Special Session 84: Regularity results of solutions of problems having nonstandard growth and nonuniform ellipticity

Relationship Between Dynamical System and Algebra

Ahmad M Alghamdi Umm Al-Qura University Saudi Arabia Co-Author(s):    -------------

Abstract: The aim of the talk is to discuss the strong relationship between dynamical system and group action on a non-empty set. Let G be a group and X be a non-empty set. The notion of the action of G on the set X is very important in mathematics. If X has also an algebraic structure then such action is more important. For instance, if X is a vector space or a group or a topological structure and so on. The symmetric structure of the notion of groups reflects a lot of behaviours and results which are related to the so-called orbits, stabilizer, fixed points and invariants. On the other hand, dynamical system meets with group action in many places as it studies the evolution and the symmetry for discrete and continuous objects. We believe that the relationship between dynamical systems and algebra is very strong and we are searching in this direction. In this talk, we shall represent examples and environment in which such relationship makes sense and try to envisage common analytical process to exploit this connexion. We shall mention also the relationship between dynamical systems with: ergodic , topology, geometry, logic, numbers, probability, analysis, and category.

Relationship Between Dynamical Systems and Algebra

Ahmad M Alghamdi Umm Al-Qura University Saudi Arabia Co-Author(s):    ---------------------------------------

Abstract: The aim of the talk is to discuss the strong relationship between dynamical system and group action on a non-empty set. Let G be a group and X be a non-empty set. The notion of the action of G on the set X is very important in mathematics. If X has also an algebraic structure then such action is more important. For instance, if X is a vector space or a group or a topological structure and so on. The symmetric structure of the notion of groups reflects a lot of behaviours and results which are related to the so-called orbits, stabilizer, fixed points and invariants. On the other hand, dynamical system meets with group action in many places as it studies the evolution and the symmetry for discrete and continuous objects. We believe that the relationship between dynamical systems and algebra is very strong and we are searching in this direction. In this talk, we shall represent examples and environment in which such relationship makes sense and try to envisage common analytical process to exploit this connexion. We shall mention also the relationship between dynamical systems with: ergodic , topology, geometry, logic, numbers, probability, analysis, and category.

Fractional Sobolev spaces and zeta functions

Emanuel Guariglia Wenzhou-Kean University Peoples Rep of China Co-Author(s):

Abstract: This talk deals with fractional Sobolev spaces and zeta functions. The main properties of this fractional derivative are given and discussed. Moreover, we prove some results linked with the introduction of a new function space. Finally, we discuss fractional calculus of zeta functions and the class of zeta functions in a fractional Sobolev space.

A rigidity result for Kolmogorov-type operators

Alessia Kogoj University of Urbino Italy Co-Author(s):    E.Lanconelli

Abstract: Let $D$ be a bounded open subset of $\mathbb{R}^N$ and let $z_0$ be a point of $D$. Assume that the Newtonian potential of $D$ is proportional outside $D$ to the potential of a mass concentrated at $z_0$. Then $D$ is a Euclidean ball centred at $z_0$. This theorem, proved by Aharonov, Shiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this talk we extend the Suzuki--Watson Theorem to a class of hypoellliptic operators of Kolmogorov-type.

Strong maximum principle and Harnack inequality for classical solutions to subelliptic partial differential equations

Sergio Polidoro Dipartimento FIM - Universit\`{a} di Modena e Reggio Emilia Italy Co-Author(s):

Abstract: Mean value formulas for classical solutions to degenerate linear second order equations in divergence form have been proved in a study in collaboration with Diego Pallara. These results heavily rely on the De Giorgi`s perimeters theory and on its extension to Carnot groups. Based on the classical PDEs theory and on the above mentioned mean value formulas, strong maximum principle and Harnack inequality for classical solutions to stationary subelliptic partial differential equations on Carnot groups have been recently proved by Giulio Pecorella in {\it Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators} (to appear on Journal of Elliptic and Parabolic Equations). Analogous results have been proved by Annalaura Rebucci in {\it Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form} JMAA 128371 (2024) for degenerate Kolmogorov equations. In this seminar we give an overview of the state of research on this subject.

On regularity results of solutions of minimizers of systems having discontinuous coefficients

Maria Alessandra Ragusa University of Catania Italy Co-Author(s):

Abstract: Is showed a problem studied in cooperation with Professor Atsushi Tachikawa. We treat the regularity problem for minimizers u(x) of quadratic and nonquadratic growth functional having integrand A(x, u, Du). We point out that concerning the dependence on the variable x is assumed only that A(x,u, p) is in the class of Vanishing Mean Oscillation class, as a function of x. Namely, is not assumed the continuity of A(x, u, p) with respect to x. Are treated partial regularity and global regularity of the minimizer u.

Existence results for some classes of nonlinear problems

Andrea Scapellato University of Catania Italy Co-Author(s):

Abstract: The talk deals with some existence results for different classes of nonlinear problems driven by $p$-Laplacian-type operators. We also examine some multiplicity results. In some cases, the Ambrosetti-Rabinowitz condition is not imposed. Moreover, we present a problem related to the nonlocal fractional $p(\cdot)$-Laplacian.

Multiplicity of solutions for certain types of nonlinear p-laplacian problems

Angela Sciammetta University of Palermo Italy Co-Author(s):

Abstract: The talk addresses various existence and multiplicity results for different classes of Laplacian problems with Dirichlet boundary conditions. Specifically, we prove the existence of two nontrivial weak solutions, both with and without assuming the Ambrosetti-Rabinowitz condition. Additionally, we establish the existence of three solutions for a specific class of Laplacian problems.