The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
A De Giorgi's theorem asserts that every W^{1,2} weak solution of uniformly elliptic equations in divergence form is Holder continuous. However, that the solution is in W^{1,1} would be sufficient in the definition of weak solutions via integration by parts. In the first part, we discuss some regularity results on such weak solutions. In the second part, we move to some degenerate elliptic equations (also in divergence form) with Neumann boundary conditions, which are used in our study of a fractional Nirenberg problem. Some existence results of this problem will be introduced in the end.
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb R^n$, and the solutions are unique in an appropriate sense. We classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmen-Lindelof result as well as a principle of positive singularities in certain Lipschitz domains. This is a joint work with S. Armstrong and C. Smart
This talk concerns the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ``mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved (2012) and Strong Maximum principles (2012) play a crucial role in the analysis. This is joint work with Guy Barles, Cyril Imbert and Emmanuel Chasseigne.
I will present a new PDE approach to obtain large time behavior of Hamilton-Jacobi equations. This applies to usual Hamilton-Jacobi equations, as well as the degenerate viscous cases, and weakly coupled systems. This is the joint work with Cagnetti, Gomes, and Mitake.
The Abelian sandpile is a deterministic diffusion process for chips placed on the integer lattice. It is well known in part because the stable sandpiles are often beautiful fractal images. The sandpile dynamics have a continuum scaling limit that is captured by certain elliptic partial differential equation. Quite unexpectedly, this PDE has the structure of an Apollonian circle packing. This latent algebraic structure, when combined with the regularity theory of elliptic equations, explains the fractal images. It also implies a previously unknown Liouville theorem for integer superharmonic functions on the lattice.
In this talk we discuss a class of mean-field games which are Euler-Lagrange equations or perturbations of Euler-Lagrange equations of certain integral functionals. These integral functionals are convex but in many interesting examples not coercive. However due to their special structure it is possible to establish various a-priori estimates which in turn yield existence of smooth solutions. We will discuss various examples, basic techniques and will present several possible extensions.
In the variational implicit-solvent approach to biomolecular interactions, one minimizes a free-energy functional of all solute-solvent (e.g., protein-water) interfaces or dielectric boundaries. The functional couples together the solute surface energy, solute-solvent van der Waals interactions, and the electrostatic contributions. The last part is described by the Poisson-Boltzmann (PB) equation in which the dielectric coefficient takes one value in the solute region and a different value in the solvent region. In this talk, I will first review some properties of the PB equation and then focus on the dielectric boundary force in the variational approach to biomolecules. Such a force is defined as the negative shape derivative of the PB energy. I will derive an explicit formula of such a force, and apply it to study the motion of a cylindrical dielectric boundary driven by the competition between the surface energy and electrostatic energy. I will discuss the implications of the mathematical findings to the understanding of biomolecular interactions.
The travel time tomography problem consists in determining the index of refraction or sound speed of a medium by making travel time measurements. We will survey what is known about this problem including some recent results on the partial data case.
A drift-diffusion equation is like the heat equation with an extra first order term. In some cases, the Laplacian is replaced by a fractional Laplacian. There are several nonlinear models in a variety of contexts that fit into this scheme. In order to understand the solvability of the non linear models, it is essential to obtain a priori estimates on the smoothness of the solution for linear drift-diffusion equations when the first order term is given by a very irregular vector field times the gradient. We will analyze different smoothness estimates in different situations. It is particularly important to understand the consequences of assuming that the drift is divergence free, given their applications to models related to incompressible fluids.
