Seminar in EDP and Applied Mathematics

November 29, 2023 – 14h (Brazil)

Streaming: Youtube Channel | SEMINARIO DE EDP E MATEMATICA APLICADA

Event

About Seminars

Our Online Seminar is one of the most important events in Brazil, it has been held since August 2020, every Wednesday at 2 pm, Brasília time, with a frequency of 14 days. Two 40-minute lectures are presented in each session. Our speakers are world-renowned mathematicians from Europe, the United States and South America.

Featured Talks & Speakers

Susanna Terracini

Dipartimento di Matematica Giuseppe Peano, Università degli Studi di Torino-Italy

14:00h – 14:40h

Boundary Harnack principles and degenerate equations on singular sets

The ratio v/u of two solutions to a second order elliptic equation in diver-
gence form solves a degenerate elliptic equation if u and v share the zero set; that is, Z(u) ⊆ Z(v). The coefficients of the degenerate equation vanish on thenodal set as u^2 .Developing a Schauder theory for such equations, we prove C^k,α-regularity of the ratio from one side of the regular part of the nodal set in the spirit of the higher order boundary Harnack principle established by De Silva and Savin in [?]. Then, by a gluing lemma, the estimates extend across the
regular part of the nodal set. Eventually, using conformal mapping in dimension n = 2, we provide local gradient estimates for the ratio which hold also across the singular part of the nodal set and depends on the highest value attained bythe Almgren frequency function.
References

[1] S. Terracini, G. Tortone and S. Vita, Higher order boundary Harnack prin-ciple on nodal domains via degenerate equations, preprint, 2022.

[2] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of
solutions to degenerate or singular problems part I: even solutions. Comm.Partial Differential Equations, 46-2 (2021), 310-361.
[3] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of

solutions to degenerate or singular problems part II: odd solutions. Mathematics in Engineering, 3-1 (2021), 1-50.

[4] D. De Silva, O. Savin. A note on higher regularity boundary Harnack inequality. DCDS-A, 35(12), (2015) 6155-6163.

Sarah Strikwerda

University of Pennsylvania, Philadelphia-USA

14:50h – 15:30h

Optimal control and multiscale coupling of elliptic-parabolic PDE

Glaucoma is the leading cause of blindness worldwide and the development is dependent on dynamics in a tissue called the lamina cribrosa. Fluid flow through the lamina cribrosa can be modeled by an elliptic-parabolic coupled differential equation. We will discuss an optimal control problem constrained by this system. Then we will discuss the well-posedness of the system when it is coupled with a nonlinear ODE through interface conditions which will account for global blood flow, a feature that impacts the development of glaucoma.

About Organization

Juan Limaco – UFF – Brazil (Coordenador);
Mauro Rincon – UFRJ – Brazil
Max Souza – UFF – Brazil;
Sandra Malta – LNCC – Brazil
Marcelo Cavalcanti – UEM – Brazil;
Rui Almeida – UBI – Portugal
Roxana Lopez – UNMSM – Peru;
Diego Souza – U Sevilla – España;
Mauricio Sepulveda – UdeC – Chile;
Roberto Capistrano – UFPE – Brazil.