Quantitative uniqueness of continuation result related to Hopf's lemma

TL;DR

This paper quantifies the classical Hopf's lemma by establishing bounds on the solution's deviation from its boundary maximum in terms of the exterior normal derivative, providing a precise continuation property for elliptic equations.

Contribution

It introduces a quantitative version of Hopf's lemma, linking boundary derivatives to interior and boundary norms of solutions to elliptic equations.

Findings

Bounded $L^1$-norms of solution deviations by exterior normal derivative

Quantitative relation between boundary derivative and solution behavior

Enhanced understanding of uniqueness in elliptic PDEs

Abstract

The classical Hopf's lemma can be reformulated as uniqueness of continuation result. We aim in the present work to quantify this property. We show precisely that if a solution $u$ of a divergence form elliptic equation attains its maximum at a boundary point $x_0$ then both $L^1$-norms of $u-u(x_0)$ on the domain and on the boundary are bounded, up to a multiplicative constant, by the exterior normal derivative at $x_0$.