The theory of nonlinear PDEs is of fundamental importance in mathematical analysis, as well as an active area of research. We consider a large class of singular, degenerate and fractional nonlinear PDEs not only in the Euclidean spaces but also in more general contexts. These equations occur, for example, in diffusion flow through a porous medium, total variation flow and minimal surfaces.
The research focuses mainly on theoretical aspects of PDEs. A crucial role in understanding nonlinear phenomena is played by regularity estimates. Their significance is driven not only by results but also by techniques from the Calculus of Variations, Functional Analysis, Harmonic Analysis, PDEs and Real Analysis that are powerful enough to solve previously unreachable problems.
The main topics are
- equations and systems of parabolic p-Laplacian type, porous medium equation, fractional equations, nonlinear potentials, obstacle problems, capacities
- maximal functions, weighted norm inequalities, bounded mean oscillation
- analysis on metric measure spaces (Sobolev spaces, functions of bounded variation, quasiminimizers, functionals with linear growth, minimal surfaces)
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