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

Tuesday, July 20, 18:00 ~ 18:50 UTC-3

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

Boyan Sirakov

PUC-Rio, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We develop a new, unified approach to the following two classical questions on elliptic PDE: (i) the strong maximum principle for equations with non-Lipschitz nonlinearities, and (ii) the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

Joint work with Philippe Souplet (Université Sorbonne Paris Nord, France).

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