This talk is concerned with a unitary 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 No.3 in Item 6 of this report [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, the author raised some open problems. The audience at this international conference raised several questions connected with the problems discussed in this talk.
B.04 Guest lectureCOBISS.SI-ID: 18129753
This is the reception talk at the Accademia given by this member of our group, who was elected in 2017 as a Socio corrispondente of this academic institution. At the request of the president of the Accademia, this talk was built on some personal achievements, 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.
B.05 Guest lecturer at an institute/universityCOBISS.SI-ID: 18215769
This talk has been delivered in a prestigious mathematical seminar, which is organized by the Analysis teams of the Stockholm University and KGH Stockholm. The content has been concerned with some recent contributions to the study of nonlinear problems with variable exponents. We have been interested to point out several nonstandard phenomena, which appear due to the presence of one or several variable exponents. Roughly speaking, under general hypotheses, we have 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 have pointed out some striking phenomena like 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, for instance possible behavior in the almost critical case (with respect to variable exponents) or classical results (like the Brezis-Kamin theorem) with possible lack of monotonicity.
B.05 Guest lecturer at an institute/universityCOBISS.SI-ID: 18248281