This talk by dr. D. Repovš presented a unified view of several classes of nonlinear problems described by Laplace-type operators on various fractal sets, including the basic setting corresponding to the Sierpinski gasket. The content of this talk included the main results established in our paper [COBISS.SI-ID 17994841], but also further qualitative properties on fractals. The analysis of nonlinear PDEs on fractal sets is a rather new mathematical field, which has a higher and higher impact due to numerous applications to phenomena arising in many applied fields. The interest for problems on fractal sets started with the pioneering contributions of Mandelbrot and Strichartz. During the talk, some open problems were raised. The audience at this international conference asked several questions connected with the problems discussed in this talk.

B.04 Guest lecture

COBISS.SI-ID: 18129753This is the talk by dr. V. Radulescu at the Academy, when he was elected in 2017 as a foreign member. At the invitation by the president of the Accademy, this talk was built on achievements of this group, which are divided into three classes: singularities, fractals and non-Newtonian fluids. The first subject is in strong connection with the Ginzburg-Landau theory, which described the formation of vortices in superconductors and superfluids. The second subject considered in this talk was related to the mathematical analysis on fractals, which are non-standard sets but which appear in many places in the nature. Finally, the third part of this talk was dedicated to non-Newtonian fluids (also called ‘smart’ fluids) and which are generally described by differential operators with variable exponent or with anisotropic behavior. The talk was built on several examples and the audience from various scientific fields appreciated the content, which was at the interplay between mathematics and other sciences. Remark: This prestigeous academy has accepted also project member dr. G. Molica Bisci.

B.05 Guest lecturer at an institute/university

COBISS.SI-ID: 18215769This talk was delivered at a prestigious mathematical seminar, which is organized by the Analysis groups of the Stockholm University and KGH Stockholm. The content was some recent contributions to the study of nonlinear problems with variable exponents. We pointed out several nonstandard phenomena, which appear due to the presence of one or several variable exponents. Roughly speaking, under general hypotheses, we considered several classes of nonlinear Dirichlet or Neumann problems driven by non-homogeneous operators. The corresponding abstract setting corresponds to Lebesgue and Sobolev spaces with variable exponent. By using variational (mountain pass, linking, Ekeland’s principle) and topological (deformation, critical groups, Morse theory) tools, we obtained some striking phenomena, e.g. existence of a continuous spectrum (this is discrete in the case of the Laplace operator), concentration properties of the eigenvalues near infinity or near the origin, lack of monotonicity in some classical results (maximum principle, singular solutions with blow-up boundary). The talk included several open problems, e.g. possible behavior in the almost critical case (with respect to variable exponents) or classical results (e.g. the Brezis-Kamin theorem) with a possible lack of monotonicity.

B.05 Guest lecturer at an institute/university

COBISS.SI-ID: 18248281In Cracow, Poland, an international conference was organized under the sponsorship of Journal of Mathematical Analysis and Applications (Elsevier), on the anniversary of dr. D. Repovš and the first issue of 2020 of the journal Opuscula Mathematica, published by the AGH University of Science and Technology was also dedicated to this occasion.

E.02 International awards