Nonlinear Analysis and Partial Differential Equations

Researchers

Juan Dávila, Manuel Del Pino, Patricio Felmer, Michal Kowalczyk, Fethi MahmoudiRaúl Manasevich, Salomé Martínez

About the Research Group

Our research efforts are primarily focused on the qualitative study of solutions to nonlinear partial differential equations with emphasis in elliptic or parabolic. In general, we work towards understanding the structure and asymptotic properties of sets of solutions to specific equations. Problems of the type that interest us arise in diverse fields of endeavor such as material science, astronomy, the theory of combustion, superconductivity and mathematical biology.

On the other hand, an understanding of these problems has lead to the development of powerful tools in different branches of mathematics, for instance, such knowledge was a key ingredient in the recent resolution of the famous Poincaré conjecture. Such problems have posed questions which, while being easy to state, are difficult to solve and have historically driven developments in mathematical analysis, giving rise in recent decades to advances in the calculus of variations and nonlinear functional analysis, subjects at the interface between analysis, geometry and topology.

In recent years, scientific computation has proved to be a powerful ally in the study of these problems, having evinced phenomena that are difficult to detect, even from a formal standpoint. Their mathematical understanding cannot be reached without facing a battery of complex challenges and the consequent necessity to develop increasingly sophisticated analytical methods to understand them.

Lines of research in which we have made important contributions are as follows:

Problems out of Equilibrium

Here, we produced a series of articles examining parabolic type equations of evolution. We have studied the role of the Sobolev-Gagliardo-Nirenberg and Gross-Sobolev optimal embeddings in the asymptotic behavior of equations for rapid diffusion in a porous medium. In addition, we have obtained new results in blow-up phenomena and vanishing in finite time.

In another line of work, we have studied models of Brownian ratchets and their transport phenomena pertinent to molecular biology. We have studied competition systems relevant to ecological modeling, namely, segregation and stability induced in spatially inhomogeneous crossed diffusions. Furthermore, we have obtained fundamental results for evolution equations in which the diffusive term is replaced by non-local dispersion operators, objects which are natural to use in models but which are not well understood mathematically.

Singular Perturbation

This deals with problems having parameters which induce the formation of solutions with patterns of concentration in the form of measures or singular solutions supported on sets of lower dimension when taken to their limits. In several works, we have examined the semi-classical limit in nonlinear Schrödinger equations. Also, a conjecture about the presence of solutions concentrated on curves was validated using methodology that was also employed in the construction of multiple interfaces for the Allen-Cahn equation.

Criticality

The role of critical exponents for Sobolev embeddings in the solvability of elliptic problems which involve the Laplacian operator has long been known to mathematicians. Much less clear is the meaning of this exponent in nonlinear problems, as well as linear problems when the form of divergence is not present. We have considered extremal Pucci operators and defined a new notion of criticality whose implications have been explored in various articles, at the same time introducing a new line of investigation dealing with the solvability of these problems by means of viscosity solutions.

Problems with p-Laplace Operators and their Extensions

This line of research focuses on equations and systems involving second-order operators having nonlinear forms of divergence, specifically the p-Laplacian and non-homogeneous extensions (j-Laplacian). These studies are motivated by problems arising in mechanics in continuous media. We recently showed that the first eigenvalue in the vector case coincides with that in the corresponding scalar problem.