UFSCar Workshop on PDEs - 2022 online edition

Opening

Sapienza Università de Roma

Let Ω be a bounded, open subset of RN. In the talk, we will consider some integral functionals. The first two will be
J(v,φ) = 1⁄2 ∫Ω|Dv| − 1⁄2 ∫Ω|Dφ| + ∫Ωv[E(x)Dφ] − ∫Ωf(x)v.
I(v,φ) = 1⁄2 ∫Ω|Dv| - 1⁄2 ∫Ω|Dφ| + ∫Ωa(x)φg(v) − ∫Ωf(x)v.
where a, f ∈ Lm, E belong to some Lebesgue spaces and g(t) is an increasing and convex real function. I is related to a Schrödinger-Maxwell system, since its critical (saddle) points are solutions of the system; the existence (thanks to a regularizing effect) of saddle points (u, ψ) of I was presented in some brazilian talks.
We will see how some cancellation properties allow us to prove the existence of u, ψ even with very singular data a, f, E. In this case a very weak definition of minimum (maximum) is needed: the T-minima, introduced by the speaker some years ago and presented in the conference “70-Djairo”.

Pontifícia Universidade Católica do Rio de Janeiro

We develop a new, unified approach to the following two classical questions on elliptic PDE: (i) the strong maximum principle for equations with non-Lipschitz nonlinearities, and (ii) the at most exponential decay of solutions in the whole space or exterior domains (Landis conjecture). Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish. If time permits, we will report on some recent C1 estimates with optimized constants and refined Landis-type results.

Universidad de Granada

We survey the results of the joint paper with Antonio Ambrosetti (Differential and
Integral equations, 33 (2020), 92-112). We consider the equation

(u'/√(1−u'2))'+F'(u)=0

modeling, if F'(u) = sin u, the motion of the free relativistic planar pendulum. Using the
critical point theory for non-smooth functionals due to Szulkin, we prove the existence of
non-trivial T periodic solutions provided T is sufficiently large.
We also show the existence of periodic solutions to the free and forced relativistic spherical pendulum, where F' is substituted by F'(u)−h2G'(u) ∼ sin u−h2cos u/sin3 u, h ∈ R.