### Session S18 - Recent progress in non-linear PDEs and their applications

Monday, July 12, 15:00 ~ 15:50 UTC-3

## a nonlocal version of the inverse problem of Donsker and Varadhan

### Erwin Topp

#### Universidad de Santiago de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

In their seminal paper of 1976, M.D. Donsker and S.R.S. Varadhan addressed the following ``inverse problem": let consider two linear, second-order, uniformly elliptic operators $L_1, L_2$ with the form \[ L_i \phi = Div(A_i(x) D \phi) + b_i(x) \cdot D\phi, \quad i =1,2. \]

If for every domain $\Omega$ and every smooth potential $V$, the operators $L_1 + V$ and $L_2 + V$ have the same principal eigenvalue in $\Omega$, then the diffusions are equal ($A_1 = A_2$), and either $L_1 \phi = L_2 (u \phi)/u$ for some $L_2$-harmonic function $u$, or $L_1 \phi = L_2^* (u \phi)/u$ for some $L_2^*$-harmonic function $u$.

In this talk we report a nonlocal a version of this problem, where both the diffusion and transport terms defining the involved operators have a fractional nature. We prove a similar conjugacy phenomena among operators having the same principal eigenvalues, by means of a minmax characterization for them, and developing new ideas to overcome the difficulties posed by the non locality.

Joint work with Gonzalo Dávila (UTFSM, Chile).