Basic elliptic estimates with optimized constants and applications to the qualitative theory of elliptic PDE 

Prof. Boyan Sirakov
Pontifícia Universidade Católica do Rio de Janeiro

Abstract: 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.