(Università di Pisa, Pisa)
(SISSA, Trieste)
(Università del Salento, Lecce)
(Università di Genova)
(Università Bocconi, Milano)
Abstract:
We obtain a compactness result for Gamma-convergence of integral functionals defined on A-free vector fields. This is used to study homogenization problems for these functionals without periodicity assumptions. More precisely, we prove that the homogenized integrand can be obtained by taking limits of minimum values of suitable minimization problems on large cubes, when the side length of these cubes tends to infinity, assuming that these limit values do not depend on the center of the cube. Under the usual stochastic periodicity assumptions, this result is then used to solve the stochastic homogenization problem by means of the subadditive ergodic theorem.
Abstract:
If we consider the topic of linearisation of finite elasticity for pure traction problems, it may happen that the limiting minimal value is strictly lower than the minimal value of standard linear elastic energy if a strict compatibility condition for external loads does not hold. This fact would exclude from the approximation all those problems in linear elasticity in which loads satisfy the usual compatibility conditions but not those of strict compatibility. We then provide an approximation result for pure traction problems in linear elasticity in terms of critical points of finite elasticity requiring only the usual compatibility conditions.
Abstract:
Dissipative evolutions of probability measures are naturally associated with monotone operators in $L^2$ spaces of random variables which are invariant with respect to measure-preserving isomorphisms. They provide an interesting example of monotone operators in Hilbert spaces which are invariant with respect to a group of linear isometries. Since maximality plays a crucial role in the theory of monotone operators and the associated contraction semigroups, it is then interesting to investigate whether every monotone operator which is invariant for a group of isometries admits a maximal extension which preserves this property, so that invariance can also be inherited by its resolvents, the Yosida approximations, and the associated contraction semigroup. Using the theory of self-dual Lagrangian representations, we prove that such an extension always exists.
A similar result holds for Lipschitz maps, thus giving an invariant version of the Kirzsbraun-Valentine extension Theorem.
(In collaboration with Giulia Cavagnari and Giacomo Sodini)
Abstract:
We study a generalized form of the Cheeger inequality by considering the shape functional $F_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q^{1/q}(\Omega)$, where the original Cheeger case corresponds to $p=2$ and $q=1$. Here $\lambda_p(\Omega)$ denotes the principal eigenvalue of the Dirichlet $p$-Laplacian. The infimum and the supremum of $F_{p,q}$ are discussed, together with the existence of optimal domains. Some open problems will be illustrated as well.
The workshop will take place at:
Aula Consiglio - 7th floor, Building 14,
Politecnico di Milano, Via Bonardi 9, 20133 Milano (Italy)
