Partial Differential Equations Conference

Abstracts of Plenary Lectures

Douglas Arnold

Complexes from complexes

Abstract: The finite element exterior calculus has highlighted the importance of Hilbert complexes to partial differential equations and their numerical solution. Hilbert complexes arise throughout mathematical physics. The fundamental partial differential operators from which most models in continuum physics are built may be realized as unbounded operators mapping between Sobolev and related Hilbert spaces, and these spaces and operators assemble into chain complexes. The resulting structure is a Hilbert complex : a finite sequence of Hilbert spaces together with closed unbounded operators from one space to the next such that the composition of two consecutive operators vanishes. This is a rich structure which combines functional analysis with homological algebra.

The most canonical and most extensively studied example of a Hilbert complex is the de Rham complex, which is what is required for application to fluid mechanics, electromagnetics, the Hodge Laplacian, and numerous other problems. This may give the impression that the finite element exterior calculus is only another way to look at these applications, which can be approached with less complicated machinery. But, in fact, there are many other important differential complexes as well, with applications to elasticity, plates, incompressible flow, general relativity, and other areas. These complexes are less well known and in many cases their properties, including properties needed to fit them into the finite element exterior calculus framework, have not been established.

In this talk I will discuss a systematic procedure for deriving such complexes and establishing their crucial properties.

Alfonso Castro

Multiplicity of solutions for quasilinear singular ordinary differential equations

Abstract: Motivated by the study of radial solutions to boundary value problems defined by perturbations of p-Laplacian equations we will discuss the role of energy functions, Pohozaev identities, phase plane analysis and blow up solutions in the existence of multiple solutions to two point boundary value problems. We will emphasize recent results where the perturbation includes a weight function such that the problem may be superlinear in a subregion but sublinear in its complement. Related results will be presented by Professors Jorge Cossio and Carlos Velez.

Mónica Clapp

Optimal partitions for the Yamabe equation

Abstract: The Yamabe equation on a Riemannian manifold (M,g) is of relevance in differential geometry. A positive solution to it gives rise to a metric on M which has constant scalar curvature and is conformally equivalent to the given metric g.

An optimal \ell-partition for the Yamabe equation is a cover of M by \ell pairwise disjoint open subsets such that the Yamabe equation with Dirichlet boundary condition has a least energy solution on each one of these sets, and the sum of the energies of these solutions is minimal.

We will present some recent results obtained in collaboration with Angela Pistoia (La Sapienza Universit‡ di Roma) and Hugo Tavares (Universidade de Lisboa) that establish the existence and qualitative properties of such partitions.

Xiaobing Feng

Recent developments in numerical fully nonlinear second-order PDEs

Abstract: Fully nonlinear PDEs are referred to as the class of nonlinear PDEs which are nonlinear in the highest order derivatives of the unknown functions appearing in the equations, they arise from many fields in science and engineering such as astrophysics, antenna design, differential geometry, geostrophic fluid dynamics, materials science, mathematical finance, meteorology, optimal transport, and stochastic control. This class of PDEs is known to be difficult to analyze analytically and to approximate numerically. In this talk, I shall first present a brief overview about some latest advances in numerical fully nonlinear PDEs and then discuss in detail a newly developed narrow-stencil finite difference and discontinuous Galerkin framework for approximating viscosity (and smooth) solutions of fully nonlinear second-order PDEs (such as Hamilton-Jacobi-Bellman and Monge-Ampere equations). The focus of the talk will be on discussing various numerical approaches/methods/ideas and their pros and cons for constructing numerical methods which can reliably approximate viscosity (and smooth) solutions of fully nonlinear second-order PDEs. Sample numerical experiments and application problems will also be presented.

Beatrice Rivière

Convergence of numerical methods for porous media flows

Abstract: Simulations of two-component single phase flows and two-component two-phase flows in heterogeneous porous media at the Darcy scale are important in the understanding of flow and transport processes in subsurface. Applications include storage of carbon dioxide in saline aquifers and production of hydrocarbons from oil and gas reservoirs. Mathematical models are based on conservation of mass for each fluid component. Models are characterized by systems of nonlinear coupled partial differential equations. In this talk, we formulate finite element and discontinuous Galerkin methods for the miscible displacement and incompressible two-phase flow problems. Challenges in the numerical analysis arise from the nonlinearity and the unboundedness or degeneracy of some of the PDE coefficients. Recent advances on the theoretical convergence of the proposed schemes for the discretization of the mathematical models in the general case are presented. Accuracy and robustness of the numerical methods for heterogeneous porous media are demonstrated.

Peter Poláčik

Quasiperiodic solutions of elliptic equations on the entire space via center manifold and KAM theorems

Abstract: In this project, we consider positive solutions of nonlinear elliptic equations on the Euclidean space of N+1 dimensions with some predetermined behavior (such as decay and symmetry) in the first N variables. We examine the behavior of these solutions in the remaining variable. Families of solutions periodic in the last variable have been found by several authors; our goal is to prove the existence of quasiperiodic solutions. In the lecture, I will discuss some general techniques based on center manifold and KAM theorems that we are using for this goal. Then I will report on recent progress. This is joint work with Dario Valdebenito.