### Seminars 2012-2013

#### Dynamic Griffith fracture evolution

#### The Yamabe Problem

This is a minicourse of 6 lectures.

- Introduction and motivation to the problem: curvature tensors on Riemannian manifolds, the Scalar curvature, conformal maps; the Poincar ́e conjecture and the min-max argument.

- The loss of compactness due to the critical exponent in the Sobolev embedding theorems.

- The relation between the best Sobolev constant in Rn and the Yamabe constant of the sphere.

- The Yamabe-Trudinger-Aubin theorem on the solvability of the problem.

- Eventually some mentions regarding generalizations to manifolds with boundary, CR-manifolds, changing-sign solutions.

#### Nonlocal nonlinear variational problems

Abstract. I would like to discuss some questions related to some semilinear equations driven by a nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace operator is replaced by a fractional Laplacian). In particular, I would like to study the qualitative properties of the solutions, such as symmetry, density estimates of the level set, asymptotic behaviors, etc. The limit interfaces of these problems are related to both the local and the nonlocal perimeter functionals on this topic, I would like to discuss some rigidity and regularity results and to present some open problems.

#### Complexity in affine control systems

Abstract. We will present the problem of the motion planning for a system which is affine in the control, focusing in particular on its complexity. We will start presenting different definitions of it in a linear system (i.e. a sub-Riemannian geometry), giving some results on their equivalence. Then we will treat the affine case, proposing similar definitions of complexity and focusing on the difficulties arising from the presence of a drift.

#### Second order Caffarelli-Kohn-Nirenberg type inequalities and related problems

Please click here for the abstract.

#### Analysis of O'Hara's knot energies

Abstract. All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk we approach this problem using gradient ows of a family of energies introduced by O'Hara in 1991-1994. We will see that this allows us to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for these evolution equations.

#### Sub-Laplacian eigenvalue bounds on CR manifolds

Abstract. I will tell about upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. The method, used to prove these, is of interest on its own; it is based on the ideas of Korevaar and their generalisations to metric measure spaces by Grigor'yan and Yau. I will describe some relevant ingredients and outline possible applications to other problems. No knowledge of CR geometry will be assumed.

#### Nonlocal Conformal Invariant Variational Problems

Abstract. Conformal invariance plays an important role in physics (conformal field theory, renormalization theory, statistical physics, percolation theory, turbulence theory) and in geometry (theory of Riemannian surfaces, Weyl tensor, Q-tensor, Yamabe problem, Yang-Mills). Here we are concerned with the study of conformal invariance in analysis. Because of the richness of the conformal group in dimension 2 (which is infinite), in the first part of the talk we will recall some known aspects of the analysis of 2-dimensional local conformal invariant variational problems of maps. Until now nonlocal variational problems have been little studied even though they appear naturally in several areas of physics and geometry. In the second part of the talk we are going to present some recent results obtained in the analysis of these problems.

#### Semi-classical states for the Nonlinear Schrodinger Equation without nondegeneracy requirements

Abstract. In this talk we will study the existence of semiclassical states for NLSE problem. Under our hypotheses on the nonlinearity it is not known if the limit problem is nondegenerate; hence, a Lyapunov Schmidt reduction is not possible here. We use variational methods to prove the existence of spikes around critical points of the potential. This is joint work with Pietro D'Avenia and Alessio Pomponio (Politecnico di Bari).

#### Quasi-periodic solutions of PDEs

Thu. April 19, 2012 @ 10.00 a.m.

Abstract. I will present a bifurcation theory for quasi-periodic solutions of Hamiltonian and reversible PDEs, like nonlinear Schrodinger and wave equations, derivative wave equations, KdV, Euler equations of hydrodynamics (water waves), etc.... In the first lecture I will present the main results and ideas. In the second lecture I will discuss some techniques of proof.

#### Regularity issues for minimizers of the Mumford-Shah functional in dimension 2

#### Optimal IncompressibleTransport

This is a minicourse of 6 lectures.

Abstract. In these lectures, we discuss the mathematical theory of inviscid incompressible fluids following Euler (1755) and Arnold (1966), in relationship with the theory of optimal transport following Monge (1780) and Kantorovich (1942). Both theories are currently very active and combine various aspects of geometry, analysis and PDEs.
More precisely, we will address the following question: is it possible to transport fluid particles of an incompressible fluid from a given configuration to another one so that the total kinetic energy spent will be as small as possible?
The formal optimality equations are just the famous Euler equation (which corresponds to the Navier-Stokes equation without viscous dissipation). A key result of the theory is the existence and uniqueness of the pressure gradient driving the fluid particles between two different configurations, no matter how rough these configurations are. This result is the incompressible counterpart of the now classical existence and uniqueness result in optimal transport
theory: given two probability densities compactly supported in the Euclidean space, there is a unique bounded map with convex potential that makes the second density the image of the first one.

#### Reduction Principles and Applications to Quasilinear Elliptic Problems Abstract

Abstract. We study Liouville theorems for coercive equations of the type $L(u)= f (x,u,\nabla_L u)\ \ \ on \Omega\subset \R^N$. By a variants of Kato’s inequality we show that the assumption that the possible solutions are nonnegative involves no loss of generality. Related consequences such as comparison principles and special a priori bounds on solutions are derived. Underlying structures we are interested in include the canonical Euclidean space and Carnot groups.

#### Existence of strong solutions for quasi-static evolution in brittle fractures

#### Entire solutions for competition-diffusion systems and optimal partitions

Abstract. We study the qualitative properties of a limiting elliptic system arising in phase separation for multiple states Bose-Einstein condensates: \[ \begin{cases} \Delta u=u v^2, \\ \Delta v= v u^2, \\ u, v>0\quad \mbox{in} \ \R^N. \end{cases} \] We first prove that all stable solutions in $\R^2$ with at most linear growth must be one-dimensional. Then we construct entire solutions with polynomial growth $|x|^d$ for any positive integer $d \geq 1$. The construction is also extended to multi-component elliptic systems. Finally, we show the connection between the existence/nonexistence of entire solutions and the qualitative properties of the optimal partitions.

#### Variational characterization of transition paths of maximal probability

Abstract. Chemical reactions can be modeled via diffusion processes conditioned to make a transition between specified molecular configurations representing the state of the system before and after the chemical reaction. In particular the model of Brownian dynamics— gradient flow subject to additive noise—is frequently used. If the chemical reaction is specified to take place on a given time interval, then the most likely path taken by the system is a minimizer of the Onsager-Machlup functional. As a first step toward the characterization of the typical behavior of transition paths the Gamma-limit of this functional is determined explicitly in the case where the temperature is small and the transition time scales as the inverse temperature.

#### On stochastic sea of the standard map

Abstract. The standard map is one of the simplest and most famous conservative transformations that is still far from being completely understood. We prove that stochastic sea (the set of orbits with non-zero Lyapunov exponents) of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters. In the proof we consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism, and show that it give birth to a hyperbolic set of large (almost full) Hausdorff dimension. The last statement has also applications in celestial mechanics, namely can be used to show that in some versions of the three body problem for many parameters the set of oscillatory motions has full Hausdorff dimension (this is a joint project with V.Kaloshin).

#### Scaling finite to linearized elasticity in quasi-static brittle fracture

Abstract. In the setting of finite elasticity, we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions, we show that for large domains the evolution converge to the quasi-static evolution obtained in the setting of linearized elasticity. After re-scaling to a fixed domain the crucial ingredient is the convergence of the non-linear to linear energy release rate, which is locally uniform with respect to crack length and time. Technically, this is proved employing several ingredients: the Γ-convergence of re-scaled energies, the strong convergence of minimizers, the Euler-Lagrange equation for non-linear elasticity and the volume integral representation of the energy release.

#### Ginzburg-Landau functionals and renormalized energy by Γ-convergence

#### Some results on the Levi-Mean Curvature

Abstract: We will introduce the Levi Curvatures for hypersurfaces in complex spaces, then we will focus in particular on the Levi-Mean Curvature. We will state some symmetry results of Alexandrov type and finally we will show some existence and regularity results for the associated Dirichlet problem.

#### Bubbling along Submanifolds for a Singular Perturbation Problem at Higher critical exponent

Abstract: We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial\Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial\Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $$ d^2 |\nabla u_{d} |^2 \rightharpoonup S, \delta_K \ass d \to 0 $$ in the sense of measures, where $\delta_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.

#### A reduced computational and geometrical framework for inverse problems in haemodynamics

The solution of inverse problems in cardiovascular mathematics is computationally expensive. Model reduction techniques might be needed to enable the solution of inverse problems also in other real-life applications. In this presentation we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less expensive reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems in haemodynamics are considered: (i) a simplified stenosed artery model for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements. Joint work with Alfio Quarteroni, Gianluigi Rozza, Toni Lassila.

#### On some critical problems for the fractional Laplacian operator

Please click here for the abstract.

#### The fundamental theorem for surfaces in 3-dimensional Heisenberg group

Abstract: The fundamental theorem for hypersurfaces in a Euclidean space says that the equations of Gauss and Codazzi are exactly the integrability conditions for finding an isometric imbedding with prescribed metric and second fundamental form. In pseudohermitian geometry, we have the analogous fundamental theorem for surfaces in 3-dimensional Heisenberg group. This is a part of recent joint work with Jenn-Fang Hwang, Andrea Malchiodi, and Paul Yang.

#### Gradient theory for plasticity as the Γ-limit of a nonlinear dislocation energy

Since the motion of dislocations is regarded as the main cause of plastic deformation, a large literature is focused on the problem of deriving plasticity models from more fundamental dislocation models. The starting point of our derivation is a semi-discrete dislocation model. The main novelty of our approach is that we consider a nonlinear dislocation energy, whereas most mathematical and engineering papers treat only a quadratic dislocation energy. Our choice of a nonlinear stress-strain relation guarantees that the dislocation strain energy is well defined also in the vicinity of the dislocations, eliminating the need of the cut-off radius that is typical of the linear theories. The Γ-limit of our nonlinear dislocation energy as the length of the Burgers vector tends to zero is a strain-gradient model for plasticity and has the same form as the limit energy obtained by starting from a quadratic dislocation energy. Our result, however, is obtained by starting from a more physical model. (work in collaboration with Caterina Zeppieri)

#### Quasistatic evolution of cavities in nonlinear elasticity

Starting with a variational static model of cavitation in nonlinear elasticity, we propose a quasistatic model also based on global minimization. We prove the existence of a quasistatic evolution, which takes into account the non-interpenetration of matter and the irreversibility of the process of cavity formation.