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## Seminar Analysis and Mathematical Physics: Alessandro Goffi (University of Padova)

**Optimal regularity for viscous Hamilton-Jacobi equations in Lebesgue spaces**

The problem of maximal regularity in Lebesgue spaces is a heavily studied question for linear PDEs that dates back to Calderón and Zygmund for the Poisson equation and the potential theoretic approach developed by Ladyzhenskaya et al for the heat equation, and represents the cornerstone in the analysis of many nonlinear PDEs. After a brief overview of the linear theory, in this talk I will focus on maximal L^q-regularity for viscous Hamilton-Jacobi equations with superlinear first order terms. I will first survey on recent results obtained for the stationary equation, which answer positively to a conjecture raised by P.-L. Lions, via a Bernstein-type argument. Then, I will discuss Lipschitz and optimal L^q-regularity for parabolic Hamilton-Jacobi equations that instead are tackled through a refinement of the Evans’ nonlinear adjoint method, thus exploiting fine regularity properties for advection-diffusion equations with “rough” drifts, providing new regularity results for systems of PDEs arising in the theory of Mean Field Games. These are joint works with Marco Cirant (Padova).

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