14:00h – 14:40h

Moment-Sum-of-Squares relaxations for variational problems

Moment-Sum-of-Squares (moment-SOS) relaxations are an established technique to compute converging sequences of lower bounds on the global minimum of finite-dimensional polynomial optimization problems. In this talk, I will discuss two recent extensions of moment-SOS relaxations to infinite-dimensional variational problems, where a (possibly nonconvex) integral functional is to be minimized over functions from a Sobolev space. The first extension optimizes so-called “null Lagrangian translations” and returns certified lower bounds on the global minimum of the variational problem. The second extension, instead, produces upper bounds by approximating minimizers of finite element discretizations of the variational problem. Conditions that ensure the convergence of these upper and lower bounds to the desired global minimum will be discussed, and current gaps between theory and practice will be illustrated by means of examples.

14:50h – 15:30h

Physics-informed neural networks for non-smooth PDE-constrained optimization problems

14:00h – 14:40h

Geometric inverse problems in 2D: a transport twistor perspective.

 I will discuss some landmark results in geometric inverse
problems in 2D from the point of view of the transport twistor space, a natural complex surface designed to be sensitive to the geodesic vector field. Towards the end of the lecture I will present some recent developments and open questions motivated by this point of view.

14:50h – 15:30h

Local and Nonlocal problems in elliptic PDEs.

  14:00h – 14:40h

A result of multiplicity of nontrivial solutions to an elliptic Kirchhoff-Boussinesq equation type in $\mathbb{R}^{N}$

In this work, using  Ljusternik-Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions  for the following class of elliptic Kirchhoff-Boussinesq type problems given by

$$\varepsilon^4 \Delta^2 u \pm \varepsilon^p \Delta_p u + V(x)u=f(u) + \beta|u|^{2_{**}-2}u ~~\text{in}~~\mathbb{R}^N,\\ u \in H^2(\mathbb{R}^N),$$

where $\varepsilon>0$, $2<p<2^{*}=\frac{2N}{N-2}$,  $2_{**}=\frac{2N}{N-4}$ and $N\geq5$, $V:\mathbb{R}^N \to\mathbb{R}$ is a continuous function and $f:\mathbb{R} \to \mathbb{R}$ is a function of $C^{1}$ class. We consider the subcritical case, i.e, $\beta=0$ and critical case, i.e, $\beta=1$.

14:50h – 15:30h

Roughness-induced effects on a reaction-diffusion equation defined in a thin domain

 In this talk we discuss a reaction-diffusion problem in a thin plane region, endowed with a Robin-type boundary condition, which describes reactions catalyzed at the boundary. Motivated by microfluidic applications we allow, in principle, resonant rough behavior in which the amplitude and period of the roughness at the boundary have the same scale as the domain thickness.

   14:00h – 14:40h

Integration with values in a Neumann space for PDE’s

We It may be useful to use elements in spaces such as L^{p}([0,T] ;L^{q}_{loc}(Ω)) or H^{-1}([0,T]; W^{m,q}(Ω)-weak) … unfortunately they were not yet defined. Theses spaces L^{q}_{loc}(Ω) and W^{m,q}(Ω)-weak are Neumann spaces, that are vector spaces provided with semi-norms that made them separated and sequentially complete.

We will define D'(Ω;E), L^{p}(Ω;E) and W^{m,p}(Ω;E) where E is a Neumann space and Ω is an open set of R^{d}, and we show some applications to PDE’s.
We insist on L^{p}(Ω;E) which is composed of measures: it cannot be composed
of class of almost equal functions because the theory fails, since Egoroff theorem
does not extend if E is not metrisable.
When E is a Banach space and \hat{f} \hat{L}^{p}(Ω;E), that is the space of p-integrable
almost equal functions, then a measure \bar{f} is defined by: for all φ \in K(Ω),
<\bar{f}, φ>=\int_{Ω}\hat{f}φ
We prove that our space L^{p}(Ω;E) is {\bar{f} : \hat{f} \in \hat{L}^{p}(Ω;E)} and that \hat{f}→\bar{f} is an isomorphism. So, our L^{p}(Ω;E) is the true Lebesgue space if E is a Banach space.

14:50h – 15:30h

Decay estimates for semilinear wave equations with time-dependent time delay

14:00h – 14:40h

Unique Continuation Principle for  Nonlocal  Nonlinear Dispersive Models.

14:50h – 15:30h

Optimal control of PDEs: Some environmental applications