Seminario del 2016

The minimizers of integral functionals of the Calculus of Variations and the solutions of PDE's in divergence form are related by the Euler's equation. The main areas of research concern the proof of the existence of minimizers/solutions and the study of their regularity. Two milestones in the regularity theory are due to De Giorgi. In 1957 he proved the local Holder continuity of solutions to linear elliptic equations in divergence form with measurable coefficients. An example by De Giorgi himself, in 1968, shows that linear elliptic systems can have solutions not only discontinuous, but even locally unbounded. Since then, the theory of regularity has been hugely developed, in many directions, both in Calculus of Variations and in PDE's. In this "Topics" lecture, I will describe some general facts and some results.