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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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