Abstract

Partial differential equations (PDE) are a rich and deep area of mathematics which is strongly motivated by problems arising in applied sciences and technology, such as physics, engineering, medicine. Most physical phenomena appearing in reality can indeed be modeled through PDE. Unfortunatley, explicit solutions to PDE can only be determined in very few special cases. On the other hand, it is often possible to describe some relevant qualitative and quantitative properties of theirs. This theoretical analysis enables, for instance, to develop effective numerical approximation methods for solutions. The workshop focuses on some fundamental aspects of the theory of PDE, including existence of solutions, their regularity and a priori estimates. Information on these issues allows for a deeper understanding of the structure of solutions and of the physical phenomena that they describe. They will be addressed in connection with some of the most advanced developments of the theory of PDE, that rely upon methods and results from Nonlinear Potential Theory, Harmonic Analysis, Geometric Analysis.