Special Session 167: Functional spaces and multiphase problems

A new class of anisotropic double phase problems: exponents depending on solutions and their gradients

Anouar Bahrouni University of Monastir Tunisia Co-Author(s):    Ala Eddine Bahrouni and Hlel Missaoui

Abstract: In this talk, we deal with two novel classes of quasilinear elliptic equations, each driven by the double phase operator with variable exponents. The first class features a new double phase equation where exponents depend on the gradient of the solution. We delve into giving various properties of the corresponding Musielak-Orlicz Sobolev spaces, including the $\Delta_2$ property, uniform convexity, density and compact embedding. Additionally, we explore the characteristics of the new double phase operator, such as continuity, strict monotonicity, and the (S$_+$)-property. Employing both variational and nonvariational methods, we deal with the existence of solutions for this inaugural class of double phase equations. In the second category, the treatment of exponents is dependent on the solution itself. This class differs from the first one due to the unavailability of suitable Musielak-Orlicz Sobolev spaces. For this reason, we employ a perturbation argument that leads to the classical double phase class. These two new classes highlight how different physical processes like the movement of special fluids through porous materials, phase changes, and fluid dynamics interact with each other.

Nonlinear scalar field equations with a critical Hardy potential

Bartosz Bieganowski University of Warsaw Poland Co-Author(s):

Abstract: We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Poho\v{z}aev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution. This is a joint work with D. Strzelecki.

Lavrentiev`s phenomenon and density of regular functions

Iwona Chlebicka University of Warsaw Poland Co-Author(s):

Abstract: Modelling materials that differ from place to place require the use of unconventional, inhomogeneous space setting. One of the key structural challenge might be the lack of density of regular functions that entails irregularity or even non-existence of solutions. I will discuss what for do we need the density, how to ensure it and examples when it is not possible

Sampling-type series and approximation of differential operators

Rosario Corso University of Palermo Italy Co-Author(s):

Abstract: Sampling series are employed in signal processing and in approximation theory to reconstruct or approximate functions from their values on discrete sets of points. One of the first results in this context is the Whittaker-Kotel'nikov-Shannon theorem for bandlimited functions. Variations of classical sampling series replace pointwise evaluations with integral averages providing flexibility, for instance, in contrast to noise. A common characteristic is the definition in terms of a so-called kernel, which may belong to various spaces of functions. In this talk we will make an overview of sampling-type series and discuss the problem of approximating differential operators.

Global small data solutions for some nonlocal semilinear evolution equations

Marcello DAbbicco University of Bari Italy Co-Author(s):

Abstract: In this talk, we discuss some global small data existence results for semilinear evolution equations with nonlocal terms in time and/or in space, and their nonexistence counterpart.

A bifurcation phenomenon for the critical p-Laplace equation in the ball

Francesca Dalbono University of Palermo Italy Co-Author(s):    Matteo Franca, Andrea Sfecci

Abstract: We study positive radial solutions for a Dirichlet problem associated with a class of quasilinear $p$-Laplace differential equations involving a critical weighted nonlinearity. We show that the existence and multiplicity of solutions exhibit a bifurcation phenomenon depending on the flatness order of the weight at zero. The existence of a second solution is new, even in the classical Laplace case. The analysis is based on a dynamical systems approach via the Fowler transformation, and the main technical contribution consists in the construction of an unstable manifold in a non-smooth setting.

Nonlinear Partial Differential Equations and Their Applications

Ki-Ahm Lee Seoul National University Korea Co-Author(s):

Abstract: This talk presents an overview of fully nonlinear partial differential equations and their applications in differential geometry, physics, engineering, and mathematical finance. We emphasize the structural relationships among different classes of nonlinear equations and the emergence of degenerate and singular phenomena, including free boundary problems. Recent developments will be discussed.

Finite-time blow-up in a Keller-Segel model with indirect signal production

Xuan Mao Hohai University Peoples Rep of China Co-Author(s):    Meng Liu and Yuxiang Li

Abstract: This talk focuses on a Keller-Segel chemotaxis system involving indirect signal production, which models the impact of phenotypic heterogeneity on population aggregation. The system exhibits a four-dimensional critical mass phenomenon, resulting in unbounded solutions, as first explored in the seminal works of Fujie and Senba (2017, 2019). Building upon this foundation, our recent research establishes the existence of solutions that explode in finite time, thereby providing a more complete picture of singularity formation and offering new insights into the dynamic behavior of population aggregation in the presence of indirect signal production.

Vitali theorems for varying measures

Valeria Marraffa Dipartimento di Matematica e Informatica Italy Co-Author(s):    A.R. Sambucini

Abstract: Some limit theorems of the type $$\int_{\Omega}f_n\,dm_n \rightarrow \int_{\Omega}f \,dm$$ are presented for scalar, (vector), (multi)-valued sequences of $m_n$-integrable functions $f_n$. Conditions for the convergence of sequences of measures $(m_n)_n$ and of their integrals $(\int f_n dm_n)_n$ in a measurable space $\Omega $ are of interest in many areas of pure and applied mathematics such as statistics, transportation problems, interactive partial systems, neural networks and signal processing.\ Sufficient conditions in order to obtain some kind of Vitali`s convergence theorems for a sequence of (multi)functions $(f_n)_n$ integrable with respect to a sequence of measures $(m_n)_n$ are considered.\ We consider the asymptotic properties of $(\int_{\Omega} f_n d m_n)_n$ with respect to varying measures, which are vaguely convergent in an arbitrary measurable spaces.\ %Bibliography \begin{thebibliography}{9} \bibitem{1} L. Di Piazza, V. Marraffa, K. Musia{\l}, A. Sambucini, {\emph Convergence for varying measures}, {\it J. Math. Anal. Appl.}, Vol. 518, N.2, Paper N. 126782, (2023). \bibitem{3} L. Di Piazza, V. Marraffa, K. Musia{\l}, A. Sambucini, {\emph Convergence for varying measures in the topological case}, {\it Annali di Matematica Pura e Applicata, (4) 203 (2024) 71-86}. \bibitem{2}V. Marraffa, B. Satco, {\emph Convergence Theorems for Varying Measures Under Convexity Conditions and Applications}, {\it Mediterr. J. Math.}, (2022), 19:274. \bibitem{4} V. Marraffa, A.R. Sambucini, {\emph Vitali theorems for varying measure}, {\it Symmetry} 2024, 16(8), 972. \end{thebibliography}

Regularity theory for a class of normalized $p-$Laplace type equations

Makson Santos University of Lisbon Portugal Co-Author(s):    Claudemir Alcantara, Jos\`{e} Miguel Urbano

Abstract: We study the regularity properties of viscosity solutions to a class of degenerate normalized $p-$Laplace type equations. In particular, we prove that the gradient of viscosity solutions is H\{o}lder continuous, and we give the optimal exponent. Moreover, we also show that viscosity solutions to equations with very general degeneracy laws are differentiable.

Curvature obstructions to concavity preservation for the porous medium flows

Asuka Takatsu The University of Tokyo Japan Co-Author(s):    Kazuhiro Ishige and Yoshiumi Tateoka

Abstract: We discuss how concavity properties are preserved under the porous medium flows on a Riemannian manifold, clarifying the effects of diffusion nonlinearity and curvature. In particular, curvature yields an obstruction: if the sectional curvature is negative at a given point, no concavity property is preserved. However, if the curvature is non-zero, concavity properties can only be preserved if they are stronger than those associated with the diffusion nonlinearity.

Anisotropic (p,q)-Laplacian equations with competing nonlinearities

Calogero Vetro University of Palermo Italy Co-Author(s):

Abstract: We consider the existence of solutions to general mathematical models for physical phenomena when different spatial directions play different roles. The mathematical formulation leads to an anisotropic elliptic system of p & q type posed in a bounded domain with smooth boundary and Dirichlet boundary condition.