Titles and Abstracts

Paolo Albano

Title: On the singularities of the distance function

Abstract: In an open bounded set, we consider the viscosity solution of the homogeneous Dirichlet problem for the eikonal equation (i.e. the distance function, \(d\), from the boundary of the given open set.)

We mainly assume that the coefficients of the eikonal equation are of class \(C^2\) while no regularity assumption is done on the boundary of the open set under exam.

We study the regularity of the distance function and the structure of some sets naturally associated with d: the singular set of \(d\), the cut locus and the \(C^1\) singular support of \(d\).

Alexandre Boritchev

Title: Exponential convergence and hyperbolicity of the minimisers for random Lagrangian systems

Abstract: We consider the stochastic Burgers equation from the Lagrangian point of view (long-time behaviour of the minimisers) as well as from the point of view of the statistical behaviour of the solutions (long-time convergence towards the stationary measure). In both cases there is a phenomenon of exponential convergence.

A part of the presentation is about a joint work with K. Khanin (University of Toronto).

Piermarco Cannarsa

Title: Generalized characteristics and singularities of solutions to Hamilton-Jacobi equations

Abstract: Viscosity solutions of Hamilton-Jacobi-Bellman equations are nonsmooth functions which may fail to be differentiable on ``small'’ singular sets. Such singularities, which play an important role for optimal control problems, have been analyzed from various viewpoints. Their dynamics can be described by generalized characteristics, which are forward solutions of the charactheristic system in Filippov's sense. This talk will survey the key points of this approach including recent results of a joint work with Wei Cheng, in which convexity estimates for the fundamental solution are used to prove the global propagation of singularities for Tonelli Hamiltonians.

Annalisa Cesaroni

Title: Homogenization of a viscous semilinear heat equation

Abstract:I will discuss the homogenization of a viscous semilinear heat equation with periodically oscillating potential depending on u/eps to a first order Hamilton Jacobi equation. According to the rate between frequency of oscillations and vanishing factor in the viscosity, we obtain different limit behaviour of the solutions. In particular in the weak diffusion regime, the limit Hamiltonian is discontinuous in the gradient entry. This is an unusual phenomenon in homogenization problems, and makes the analysis of the limit more challenging.

Joint work with Nicolas Dirr (University of Cardiff, UK) and Matteo Novaga (University of Pisa, Italy).

Wei Cheng

Title: Dynamic and asymptotic behavior of singularities of weak KAM solutions on the torus

Abstract: For mechanical systems on torus, we develop a theory of generalized characteristics semiflows associated with certain Hamilton-Jacobi equations which also leads to a homotopy equivalence result on the Aubry sets and singular sets.

This is a joint work with Piermarco Cannarsa.

Xiaojun Cui

Title: Calibrations and Laminations

Abstract: On a closed Riemannian manifold, the structures of minimal closed 1-currents and of 1-calibrations are completely resolved by Aubry-Mather theory and weak KAM theory. Partial results for codimension case are also known. In this talk I will discuss some results for the intermediate-dimensional case. It based a joint work with Victor Bangert.

Albert Fathi

Title: Topology of the set of singularities of viscosity solutions of the Hamilton-Jacobi equation

Abstract: We will mainly report on the progress done recently on the connectedness properties of the set of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation.

To make the lecture accessible to people with no previous knowledge in the subject, after stating the main result in its full generality, we will restrict to distance functions to closed subsets of Euclidean space, which contain all the relevant aspects of the problem.

Roberto Guglielmi

Title: Sensitivity analysis of the value function for semilinear parabolic optimal control problems

Abstract: We derive sufficient conditions for expressing the first and second order sensitivities of the value function associated with a class of optimal controls problems constraint by parabolic semilinear equations. In particular, we relate the first order sensitivity with the adjoint equation and the second order sensitivity with a suitably defined Riccati equation.

This is a joint work with Karl Kunisch (University of Graz).

Mingshang Hu

Title: Stochastic Maximum Principle for Optimization with Recursive Utilities

Abstract: We obtain the variational equations for backward stochastic differential equations in recursive stochastic optimal control problems, and then get the maximum principle which is novel. The control domain need not be convex, and the generator of the backward stochastic differential equation can contain z.

Katsuyuki Ishii

Title: Convergence of a threshold-type algorithm for curvature-dependent motion of hypersurface

Abstract: Let \( \{\Gamma(t)\}_{t\in [0,T)} \) be a family of compact hypersurfaces in \(\mathbb{R}^N\). We call it a curvature-dependent motion (CDM for short) if \(\Gamma(t) \) moves by the following equation

\[
V=\kappa+\langle\mathbf{b},\mathbf{n}\rangle+g\quad\mbox{on }\quad\Gamma(t),\quad\ t\in (0,T).
\]
Here \(T>0\), \(\mathbf{n}=\mathbf{n}(t,x)\) is the inner unit normal vector field on \(\Gamma(t)\), \(V=V(t,x)\) is the velocity of \(\Gamma(t)\) in the direction of \(\mathbf{n}\), \(\kappa=\kappa(t,x)\) is the (\((N-1) \)-times) mean curvature of \(\Gamma(t) \), \(\mathbf{b}=\mathbf{b}(t,x)\) denotes a given vector field, \(g=g(t,x)\) is a forcing term and \(\langle\cdot,\cdot\rangle\) denotes the inner product in \(\mathbb{R}^N\). As well known, the mean curvature flow is the case of \(\mathbf{b}\equiv\mathbf{0}\) and \(g\equiv 0\). The CDM arises in various fields such as two-phase problems, image processing, and so on.

In this talk we introduce a threshold-type algorithm for the above motion. Roughly speaking, we iteratively use the solutions of the following initial value problem to construct an approximate flow to the CDM. Let \(C_0\) be a compact subset of \(\mathbb{R}^N\). For \(k=0,1,2,\ldots\), set \(\mathbf{b}_k(t,x):=\mathbf{b}(t+kh,x)\) and \(g_k(t,x):=g(t+kh,x)\). Let \(w_k=w_k(t,x)\) be a unique solution of the initial value problem for the linear parabolic equation:
\begin{eqnarray*}
& & w_t-\Delta w+\langle\mathbf{b}_k,Dw\rangle+g_k=0\quad\mbox{in }(0,h]\times\mathbb{R}^N, \\
& & w(0,x)=d(x,C_k)\quad\mbox{for }x\in\mathbb{R}^N.
\end{eqnarray*}
Here \(d(x,D)\) is the signed distance function to \(\partial D\) defined by
\[
d(x,D):=\left\{\begin{array}{ll}
\mbox{dist}\hspace{0.5mm}(x,\partial D) & \mbox{for }x\in D,\\
-\mbox{dist}\hspace{0.5mm}(x,\partial D) & \mbox{for }x\notin D.
\end{array}\right.
\]
Set
\[
C_{k+1}:=\{w_k(h,\cdot)\geq 0\}.
\]
Then we have a sequence \(\{C_k\}_{k=0}^{+\infty}\) of compact subsets of \(\mathbb{R}^N\). The main purpose of this talk is to discuss the convergence to the generalized CDM and its optimal rate to the smooth and compact CDM.

This is based on my joint work with Professor Masato Kimura (Kanazawa University) and Mr. Takahiro Izumi (Yasuna Machine Designing).

Wenjia Jing

Title: Front propagations in dynamic random environments

Abstract: We study the behavior of surfaces moving with positive normal velocity that is determined by a space-time dependent environment. Mathematically, this is modeled by a first order Hamilton-Jacobi equation with a space-time dependent Hamiltonian that grows linearly with respect to the amplitude of the momentum. We study the homogenization behavior of the environment, assuming that it satisfies structural conditions like periodicity, and or stationary ergodicity. I will discuss why the combination of the time-dependence of the environment and the linear growth of Hamiltonian makes the homogenization problem non-trivial, and how we can resolve the issue in certain settings. This is based on joint works with P. E. Souganidis and H. V. Tran.

Taiga Kumagai

Title: A perturbation problem involving singular perturbations of domains for Hamilton-Jacobi equations

Abstract: We consider the problem
\begin{align*}
\begin{cases}
u^\varepsilon - \cfrac{b \cdot Du^\varepsilon }{\varepsilon } + |Du^\varepsilon | = f \quad &\text{ in } \Omega, \\
u^\varepsilon = 0 \quad &\text{ on } \partial \Omega,
\end{cases}
\end{align*}
where \(\varepsilon\) is a positive parameter, \(\Omega\) is an open subset of \(\mathbb{R}^2\) determined through a Hamiltonian function \(H\), \(u^\varepsilon : \overline{\Omega} \to \mathbb{R}\) denotes the unknown function, \(f : \overline{\Omega} \to \mathbb{R}\) is a given, continuous, and nonnegative function, and \(b: \mathbb{R}^2 \to \mathbb{R}^2\) is a Hamiltonian vector field.

We study the asymptotic behavior of solutions \(u^\varepsilon\) as \(\varepsilon\) goes to zero. The limit of the solutions is described as solutions of a system of ODEs on a graph. Freidlin-Wentzel, Freidlin-Weber, Sowers, by probabilistic techniques, and Ishii-Souganidis, by PDE techniques, studied stochastic perturbation problems for Hamiltonian flows. Our problem can be seen as a deterministic control version of such perturbation problems.

Hongwei Lou

Title: Second-Order Necessary/Sufficient Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

Abstract: We consider optimal control problems governed by elliptic partial differential equations of second order in divergence form. The equation is semilinear with both leading term containing controls. The main aim is to get second-order necessary optimality condition for optimal control, where the ``second-order" is in the sense of that Pontryagin's maximum principle be looked as the first-order necessary optimality condition.

Joint work with Jiongmin Yong (University of Central Florida).

Marco Mazzola

Title: Discontinuous Solutions of Hamilton-Jacobi Equations under State Constraints

Abstract: This talk deals with Hamilton-Jacobi equations of the form
\begin{equation}
\tag{HJ}\left\{
\begin{array}{ll}
\frac{\partial V}{\partial t} = H(t,x,- \frac{\partial V}{\partial
x}) & \;\; \mbox{ on } \;\; [0,1]\times K
\\[1 mm]
V(1,x) \; = \; g(x) & \;\; \mbox{ on } \;\; K,
\end{array}
\right.
\end{equation}
where the Hamiltonian \(H:[0,1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\) is convex and positively homogeneous with respect to the last variable, \(K\) is any closed subset of \(\mathbb{R}^n\) and \(g:\mathbb{R}^n \to \mathbb{R} \cup\{+ \infty\}\) is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We provide conditions on \(H\) and \(K\) that guarantee the uniqueness of lower semicontinuous solutions of (HJ).

This is a joint work with Helene Frankowska.

Marco Mazzucchelli

Title: A subcritical waist in every tonelli lagrangian system

Abstract: A waist of a Tonelli Lagrangian system is a periodic orbit that is a local minimizer of the free period action functional. A celebrated result of Taimanov says that, for electromagnetic Lagrangians, there is always a waist on every sufficiently low energy level. In this talk, we will discuss the extension of this result to general Tonelli Lagrangian systems: we will show that such a waist exists on every energy level in the interval \((e_0(L),c_0(L))\); here, \(e_0(L) \)is the minimal energy such that the corresponding energy hypersurface projects onto the whole configuration space, while \(c_0(L)\) is the Mañé critical value of the abelian cover. We will also show that the result is sharp: waists do not necessarily exist outside the range of energies \((e_0(L),c_0(L))\). Time permitting, we will also mention the important consequences of the existence of a waist. This talk is based on a joint work with Luca Asselle.

Atsushi Nakayasu

Title: Homogenization and cell problems for noncoercive quasiconvex Hamiltonians

Abstract: We consider the so-called cell problem arising in the context of homogenization of Hamilton-Jacobi equations whose Hamilton is not necessarily coercive or convex. One of the main subject of this talk is solvability of the cell problem and several observation will be shown by introducing a generalized notion of effective Hamiltonian. We will also focus on inf-sup type formulas of the effective Hamiltonian and provide a new proof based on convergence of derivatives of mollified Lipschitz functions, which can be applied to quasiconvex Hamiltonians. This talk is based on joint work with Nao Hamamuki (Hokkaido U.) and Tokinaga Namba (U. Tokyo).

Tokinaga Namba

Title: Well-posedness of Hamilton-Jacobi equations with Caputo's fractional time derivative

Abstract: We introduce a notion of viscosity solutions for evolutionary Hamilton-Jacobi equations in a periodic domain, whose time-derivative is Caputo's fractional one and its order is larger than 0 and less than 1. Existence and uniqueness results are proved via the comparison principle and Perron's method. Here they do not require any special assumption for Hamiltonians and initial data against the usual setting, that is, time-derivative order is 1. Our results by a viscosity approach are completely new although there are many previous studies by a non-viscosity approach, in particular, for linear pdes. This talk is based on a recent work with Yoshikazu Giga(UTokyo, Japan).

Teresa Scarinci

Title: Second-order sensitivity relations for Hamilton-Jacobi equations arising in optimal control

Abstract: In this talk we investigate a class of Hamilton-Jacobi-Bellman equations arising in optimal control. More precisely, we consider a Mayer optimal control problem, whose admissible trajectories are all solutions of a differential inclusion defined by a locally Lipschitz multifunction with convex and compact values. We mainly focus on the sensitivity analysis of the optimal value function, \(V\), associated with such optimal control problem. In the literature, the sensitivity analysis provides a “measure” of the robustness of optimal strategies with respect to variations of the state variable.

We present some sensitivity relations connecting the “jets” of \(V\) along optimal trajectories with the pair \((−p,−R)\), where \(p\) is the dual arc appearing in the Maximum Principle and \(R\) is the solution of a suitable Riccati equation. Essentially, the superjets and the subjets of a function are second-order local approximations from above and below of its epigraph and hypograph, respectively. These notions were introduced by Crandall and Lions for the study of viscosity solutions of second order partial differential equations. Moreover, we propose an application and show results regarding the propagation of the regularity of the optimal value function \(V\) along optimal trajectories.

This is part of a joint work with P. Cannarsa and H. Frankowska.

Antonio Siconolfi

Title: Hamilton--Jacobi equations on networks I

Abstract: This is the first of two interrelated talks, the second will be delivered by Alfonso Sorrentino. We present the outputs of a joint research project, still in progress, regarding Hamilton Jacobi equations on connected networks immersed in an ambient manifold and related asymptotic problems. The topic is nowadays attracting a good deal of interest in the mathematical community. It involves, in fact, a number of subtle theoretical issues and has a great impact in the applications. To describe the problem, we consider Hamiltonians on any arc of the network, which are mutually unrelated, and look for solutions of the corresponding Hamilton-Jacobi equations on arcs, joining them together continuously at the vertices. The coupling term is, roughly speaking, the topological structure of the network itself. The novelty of our method is to combine the partial differential equation on the immersed network with a discrete functional equation on the underlying abstract graph. This allows simplifying to some extent the analysis, in particular for the establishment of comparison principles. My presentation will be focused on a general presentation of the method, with examples, in the framework of the existing literature.

Alfonso Sorrentino

Title: Hamilton--Jacobi equations on networks II

Abstract: This is the second of two interrelated talks, the first being given by Antonio Siconolfi. We continue the discussion of our novel and more global approach to the study of the Hamilton-Jacobi equation on networks, based on the transposition of the problem to an abstract graph. This graph encodes all of the important information on the local/global structure of the network and on its topological features.

In this lecture we will focus on the analysis of a class of Eikonal equations and on the description of how tools and techniques from weak KAM theory and Aubry-Mather theory can be suitably adapted to this setting.

Time permitting, we will also discuss how this approach can be used to deal with other interesting related topics, such as the homogenization of HJ equation on topological crystals, the analysis of the discounted version of the HJ equation, as well as the convergence of discounted solutions in the limit as the discount vanishes.

Xifeng Su

Title: The Aubry-Mather model in the continuous limit and related topics

Abstract: This talk will introduce several models of classical mechanics (especially solid state physics) and quantum mechanics for crystals and quasi-crystals. The models in solid state physics are the generalized Frenkel-Kontorova models on the crystals and quasi-crystals while the models in quantum mechanics will be related to the spectrum of the Schodinger operators.

After surveying on these models, I will concentrate on the Aubry-Mather models (thermodynamic formalism after freezing the system) and talk about the corresponding discrete weak KAM theory. The existence of the discrete weak KAM solutions are related to the additive eigenvalue problem in ergodic optimization. I will show that the discrete weak KAM solutions converge to the weak KAM solutions for the autonomous Tonelli case as the time step goes to zero.

This is a joint work with P. Thieullen.

Shanjian Tang

Title: Systems of Quadratic Backward Stochastic Differential Eqautions

Abstract: In this talk I will introduce some backgrounds of systems of quadratic backward stochastic differential eqautions (QBSDEs), and some known results. In particular,
I will sketch the recent results of my joint paper with Ying HU (SPA, 2016) on systems of diagonally QBSDEs.

Nicolas Vichery

Title: Homological subgradient, applications and computations

Abstract:We will give a definition of the homological subgradient of a real function and compare it to other subdifferentials. We will recall its main properties and its interest to study Hamilton-Jacobi equations and symplectic geometry. Finaly, we will study the subdifferential in a particular case of functions: the difference of convex functions. This work is based on the microlocal theory of sheaves as defined and developped by Kashiwara and Schapira.

Falei Wang

Title: Stochastic optimal control problem with infinite horizon driven by \(G\)-Brownian motion

Abstract: The present paper considers a stochastic optimal controlproblem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by \(G\)-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle.

Moreover, we prove that the value function is the uniqueness viscosity solution of the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation.

Kaizhi Wang

Title: A necessary and sufficient condition for convergence of the Lax-Oleinik semigroup for reversible hamiltonians on \(\mathbb{R}^n\)

Abstract: This is a joint work with Qihuai Liu, Lin Wang and Jun Yan. The present paper is devoted to the study of the convergence of the Lax-Oleinik semigroup associated with reversible Hamiltonians \(H(x,p)\) on \(\mathbb{R}^n\). We provide a necessary and suffcient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on \(\mathbb{R}^n\), even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax-Oleinik semigroup may not converge.

Lin Wang

Title: Large time behavior of viscosity solutions of Hamilton-Jacobi equations depending on unknown functions

Abstract: We consider the evolutionary Hamilton-Jacobi equations with contact Hamiltonians \(H(x,u,p)\) and continuous initial conditions on a closed manifold.

Under certain assumptions on \(H(x,u,p)\) with respect to \(u\) and \(p\), we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set \(\mathcal{C}\), we discuss the large time behavior of viscosity solutions of Hamilton-Jacobi equations depending on unknown functions. This talk is based on joint work with Xifeng Su and Jun Yan.

Qiaoling Wei

Title: Viscosity solutions of Hamilton-Jacobi equation and minmax

Abstract: Given a Hamilton-Jacobi equation, the minmax solution is a type of weak solution defined geometrically: it takes the minmax of a generating family of the geometric solution which is a Lagrangian submanifold in the cotangent bundle. In the case where the Hamiltonian \(H(t,x,p)\) is convex on \(p\), it is the classical Lax-Oleinik semi-group in weak KAM theory and coincides with the viscosity solution. But for nonconvex Hamiltonian, the minmax and the vicosity solution may differ. Replacing minmax by ''iterated minmax", we will see that, as the length of steps used to iterate tend to zero, the iterated minmax will converge to the viscosity solution.

Yifeng Yu

Title: Non-roundness and flat pieces of the effective burning velocity from an inviscid quadratic Hamilton-Jacobi model

Abstract: I will talk about some finer properties of the effective burning velocity from a combustion model introduced by Majda and Souganidis in 90´s. We proved that when the dimension is two and the flow of the ambient fluid is either weak or very strong, the level set of the effective burning velocity has flat pieces. Implications on the effective flame front and other related inverse type problems will also be discussed. This is a joint work with Wenjia Jing and Hung Tran.

Maxime Zavidovique

Title: Aubry Mather theory for weakly coupled systems of Hamilton-Jacobi equations

Abstract: The purpose of this talk will be to describe results showing how weakly coupled systems of Hamilton—Jacobi equations present strong similarities with a single Hamilton—Jacobi equation. This allows to introduce a critical value, for which weak KAM solutions (or solutions to the cell problem) exist and introduce Aubry—Mather sets. As for a single equation, we will explain how these sets help understand the structure of solutions to the cell problem and how they can be interpreted from a dynamical point of view.

Qi Zhang

Title: Stochastic Recursive Control Problem in Non-Markovian Framework and Associated Stochastic HJB Equation

Abstract: The stochastic recursive control problem is introduced by the solution of backward stochastic differential equation. It includes some control problems revelent to the stochastic recursive differential utilities and thus plays a big role in the mathematical finance problems. When the coefficients in the system are random rather than deterministic functions, things are much different. For example, the associated HJB equation for the stochastic recursive control problem is no longer a PDE but a backward stochastic partial differential equation. In this talk we introduce some recent studies for this topic. This is a joint work with Qingxin Meng.