# Seminario di equazioni differenziali e applicazioni: “Fractional semiconcavity and some applications to state constrained problems”

## Lunedì 22 Ottobre 2018, ore 13:30 - Aula 2AB45 - Rossana Capuani

ARGOMENTI: Seminari

Lunedì 22 Ottobre 2018 alle ore 13:30 in Aula 2AB45, Rossana Capuani (Università di Verona) terrà un seminario dal titolo “Fractional semiconcavity and some applications to state constrained problems”.

Abstract
For problems in the calculus of variations or optimal control, very few semiconcavity results are available in the presence of state constraints. It was known so far that global linear semiconcavity should not be expected [1]. In this talk we show that semiconcavity actually holds, but with a fractional modulus of semiconcavity. More precisely, let u be the value function associated with a state constraint problem of calculus of variation. We denote by $T$ the finite horizon, by $\Omega$ the state constraint and by $H$ the associated Hamiltonian. Then, under suitable smoothness assumptions on the data, for any $\varepsilon > 0$, there exists a constant $c_{\varepsilon} \geq 1$ such that, for any $(t, x) \in [0, T - \varepsilon] \times \overline{\Omega}$, there exists $p \in \mathbb{R}^d$ with $$u(t + \sigma, x + h) - u(t, x) \leq \sigma H(t, x, p) + \langle p,h \rangle + c_{\varepsilon} (|h| + |\sigma|)^{\frac{3} {2}} \qquad (0.1)$$
for any $(\sigma, h) \in \mathbb{R} \times \mathbb{R}^n$ such that $(t + \sigma, x + h) \in [0, T - \varepsilon] \times \overline{\Omega}$. A key argument for the proof is a maximum principle for calculus of variation with state constraints introduced in [2]. Note that $(0.1)$ immediately yields a fractional semiconcavity for the value function. Finally, using this semiconcavity property, we give an interpretation of MFG system in the presence of state constraints.
These results have been obtained in collaboration with Piermarco Cannarsa (Rome Tor Vergata) and Pierre Cardaliaguet (Paris-Dauphine).

References
[1] Cardaliaguet, P., and Marchi, C. Regularity of the eikonal equation with Neumann boundary conditions in the plane: application to fronts with nonlocal terms SIAM journal on control and optimization, 45(3), 1017-1038, 2006.
[2] Cannarsa, P., Capuani, R., and Cardaliaguet, P., C 1,1 –smoothness of constrained solutions in the calculus of variations with application to mean field games. arXiv preprint arXiv:1806.08966, 2018.