## PhD research topic - Andrea Davini

#### Research areas

PDEs, Control Theory, Dynamical Systems

#### Keywords

Hamilton-Jacobi equations, weak KAM Theory, viscosity solutions

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Research topics

My research interests in latest years concern the study of Hamilton-Jacobi equations and related asymptotic problems,

mainly addressed by making use of methods coming from weak KAM Theory, see [F] for an introduction to

the topic. The mathematical tools involved are PDE methods (issued in particular from viscosity solution theory), as well as variational, dynamical and

probabilistic techniques. A list of topics of my interest is the following:

-Selection principles.

We are interested in the asymptotics of solutions to a HJ equation, perturbed through a regularizing term, as the perturbation parameter tends to zero.

Typically, the solutions of the perturbed equations converge, along subsequences, to solutions of the unperturbed equation, that are usually nonunique. The issue is understanding whether this procedure selects a specific solution in the limit and possibly characterizing it. In [DFIZ] we have proved a selection principle of this kind for the ergodic approximation of the HJ equation when the ambient space is a closed manifold and the Hamiltonian is convex and coercive in the momentum. We propose to extend the analysis to more general cases, for instance by considering some models when the ambient space is noncompact and/or the Hamiltonian is non coercive.

-Systems of weakly coupled HJ equations.

A weak KAM theory for systems of this kind was provided in [DZ] by making use of PDE tools and viscosity solution techniques. A dynamical and variational point of view of the matter was brought in by [MSTY] and further developed in [DSZ], where a Lax-Oleinik formula adapted to evolutive systems of HJ equations is defined and the existence of random minimizing curves is proved. This angle detects the stochastic character of the problem, displayed by the random switching nature of the dynamics related to systems. This approach provides the tools for an extension of the dynamical and geometrical counterpart of the weak KAM Theory to systems, in analogy with the scalar case. We propose to carry on this study, aiming to complete the picture of analogies/differences between the vectorial and scalar case and to shed light on some open questions.

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Bibliography

[DFIZ] A. Davini, A. Fathi, R. Iturriaga, M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation. Invent. Math. 206 (2016), no. 1, 29-55.

[DSZ] A. Davini, A. Siconolfi, M. Zavidovique, Random Lax-Oleinik semigroups for Hamiltoni-Jacobi systems. J. Math. Pures Appl., to appear.

[DZ] A. Davini, M. Zavidovique, Aubry sets for weakly coupled systems of Hamilton-Jacobi equations, SIAM J. Math. Anal. 46 (2014), no. 5, 3361-3389.

[F] A. Fathi, Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobi equation and Aubry set. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1193-1236.

[MSTY] H. Mitake, A. Siconolfi, H. Tran, N. Yamada, A Lagrangian Approach to Weakly Coupled Hamilton-Jacobi Systems. SIAM J. Math. Anal. 48 (2016), no. 2, 821-846.