## Past Seminars- Nonlinear PDEs

• Olga Turanova
Tue Apr 24, 2018
3:00 pm
This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises via the incompressible limit of this model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.
• Jeremy LeCrone
Tue Feb 13, 2018
3:00 pm
In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in...
• Hongzi Cong
Thu Jan 25, 2018
3:00 pm
In this talk, we will show  that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62–94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.
• Yi Zhang
Tue Jan 16, 2018
3:00 pm
Given a planar infinity harmonic function u, for each $\alpha>0$ we show a quantitative $W^{1,\,2}_{\loc}$-estimate of $|Du|^{\alpha}$, which is sharp when $\alpha\to 0$.  As a consequence we obtain an $L^p$-Liouville property for infinity harmonic functions in the whole plane
• Gautam Iyer
Thu Jan 11, 2018
4:00 pm
Consider a diffusive passive scalar advected by a two dimensional incompressible flow. If the flow is cellular (i.e.\ has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. We show that on intermediate time scales, the effective behaviour...
• Albert Fathi
Tue Dec 5, 2017
3:00 pm
This is a joint work with Piermarco Cannarsa and Wei Cheng.  We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.  The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property...
• Dejan Slepcev
Tue Oct 17, 2017
3:00 pm
The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov,  asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and...