Research Statement

Variational problems and weak convergence methods for Nonlinear PDE

My motivation for this study is condensed matter Physics. A governing principle for many physical systems is the minimization of an appropriate free energy. This energy usually has many scales reflecting the multiplicity of the relevant physical processes. Energy minimization can then lead to a variety of multiple scale structures including point and line singularities (vortices), microstructure, domain wall branching and multiple scale oscillations. In recent work (Papers # 1, 2, 3 and preprints #2, 3 in my publication list) my collaborators and I have studied the development of singularities and microstructure in thin elastic sheets.

In the long term, I am interested in the general principles that underlie the spontaneous formation of structures in the minimizers for variational problems. In this context I am studying the problem of minimizing the functional $\mathcal {E}$^$\epsilon$,

$$\mathcal {E} \epsilon \int_{{\mathcal{S}}}^{} \epsilon^{2}_{} \int_{{\mathcal{S}}}^{}$$ (R-VP)

where W[.] is nonconvex, F is convex in D²u and $\epsilon$ is a small parameter reflecting the multiple energy scales in the problem.

For $\epsilon$ = 0, we have a nonconvex variational problem which, in general, does not have a minimizer. Minimizing (sub)sequences could develop rapid oscillations or could concentrate energy on sets of measure zero.

For any $\epsilon$ > 0 however, the second term in $\mathcal {E}$^$\epsilon$ introduces a small scale cut-off and regularizes the nonconvex variational problem by a singular perturbation. We thus have a minimizer u^$\epsilon$ for all $\epsilon$ > 0. We want to characterize the singular limit of the variational problems $\mathcal {E}$^$\epsilon$ as $\epsilon$ $\rightarrow$ 0. This leads to the following questions, whose answers depend on the precise functional forms of W and F:

Morphology

Do the minimizers u^$\epsilon$ display oscillations or concentration effects? Describe the oscillatory/singular regions in the minimizers.

Scaling laws

How do the energy $\mathcal {E}$^$\epsilon$(u^$\epsilon$), and the length scales associated with the oscillations/singular regions in the minimizer u^$\epsilon$ depend on $\epsilon$?

Variational Convergence

Does $\lim_{{\epsilon \rightarrow 0}}^{}$u^$\epsilon$ exist, in some appropriate sense? If so, is there a limiting energy $\bar{{\mathcal E}}$ whose minimizer is $\lim_{{\epsilon \rightarrow 0}}^{}$u^$\epsilon$?

The model problem (R-VP) is related to fundamental questions in geometric analysis, and the regularity of solutions to elliptic equations. Answering the three questions above, even for specific W and F, will represent important advances in the study of variational problems. Finally, a successful analysis of (R-VP) will directly impact our understanding of the structures that arise in condensed matter systems.

Extended dynamical systems and phase transitions

Pattern formation is an ubiquitous phenomenon in extended nonlinear systems. Patterns result from the interplay of many factors including nonlinearities, external forcing and/or excitability of the medium, spatial interactions, and internal dissipation. They are therefore prototypical ``multiple scale'' phenomena.

In recent work (Paper # 9) we introduced Continuum Coupled Maps, a new framework for studying pattern formation in periodically forced systems. A continuum coupled map (CCM) is a dynamical system defined on the space of smooth functions $\xi$ : D $\rightarrow$ $\mathbb {R}$ by

$$\xi_{{n+1}}^{}$$ = G*[Mo$$\xi_{n}^{}$$],

where D is a continuous spatial region, M is a nonlinear map, G is a smooth kernel, o denotes functional composition and * denotes convolution. M and G introduce nonlinearity and spatial coupling respectively. This is a powerful framework and allows one to model non-variational dynamical systems, i.e. systems not governed by an energy functional. I am currently investigating domain walls, coarsening and other multiple scale behaviors using this framework.

The statistical mechanics of extended non-equilibrium systems is an outstanding open problem in physics. Pattern formation is a non-equilibrium process that forms coherent structures despite thermal fluctuations, and therefore gives an approach to studying non-equilibrium statistical mechanics. This is a difficult problem, since one has to account for multiple scale behavior coupled with random fluctuations. It often has to be treated outside a perturbation framework since the interactions between the fluctuations can give rise to large effects (fluctuation renormalization).

My collaborators and I are guiding Rui Zheng in thesis research. He is investigating the interplay between pattern formation and thermal fluctuations in the Swift-Hohenberg equation, using numerical methods that were recently developed by Kevrekidis and coworkers for the analysis of complex systems with multiple spatial/temporal scales.

My research in this area has two main directions. One direction is the analysis of phenomenological equations that arise in statistical mechanics (e.g. papers # 4,5,8,9). Another direction is a mathematical justification for Renormalization Group (RG) methods, that are non-rigorous but extremely successful in analyzing phase transitions. My long term interests are a rigorous analysis of statistical mechanical systems and phase transitions, especially in an non-equilibrium setting.

The formation and dynamics of singularities

My collaborators and I (Paper # 7) investigated the parabolic-elliptic system of equations

$$\partial_{t}^{} \rho \Delta \rho \nabla \rho \nabla$$

$$\Delta \rho$$

This is a version of the Keller-Segel system that was first introduced to model bacterial chemotaxis, and is a ``generic'' model for a parabolic system governing a locally conserved density $\rho$ with the competing effects of diffusion and nonlinear growth.

This system is a gradient flow, and it has a non-increasing energy. The equations support steady solutions as well as solutions that blowup in finite time. In analyzing the asymptotic states for generic initial conditions, we discovered a surprising similarity between the infinite dimensional PDE and Morse theory for finite dimensional dynamical systems. For the PDE, the asymptotic states are described by the various blowup modes, the steady solutions, and their ``unstable manifolds''. This is analogous to the $\omega$-limit sets for finite dimensional gradient flow which consists of equilibrium points and their unstable manifolds. This analogy is not just qualitative, but also gives quantitative predictions, e.g. for the blowup time of a solution.

A similar connection with finite dimensional dynamical systems is also observed in the numerical simulations of Einstein's equations for initial conditions at the threshold of collapse to black-hole formation. This phenomenon gives us an approach to investigate the important cosmic censorship conjecture which postulates that there are no ``naked'' black hole singularities.

A similar connection with finite dimensional dynamical systems is also observed in the numerical simulations of Einstein's equations to test the important cosmic censorship conjecture which postulates that there are no ``naked'' black hole singularities.

The formation and the dynamics of singularities are important in many contexts ranging from geometry (e.g. recent work by Perleman on the Geometrization conjecture) to physics (e.g. Black holes) and even technology (e.g. droplet breakoff in inkjet printers). A rigorous relationship between singularity formation and finite dimensional dynamics will therefore have a significant impact on a variety of problems.

Isometric immersions and underdetermined PDEs

I am interested in the rigidity of underdetermined PDEs, i.e. the global consequences of local differential constraints. Along with my collaborators (paper # 6), I analyzed the existence of smooth (C³) isometric immersions $\phi$ : B₁^m $\rightarrow$ B_$\epsilon$^d of the unit ball in $\mathbb {R}$^d into an $\epsilon$-ball in $\mathbb {R}$^d, and proved that such immersions exist for all $\epsilon$ > 0 if d $\geq$ 2m, but no such immersions exist if $\epsilon$ < 1/2 and d < 2m. At the present time, I am examining similar questions for isometric immersions of portions of the hyperbolic plane $\mathbb {H}$² into $\mathbb {R}$³, with certain curvature restrictions.

The variational problem (R-VP) is intimately connected with the underdetermined PDE W(Du) = 0. I am investigating the relation between bounds on D²u and the rigidity of solutions to W(Du) = 0. In many cases, including the isometric immersion problem, Gromov's theory of convex integration gives a large number of C¹ solutions to W(Du) = 0. However, the solution set can become empty if we impose restrictions on the size of the curvature (paper # 6 and preprint # 3).

Multiple tools for Multiple scales

As I have detailed above, my long term research interest is the analysis of PDEs (and stochastic PDEs) that display multiple scale behavior. Such problems arise in a variety of contexts and are studied in a wide range of disciplines.

The techniques for rigorous multiple-scale analysis are functional analytic, and include Compensated Compactness, Relaxation, Young Measures, Homogenization, H-measures and $\Gamma$-convergence. More recently, geometric techniques such as convex integration have found application in the analysis of variational and under-determined nonlinear PDE, and the regularity theory for variational/elliptic PDE.

In the applied mathematics and physics communities, the methods of choice include matched asymptotics, similarity solutions, scaling arguments, averaging methods, renormalization and modeling/numerical simulations.

In my work, I have found it fruitful to exploit many of these approaches, and not restrict myself to particular ways of thinking. A combination of numerical simulations and ``non-rigorous'' analysis often provides useful conjectures and guides the rigorous analysis of a problem. Conversely, a rigorous understanding of the types of possible behaviors leads to ``good'' choices for approximate solutions or the appropriate method for numerical simulation. In addition, much of my work has been motivated by experiment, and in turn has sparked new experiments. My long term approach to research will be along these lines, integrating analytic, geometric and numerical techniques, and closely collaborating with people in other disciplines.