# Past Partial Differential Equations Seminar

26 May 2014
17:00
Abstract
We present a new method to establish the rotational symmetry of solutions to overdetermined elliptic boundary value problems. We illustrate this approach through a couple of classical examples arising in potential theory, in both the exterior and the interior punctured domain. We discuss how some of the known results can be recovered and we introduce some new geometric overdetermining conditions, involving the mean curvature of the boundary and the Neumann data.
• Partial Differential Equations Seminar
19 May 2014
17:00
Abstract
In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem $$- div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$ $$u = 0 \; {\rm on} \; \partial \Omega,$$ (namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition), where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type $$F (x, u) = {f(x) \over u^\gamma} + g(x)$$ with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative). The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$. We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.
• Partial Differential Equations Seminar
12 May 2014
17:00
Jean Van Schaftingen
Abstract
I will show how families of concentrating stationary vortices for the shallow water equations can be constructed and studied asymptotically. The main tool is the study of asymptotics of solutions to a family of semilinear elliptic problems. The same method also yields results for axisymmetric vortices for the Euler equation of incompressible fluids.
• Partial Differential Equations Seminar
5 May 2014
17:00
Donatella Danielli-Garofalo
Abstract
Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions.
• Partial Differential Equations Seminar
28 April 2014
17:00
Jean-Philippe Nicolas
Abstract
The conformal approach to scattering theory goes back to the 1960's and 1980's, essentially with the works of Penrose, Lax-Phillips and Friedlander. It is Friedlander who put together the ideas of Penrose and Lax-Phillips and presented the first conformal scattering theory in 1980. Later on, in the 1990's, Baez-Segal-Zhou explored Friedlander's method and developed several conformal scattering theories. Their constructions, just like Friedlander's, are on static spacetimes. The idea of replacing spectral analysis by conformal geometry is however the door open to the extension of scattering theories to general non stationary situations, which are completely inaccessible to spectral methods. A first work in collaboration with Lionel Mason explained these ideas and applied them to non stationary spacetimes without singularity. The first results for nonlinear equations on such backgrounds was then obtained by Jeremie Joudioux. The purpose is now to extend these theories to general black holes. A first crucial step, recently completed, is a conformal scattering construction on Schwarzschild's spacetime. This talk will present the history of the ideas, the principle of the constructions and the main ingredients that allow the extension of the results to black hole geometries.
• Partial Differential Equations Seminar
10 March 2014
17:00
Valeriy Slastikov
Abstract
We study liquid crystal point defects in 2D domains. We employ Landau-de Gennes theory and provide a simplified description of global minimizers of Landau- de Gennes energy under homeothropic boundary conditions. We also provide explicit solutions describing defects of various strength under Lyuksutov's constraint.
• Partial Differential Equations Seminar
3 March 2014
17:00
Paolo Marcellini
Abstract
Motivated by integrals of the Calculus of Variations considered in Nonlinear Elasticity, we study mathematical models which do not fit in the classical existence and regularity theory for elliptic and parabolic Partial Differential Equations. We consider general nonlinearities with non-standard p,q-growth, both in the elliptic and in the parabolic contexts. In particular, we introduce the notion of "variational solution/parabolic minimizer" for a class of Cauchy-Dirichlet problems related to systems of parabolic equations.
• Partial Differential Equations Seminar
24 February 2014
17:00
Didier Bresch
Abstract
In this talk, we will discuss low Weissenberg number effects on mathematical properties of solutions for several PDEs governing different viscoelastic fluids.
• Partial Differential Equations Seminar
17 February 2014
17:00
Christoph Thiele
Abstract
An old conjecture by A. Zygmund proposes a Lebesgue Differentiation theorem along a Lipschitz vector field in the plane. E. Stein formulated a corresponding conjecture about the Hilbert transform along the vector field. If the vector field is constant along vertical lines, the Hilbert transform along the vector field is closely related to Carleson's operator. We discuss some progress in the area by and with Michael Bateman and by my student Shaoming Guo.
• Partial Differential Equations Seminar
10 February 2014
17:00
Abstract
We consider equations with the simplest hysteresis operator at the right-hand side. Such equations describe the so-called processes "with memory" in which various substances interact according to the hysteresis law. The main feature of this problem is that the operator at the right-hand side is a multivalued. We present some results concerning the optimal regularity of solutions. Our arguments are based on quadratic growth estimates for solutions near the free boundary. The talk is based on joint work with Darya Apushkinskaya.
• Partial Differential Equations Seminar