In the first part of the talk we review basic partial differential equations modeling wave propagation in acoustic and elastic media showing the exact solution. Then, we construct stable discontinuous Galerkin formulations for first order hyperbolic systems and introduce numerical fluxes. In the second part of the talk we discuss wave propagation in inhomogeneous media and present exact solutions with both p- and s-waves and associated interface (transmission) conditions. We finish by presenting a stable Petrov-Galerkin discontinuous finite element method for solving acoustic-acoustic, elastic-elastic and acoustic-elastic interface problems. We show numerical results for several examples to demonstrate the robustness and efficiency of the proposed method. We conclude with a list of few possible future extensions of this work.

The stability of an approximating sequence $(A_n)$ for an operator $A$ usually requires, besides invertibility of $A$, the invertibility of further operators, say $B, C, ...$, that are well-associated to the sequence $(A_n)$. We study this set, $\{ A, B, C, ...\}$, of so-called stability indicators of $(A_n)$ and connect it to the asymptotics of $\Vert A_n \Vert$ and $\Vert A^{-1}_n \Vert$ as well as to spectral pollution by showing that $\lim \sup Spec_c(A_n) = Spec_c(A) \cup Spec_c(B) \cup Spec_c(C) \cup ...$.

In this talk, we introduce and analyze the logarithmic p-Laplacian $L_{\Delta_p}$, a nonlocal operator of logarithmic order that arises as the formal derivative of the fractional p-Laplacian $(-\Delta_p)^s$ at $s=0$. This operator serves as a nonlinear extension of the recently developed logarithmic Laplacian operator. We present a variational framework to study Dirichlet problems involving $L_{\Delta_p}$ in bounded domains, which enables us to explore the relationship between the first Dirichlet eigenvalue and eigenfunction of both the fractional p-Laplacian and the logarithmic p-Laplacian. As a key result, we obtain a Faber–Krahn-type inequality for the first Dirichlet eigenvalue of $L_{\Delta_p}$. Additionally, we examine the validity of maximum and comparison principles, showing that these depend on the sign of the first Dirichlet eigenvalue of $L_{\Delta_p}$. If time permits, we discuss a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic p-Laplacian. The talk is based on joint work with B. Dyda and S. Jarohs.

In this talk, we start by a review on mathematical modeling in experimental sciences, show how to derive, analyse and simulate mathematical models able to predict required useful results. The second part will focus on applications in modeling biological cell migration and proliferation. These models provide a mathematical framework to analyze and simulate the intricate dynamics of tumor growth, considering factors such as cell proliferation, angiogenesis, and interactions with the surrounding microenvironment. This can help to develop new therapies, design biomaterials, and engineer tissues for regenerative medicine. When the cell density is large enough, the continuous medium assumption is a good approximation and partial differential equations are suitable to model cell systems.

In this talk, we present a genesis of the Banach contraction principle, including its generalizations and applications.

Selected References

1. S. Almezel, H. Ansari, M. A. Khamsi, Topics in Fixed Point Theory, Springer, 2014.
2. H. Aydi, The Banach contraction principle: generalizations, extensions and its inverse, Lett. Nonl. Anal Appl. 2 (4) (2024), 184-199.
3. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3 (1922), 133-181.
4. L.E.J. Brouwer, Über Abbildungen von Mannigfaltigkeiten. Math. Ann. 71 (1912), 97-115.
5. J. Jachymski, I. Józwik, M. Terepeta, The Banach fixed point theorem: selected topics from its hundred-year history. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 (2024), no. 4, Paper No. 140, 33 pp.
6. A. Tarski, A lattice theoretical fixed point theorem and its applications. Pacific J. Math. 5 (1955), 285-309.