Organizers: Giovanni Catino and Fabio Cipriani

**Colette De Coster**, Univ. Valenciennes,

*A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis*, Tuesday, July 02, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk, we survey some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation
\begin{equation*}
{\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},
\end{equation*}
in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, with $a,b>0$ parameters.
This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids.
In this talk, we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.
This seminar is organized within the PRIN 2015 Research project «Partial Differential Equations and related Analytic-Geometric Inequalities» Grant Registration number 2015HY8JCC _003, funded by MIUR – Project coordinator Prof. Filippo Gazzola
**Franco Tomarelli**, Politecnico di Milano,

*Pure Traction Problems between Linear and Finite Elasticity*, Wednesday, June 19, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energies of a hyperelastic material body subject to an equilibrated force field.
We show that the strains of minimizing sequences associated to re-scaled nonlinear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy functional exhibits a gap that makes it different from the classical linear elasticity functional; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded
from below, while a mild violation may produce unboundedness of strains and a limit energy which has infinitely many extra minimizers which are not minimizers of standard linear elastic energy. A consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that infringe such compatibility condition.
**Francesca Colasuonno**, Università degli Studi di Torino,

*Symmetry preservation for fourth order eigenvalue optimization problems*, Wednesday, June 12, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk, we will discuss some recent results on two eigenvalue optimization problems governed by the biharmonic operator, under Dirichlet or Navier boundary conditions. From a physical point of view, in two dimensions our problem corresponds to building a plate, of prescribed shape and mass, out of materials with different densities --varying in a certain range of values-- in such a way to minimize the lowest frequency of the body. This problem is also referred to as composite plate problem. Both for the clamped plate (i.e., Dirichlet b.c.) and for the hinged plate (i.e., Navier b.c.), we will prove the existence of an optimal configuration and give an explicit representation of the minimizing densities in terms of sublevel sets of their corresponding first eigenfunctions. Finally, we will discuss symmetry preservation properties of the optimal configurations, in the presence of some symmetry and convexity of the domain. The tools used differ depending on the boundary conditions: while the hinged plate problem inherits the maximum principles for second order elliptic systems, allowing us to exploit the moving plane method to get symmetry preservation in more general domains, the situation is more complicated for the clamped plate problem, where we will use the polarization technique and the properties of the Green's function to deal with radial symmetry preservation in a ball.
This talk is based on two joint papers with Eugenio Vecchi (Trento).
**Filippo Riva**, SISSA,

*On the quasistatic limit for a debonding model in dimension one; a vanishing inertia and viscosity approach*, Thursday, June 06, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
In the theory of linearly elastic fracture mechanics one-dimensional debonding models, or peeling tests, provide a simplified but still meaningful version of crack growth models based on Griffith's
criterion. They are both described by the wave equation in a time-dependent domain coupled with suitable energy balances and irreversibility conditions.Unlike the general framework, peeling tests allow to deal with a
natural issue of great interest arising in fracture mechanics. It can be stated as follows: although all these models are dynamic by nature, the evolution process is often assumed to be quasistatic (namely the
body is at equilibrium at every time) since inertial effects can be neglected if the speed of external loading is very slow with respect to the one of internal oscillations. Despite this assumption seems to
be reasonable, its mathematical proof is really far from being achieved.In this talk we validate the quasistatic assumption in a particular damped debonding model, showing that dynamic evolutions converge to the quasistatic one as inertia and viscosity go to zero. We also highlight how the presence of viscosity is crucial to get this kind of convergence.
**Hynek Kovarik**, Università degli studi di Brescia,

*Robin eigenvalues on domains with cusps*, Thursday, June 06, 2019, time 14:00, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the Laplace operator with the Robin boundary condition with negative coefficient
on bounded domains with cusps. We show that if the cusps is not too strong, then the operator is bounded from below and calculate the leading term in the asymptotic expansion of its negative eigenvalues as the coefficient of the boundary condition tends to infinity. This is a joint work with Konstantin Pankrashkin.
**Fabio Cipriani**, Politecnico di Milano,

*How to hear the shape of a drum*, Tuesday, May 14, 2019, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
In a iconic 1912 paper Hermann Weyl, motivated by problems posed by the physicist H.A. Lorentz about J.H. Jeans's radiation theory, showed that the dimension and the volume of a Euclidean domain may be traced from the asymptotic distribution of the eigenvalues of its Laplace operator.
In a as much famous 1966 paper titled "Can one hear the shape of a drum" Marc Kac popularized this and related problems connecting geometry and spectrum. He noticed that the hope to characterize {\it isometrically}, Euclidean domains or compact Riemannian manifolds by the spectrum of the Laplace operator, is vain: John Milnor in 1964 had showed the existence of non isometric 16 dimensional tori sharing a common (discrete) spectrum.
The aim of the talk is to show how to recognize {\it conformal maps} between Euclidean domains as those homeomorphisms which transform multipliers of the Sobolev-Dirichlet spaces of a domain into multipliers of the other and leave invariant the {\it fundamental tone} or {\it first nonzero eigenvalue} of the Dirichlet integral with respect to the energy measures of any multiplier. Related results hold true for {\it quasiconformal and bounded distortion maps}.
In the opposite direction, we prove that the trace of the Dirichlet integral, with respect to the energy measure of a multiplier, is a Dirichlet space that only depends upon the orbit
of the conformal group of the Euclidean space on the multiplier algebra.
The methods involve potential theory of Dirichlet forms (changing of speed measure, multipliers) and the Li-Yau conformal volume of Riemannian manifolds.
This is a collaboration with Jean-Luc Sauvageot C.N.R.S. France et Universit\'e Paris 7.