Research interests

Calculus of Variations and Optimization:
  • minimal surfaces, surfaces of prescribed mean curvature;
  • geometric and functional inequalities, isoperimetric stability;
  • small-strain rate-dependent theories in deformable solid mechanics;
  • shape optimization, spectral optimization;
  • geometric measure and integration theory;
  • variational problems in a geometric measure-theoretic setting.
Qualitative properties of solutions to PDEs:
  • variational methods for second-order elliptic equations;
  • maximum and comparison principles for solutions;
  • smoothness and regularity of solutions;
  • elliptic equations on manifolds;
  • nonlinear spectral theory for the p-laplacian
  • properties of eigenfunctions of non–local operators.

Publications

Articles appeared on international journals.

  • De Philippis, Franzina, Pratelli. Existence of isoperimetric sets with densities converging from below on ${\mathbb {R}}^N$ , Journ. of Geom. Anal. (2016).
  • Brasco, Franzina. Convexity properties of Dirichlet integrals and Picone- type inequalities, Kodai Math. Journal (2014).
  • Franzina, Palatucci. Fractional p-eigenvalues, Riv. Mat. Univ. Parma (2014).
  • Brasco, Franzina. An isotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities, NoDEA (2013).
  • Franzina, Lindqvist. An eigenvalue problem with variable exponents, Nonlinear Anal., (2013).
  • Brasco, Franzina. On the Hong-Krahn-Szego inequality for the p-Laplace operator, Manuscripta Math. (2013).
  • Brasco, Franzina. A note on positive eigenfunctions and hidden convexity, Arch. der Mathematik (2012).
  • Franzina, Lamberti. Existence and Uniqueness for a p≠laplacian eigenvalue problem, Electron. J. Diff. Equations (2010).
  • Proceedings of international conferences.

  • Franzina, Valdinoci. Geometric analysis of fractional phase transition interfaces, in ''Geom. Prop. for Parabolic and Elliptic PDE's'', Springer (2013).
  • PhD Thesis.

  • Franzina. Existence, Uniqueness, Optimization and Stability for Low Eigenvalues of some Nonlinear Operators, Repo UniTn-eprints.PhD (2012).
  • Work in progress

  • Franzina, Luu, Pisante, Ponsiglione. Phase Transitions and hypersurfaces of constant mean curvature in the hyperbolic space, in preparation.
  • Dal Maso, Franzina, Zucco. Gamma–limits of Neumann problems for the Laplace operator, in preparation.