Functional inequalities and nonlocal equations

One part of this project concerns functional inequalities in Sobolev spaces. These problems have a long history and arise in a variety of applications. We intend to develop methods in order to study existence, uniqueness and stability of extremals. This will result in new results, methods and tools.

The focus of our studies will be functional inequalities of Sobolev- Morrey- and Hardy-type.

Typical methods: linearization, blow-up, symmetrization and convexity arguments.We will also develop various tools along with the regularity theory of related nonlinear nonlocal equations. In most classical models of diffusion, the diffusive process is assumed to occur infinitesimally. This gives rise to partial differential equations.

However, there are many cases in which the diffusion interaction occurs at a long range. Models of such phenomena therefore give rise to nonlocal or fractional equations.

Typical methods: Moser-type iterations, linearization, approximation, symmetrization and viscosity methods.The time frame is four years and the research will be conducted by the PI, a PhD student and other possible collaborators.