| Abstract: |
| Two classical research topics in the calculus of variations are the study of regularity results for minimizers, with values in a manifold, of functionals with $p$-growth, whose foundations date back to the works of Eells \& Sampson, Schoen \& Uhlenbeck, Fuchs, Hardt \& Lin and Luckhaus, and the density of smooth maps between compact manifolds in Sobolev spaces, characterized by Bethuel, Hang \& Lin, and Hajlasz in terms of the topological properties of the target.
In this talk we extend these themes to the nonuniformly elliptic setting of functionals with $(p,q)$-growth, with particular emphasis on double phase energies. We show that vector-valued minimizers subject to certain manifold constraints enjoy partial regularity of the gradient, that is, regularity of the solutions outside a negligible closed subset. We further investigate necessary and sufficient conditions for the density of smooth maps between suitable compact manifolds in nonhomogeneous spaces characterized by the finiteness of anisotropic energies such as those of double phase type. From joint works with C. A. Antonini, A. Nastasi, and C. Pacchiano Camacho. |
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