B. Sirakov | Basic elliptic estimates with optimized constants and app. to the qualitative theory...
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Boyan Sirakov (PUC-Rio) 
Basic elliptic estimates with optimized constants and applications to the qualitative theory of elliptic PDE

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 (Landis conjecture). 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. If time permits, we will report on some recent C1 estimates with optimized constants and refined Landis-type results. These are based on a new
boundary weak Harnack inequality which also has applications in the boundary regularity theory of equations in divergence form.
References
[1] B. Sirakov, P. Souplet, The V´azquez maximum principle and the Landis
conjecture for elliptic PDE with unbounded coefficients, Adv. Math. 387
(2021), 107838.