In a series of collaborations, the PI and Russell Schwab studied nonlinear problems involving highly oscillatory boundary conditions by reducing it to a lower dimensional, nonlocal problem. The first result in the series "Neumann Homogenization via Integro-Differential Operators" Part I dealt with the case of a translation invariant equation posed on a trip, providing a new proof of a result of Choi, Kim, and Lee. The next paper in the series, "Neumann Homogenization via Integro-Differential Operators, Part 2: singular gradient dependence", takes the method one step further to deal with linear, but variable coefficient operators, answering in particular a problem posed in previous work of Barles,Da Lio,Lions, and Souganidis.  The integro-differential method to these boundary layer problems is such that further advances in the theory of integro-differential operators will immediately translate into new results for these oscillatory boundary layer problems.

In a different work with Schwab, we provide an answer to an unresolved question in the general theory of integro-differential equations, namely: how large is the class of nonlocal "elliptic" operators? We reformulate this question by determining which is the class of (possibly nonlinear) operators satisfying the global comparison principle (GCP). Our main result, presented in "Min-max formulas for nonlocal elliptic operators" says that the operators satisfying the GCP are exactly those that may be represented as a min-max of operators of Lévy type. As such, our result provides a nonlinear analogue to a classical linear result by Courrège. Besides resolving a gap in the integro-differential theory, this result opens the way for applications of integro-differential methods in various settings: particularly (local) boundary value problems --such as those involving boundary layers, as well as free boundary and interfacial problems arising in fluid mechanics and material science.

In an entirely different direction, work with Maria Gualdani was completed leading to a new estimate for radial solutions to the (spatially homogeneous) Landau equation with Coulomb potential. The Landau equation is a (nonlocal) partial differential equation which plays an important role in plasma physics,the quantitative understanding of its weak solutions is great importance in understanding instabilities and long time behavior for a plasma, and it has posed a significant mathematical challenge even to this day. The solution to the equation represents the mass distribution for a particles in a plasma (thus, the Coulomb interaction), and the dynamics display a competition between the the tendency to diffuse and to concentrate. An important unresolved question is whether the mass density may become arbitrarily large at a given point. Analogues of this equation where the particle interaction is taken to be not Coulombic but some other stronger repulsive force (Maxwell spheres, moderately soft potentials) have been understood for a long time.  The work completed with Gualdani, "Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential" is one further step towards showing the immediate regularization and global well posedness for this equation. We apply barrier arguments and tools from the theory of parabolic equations to provide estimates on the solution conditioned on the validity of a certain "borderline" inequality. At the same time, an unconditional result is proved for an isotropic analogue of the Landau equation first studied by Krieger and Strain. This result, although far from an answer to the question or regularity for the Landau equation, provides strong evidence that general solutions to the original equation become regular.

 Lastly, in work with Jun Kitagawa we prove pointwise estimates for a class of equations introduced by Neil Trudinger, known as "Generated Jacobian Equations". This is a very broad class of degenerate elliptic equations that encompass problems in optimal transport, prescription of curvature, but which goes beyond optimal transport theory and contains equations that play a role in the design of reflective/refractive surfaces in geometric optics, as well as in certain problems in optimal matching theory with non quasilinear utility functions. The results with Kitagawa complement in the case of near field reflectors earlier work of Wang and Karakhanyan and extends previous work by Figalli, Kim, and McCann in optimal transport.

Last Modified: 03/09/2017
Modified by: Nestor Guillen