For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.[-]

In many mechanical systems where energy is conserved, the phenomenon of resonance can occur, meaning that for certain time-periodic forces, the solution of the system becomes unbounded. Examples of partial differential equations describing such systems include the wave equation and equations of linearized elasticity (Lamé system). On the other hand, resonance does not occur in systems with strong dissipation, such as systems described by the heat equation. More precisely, in such a system, there exists a unique time-periodic solution for each time-periodic right-hand side. In this lecture, we will address the question "how much dissipation is necessary to prevent the occurrence of resonance?". We will analyze periodic solutions to the so-called heat-wave system, where the wave equation is coupled with the heat conduction equation via a common boundary. In this system, dissipation only exists in the heat component, and the system can be viewed as a simplified model of fluid-structure interaction. We will demonstrate that in certain geometric configurations, there exists a unique time-periodic solution for each time-periodic right-hand side, assuming sufficient regularity of the forcing term. A counterexample illustrates that this regularity requirement is stronger than in the case of the Cauchy problem. Finally, we will discuss the open question of whether the result is valid for arbitrary geometry or if there exists a geometry where resonance can occur.[-]

We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get rapid stabilization for Korteweg-de Vries equations and how to stabilize in finite time $1-D$ parabolic linear equations by means of periodic time-varying feedback laws.[-]