**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.

**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).
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.