic field is governed by the Lorentz force equation (LFE), a classical model independently introduced by Poincar\`{e} and Planck in the early twentieth century. It has been known since then that the periodic solutions -- corresponding to particles traveling in closed orbits -- can be formally obtained as critical points of the relativistic action functional. However, the action functional becomes nonsmooth as the speed of the particle approaches the speed of light. Thus, a rigorous critical point theory for the nonsmooth action functional has only recently been developed. In this talk, I will present some recent advances in the variational study of the LFE in collaboration with Manuel Garz\`{o}n (ICMAT, Madrid).

The regularizing effect of superlinear terms in Elliptic and Parabolic Equations

Martina Magliocca Universidad de Sevilla Spain Co-Author(s):

Abstract: We will discuss the regularizing effect induced by superlinear terms in some class of Elliptic and Parabolic Equations. Our model equations are \begin{equation*}\label{P} u_t-\Delta u=g(u)|\nabla u|^q+f \qquad\text{in }(0,T)\times \Omega, \end{equation*} and \begin{equation*}\label{E} -\Delta u=g(u)|\nabla u|^q+f\qquad \text{in }\Omega, \end{equation*} being $\Omega\subset \mathbb{R}^N$ for $N\ge 2$, $g(u)>0$ and $q < 2$. We assume that the term \begin{equation*}\label{super} g(u)|\nabla u|^q \end{equation*} behaves in a superlinear way. Roughly speaking, if $g(u)\equiv \text{const}.$, then we are asking for $1 < q < 2$. In the more general case $g(u)\not\equiv \text{const}.$, the $q$ threshold is influenced by this perturbation term and the superlinear $q$ range depends on its growth. An important remark on this kind of problems concerns the data assumptions, which have to satisfy well precises compatibility conditions in order to have existence of solutions. We will show that, under certain growth assumptions on $g(u)$, we can relax the regularity needed on the data w.r.t. the case $g(u)=\text{const}.$.

Singular elliptic equations having a gradient term with natural growth

Anna Mercaldo University of Naples Federico II Italy Co-Author(s):

Abstract: This talk will present existence results for a class of homogeneous Dirichlet boundary value problems for equations whose prototype is $$ -\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, \quad\hbox{in } \Omega\,, $$ where $\Omega$ an open bounded subset of $\mathbb R^N$, $0

L$^\\infty$ estimates for a class of nonlinear elliptic systems with nonstandard growth

Antonella Nastasi University of Palermo Italy Co-Author(s):    Elvira Mascolo (University of Florence) and Cintia Pacchiano Camacho (Calgary University).

Abstract: The talk shall focus on energy integral functionals of the form $$\int_{\Omega} F(x,Du)\,\mathrm{d} x,$$ where the integrand is characterized by nonstandard growth conditions with respect to the gradient. We prove the local boundedness of solutions of partial diferential systems in divergence form. The systems under consideration include the first variations of functionals depending on the space variable and having nonstandard growth with respect to the gradient, like for instance the model with growth depending on the point, but without assuming the usual $\Delta_2$ condition. The results are part of a joint project with Elvira Mascolo (University of Florence) and Cintia Pacchiano Camacho (Calgary University).

Fractional capacitary potentials at the critical threshold: decay, breakdown, and vanishing capacity

Giampiero Palatucci University of Parma Italy Co-Author(s):    Matteo Focardi; Caterina Ida Zeppieri

Abstract: The $p$-fractional capacitary problem provides a natural nonlocal counterpart of classical capacity for arbitrary compact sets. A central issue is the behavior at infinity of the associated equilibrium potentials and its relation with the scaling of the energy. In this talk I will first discuss the decay at infinity in the subcritical regime. I will then turn to the critical threshold, where the picture changes dramatically. I will show that the corresponding fractional capacity vanishes, revealing a sharp degeneracy of the critical regime. This contrast highlights the special role of the threshold case and the need for a different potential-theoretic perspective at criticality.

The asymptotic behavior of Quasilinear CalderOn problem w.r.t. the boundary data

Gianpaolo Piscitelli Universita` degli studi di Napoli Parthenope Italy Co-Author(s):    A. Corbo Esposito, L. Faella, V. Mottola, G. Piscitelli, R. Prakash, and A. Tamburrino

Abstract: We consider a quasilinear version of the Calderon problem on a region which, at a given point in the space, present a behavior for large (or small) arguments that is described by monomials of order $p$ and $q$. The original contribution works [1,2]make is that the nonlinear problem can be approximated by a weighted $p-$Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p-$Laplacian in inverse problems with nonlinear materials. Moreover, when $p=2$, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of[1,2]is that for ``large`` or ``small``Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted $p-$Laplace problem. [1] A. Corbo Esposito, L. Faella, G. Piscitelli, R. Prakash, and A. Tamburrino, The p-Laplace ``signature`` for Quasilinear Inverse Problems with Large boundary data, SIAM Journal on Mathematical Analysis, 56 (2024), pp. 275-303. [2] A. Corbo Esposito, L. Faella, V. Mottola, G. Piscitelli, R. Prakash, and A. Tamburrino, The p0-laplace ``signature`` for Quasilinear Inverse Problems, SIAM J. Imaging Sci., 17 (2024), pp. 351-388.