Two types of new results of the presenter will be discussed:
1. Holder and Lipschitz stability estimates for coefficient inverse problem and inverse source problem with the final overdetermination [1]. The solution of the parabolic equation is known at $t=0$ and $t=T$. Both Dirichlet and Neumann boundary conditions are known either on part of the boundary or on the entire boundary. A new Carleman estimate for the parabolic operator is the key here. Unlike standard Carleman estimates in this one, the Carleman Weight Function is independent on $t$. The Holder stability estimate is in the case of incomplete boundary data and the Lipshitz stability is in the case of complete boundary data. Both results and the methodology are significantly different from previous ones.
2. Stability estimates and uniquness theorems for some inverse problems for the Mean Field Games system [2]. These results are also new. The Mean Field Games system is a system of two parabolic equations, which was originally proposed by J.-M. Lasry and P.-L. Lions in 2006-2007 and became quite popular nowadays due to a number of very exciting applications. The main challenge here is that the time t is running in two opposite directions in these equations. Therefore, the Volterra-like property of conventional systems of parabolic PDEs is not kept here.
[1] M. V. Klibanov, Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate, arxiv: 2301.09735, 2023.
[2] M. V. Klibanov, Yu. V. Aveboukh, Stability and uniqueness of two inverse problems for the Mean Field Games system, in preparation.