## PhD research topic - Emanuele Spadaro

#### Research areas

Calculus of Variations, Geometric Measure Theory, PDEs

#### Keywords

Minimal Surface Theory, Free Boundary Problems, Geometric Flows

#### Research topics

My research interests concern the analysis of solutions to partial differential equations arising in the geometric calculus of variations and in the variational models in applied mathematics. For example, the following are questions I am interested in.

– The minimal surface theory, whose prototypical example is the study of the solutions to the Plateau problem of finding the surfaces with minimal area spanning a given contour. There are many aspects of this theory which have been considered in the last decades and many remain to be investigated; moreover, several important results have been achieved recently through minimal surface theory, such as the solution to the positive mass conjecture in general relativity in any dimension by Schoen and Yau and the solution to the Willmore conjecture by Marques and Neves.

– Phase separations and evolutions arising in material science. Many interesting models in mathematical physics are described by order parameters which undergo phase separations (e.g., diblock co-polimers’ melts, solid-solid phase transitions, etc…) The analysis of these systems and of the geometric flows arising in their evolutions represent a big challenge for the mathematical analysis.

– Free boundary problems describe the behavior of systems whose boundary conditions are not explicitly described but are determined by the equilibrium configurations themselves. For example, this is the case of an elastic body at rest on a surface (the Signorini problem) or the phase transition in matter, as in ice formation (the Stefan problem). The analysis of the solutions to these partial differential equations is particularly challenging because of the free boundary conditions, which are not known a priori but need to be determined and geometrically analyzed.

####

Bibliography

[1] L. Simon. Lectures on geometric measure theory, (1983).

[2] E. Spadaro. Regularity of higher codimension area minimizing integral currents. Geometric measure theory and real analysis, (2014).

[3] C. De Lellis, E. Spadaro. Q-valued functions revisited. Mem. Amer. Math. Soc. (2011).

[4] L. Caffarelli. The obstacle problem revisited. J. Fourier Anal. Appl. (1998).

[5] L. A. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems, (2005).