Organizers: Giovanni Catino and Fabio Cipriani

**Giovanni Molica Bisci**, Università degli Studi Mediterranea,

*On nonlocal critical equations*, Friday, January 29, 2016, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
A very interesting area of nonlinear analysis lies in the study of elliptic equations involving fractional operators. Recently, a great attention has been focused on these problems, both for the pure mathematical research and in view of concrete real-world ap- plications. Indeed, this type of operators appear in a quite natural way in different contexts, such as the description of several physical phenomena. In particular, nonlocal critical equations are relevant for their relations with problems arising in differential geometry and in physics, where a lack of compactness occurs (see [4, Part III]). Motivated by this wide interest in the current literature, also in connection with the celebrated Brezis-Nirenberg problem (see [1]), in the first part of the talk we will describe the state of the art for nonlocal critical problems involving the fractional Laplacian operator or its generalizations. Successively, some recent existence and multiplicity results will be discussed [2, 3, 5]. In conclusion, certain open problems will be briefly presented.
References:
[1] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equa- tions involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
[2] A. Fiscella, G. Molica Bisci and R. Servadei, Bifurcation and multi- plicity results for critical nonlocal fractional problems, Bull. Sci. Math. 140 (2016), 14–35.
[3] J. Mawhin and G. Molica Bisci, A Brezis-Nirenberg type result for a non- local fractional operator, preprint 2016.
[4] G. Molica Bisci, V. Ra ?dulescu, and R. Servadei, Variational Meth- ods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, No. 142, Cambridge University Press, Cambridge, 2016.
[5] G. Molica Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differential Equations 20 (2015), 635–660.
**Ilaria Lucardesi**, SISSA,

*The wave equation on domains with cracks growing on a prescribed path*, Friday, January 15, 2016, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk I analyze a scalar wave equation in a time varying domain of the form "set minus a growing crack", when the crack develops along a given path. Under suitable regularity assumptions, I show existence, uniqueness and continuous dependence on the cracks of the weak solution of the wave equation under study. This is a joint work with Gianni Dal Maso.
**Alain Miranville**, Université de Poitiers,

*Cahn-Hilliard inpainting*, Friday, November 27, 2015, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
Our aim in this talk is to discuss variants of the Cahn-Hilliard equation in view of applications to image inpainting. We will present theoretical results as well as numerical simulations.
**Dario D. Monticelli**, Politecnico di Milano,

*On the Dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas*, Friday, November 13, 2015, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk we will describe some recent results concerning the Dirichlet problem for a class of second order differential equations of mixed elliptic-hyperbolic type on suitable bounded domains of R^2, which is used as a model to describe possible heating in axisymmetric cold plasmas subjected to high frequency electromagnetic waves near certain frequencies. The presence of hyperbolicity makes the problem overdetermined for classical solutions. We will show that the problem is well-posed for weak solutions belonging to a weighted version of the classical Sobolev space H^1_0, when the datum is chosen in a suitable weighted L^2 space. We will also provide a complete spectral theory for the Dirichlet problem in the setting of weighted Lebesgue and Sobolev spaces, with some applications to semilinear equations and to equations with lower order terms. Finally we will give a variational characterisation of weak solutions, which are shown to be saddle points of an associated strongly indefinite functional.
These results are joint work with D. Lupo (Politecnico di Milano) and K.R. Payne (Università degli Studi di Milano).
**Davide Buoso**, Politecnico di Torino,

*Shape sensitivity analysis for vibrating plates*, Friday, October 30, 2015, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk we will consider the eigenvalues of the biharmonic operator subject to various boundary
conditions, namely Dirichlet, Neumann, Navier and Steklov ones. Note that such problems arise in the theory of linear elasticity, within the so-called Kirchhoff-Love model for the vibration of a plate. We will show that the eigenvalues of the Bilaplacian are analytic with respect to the shape, and compute Hadamard-type formulas for their differentials. Then, using the Lagrange Multiplier Theorem, we are able to show that the ball is a critical domain for any eigenvalue (under any of the boundary conditions considered). In the last part of the talk we will focus on eigenvalue shape optimization results for Neumann and Steklov problems. We will provide isoperimetric inequalities in quantitative form for the fundamental tones (i.e., the first non-trivial eigenvalues), and discuss the limiting cases.
**Corrado Mascia**, Universita' di Roma "La Sapienza",

*Hyperbolic traveling fronts: the bistable equation with relaxation*, Thursday, June 25, 2015, time 14:00 o'clock, Aula Seminari III Piano

**Abstract:****Abstract:**
The main concern of the talk is to discuss a class of hyperbolic equation
in the presence of a reaction term of Allen-Cahn type, motivated by
the assumption that the alignment of the flux term with the gradient of the unknown function is not istantaneous but delayed by the presence of a relaxation time.
After a brief overview on the derivation of such class of equations
starting from some appropriate modelling assumptions, emphasis will be given to the topic of front propagation in one dimension.
Rigorous results concerning existence and stability of planar fronts will be presented, comparing it with the corresponding results for the
standard parabolic Allen--Cahn equation.
(joint collaboration with C.Lattanzio, R.G.Plaza, C.Simeoni)