Functional Analysis and Applications Seminar

 

(organized by the Functional Analysis and Applications Group)

 

Department of Mathematics, University of Aveiro

 


 

 

 

Next talk of the Seminar (11:00, Room 11.2.21)

 

29/05/2025: “Existence results for the parabolic p(x, t) and p(u)-Laplacian problems

 

Hermenegildo Borges de Oliveira (Universidade do Algarve)

 

Variational problems involving integrands with variable exponents of nonlinearity were introduced at the beginning of 1980's by Zhikov, but functionals with variable exponents were already used, in the context of Functional Analysis, by several authors such as Orlicz, Musielak and Rákosník, among others. However, it was only with the advent of the application of this theory in the mathematical modeling of complex real world phenomena that this area had a great boost. By that time, problems with variable exponents were already used in engineering applications to model a large class of smart fluids. A more recent application is the image processing where the variable exponent is used to underline the borders of the distorted image and to eliminate the noise. In this seminar, we will take a look at the results of Fan and Zhang in the early 2000s on the existence, uniqueness and regularity for the elliptic p(x)-Laplacian problem. We continue with the results of Chipot and Oliveira around 2020 on the existence for the elliptic p(u)-Laplacian problem. Then we will address the results proved by Alkhutov and Zhikov and by Diening, Nägele and Růžička in the early 2010s on the existence for the parabolic p(x,t)-Laplacian problem. Regarding this problem, we will show a recent result by Bae, Oliveira and Wolf, in which we prove the existence of weak solutions under weaker assumptions on the continuity and boundedness of the nonlinearity exponent p. If time permits, we will make a short digression on open problems, especially for the p(u)-Laplacian.

 

This seminar is supported in part by FCT – Portuguese Foundation for Science and Technology, under the FCT Multi-Annual Financing Program, through the Center for Research and Development in Mathematics and Applications (CIDMA) of Universidade de Aveiro.

 

 

 

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