Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

Abstract

Let Γ be a portion of a C 1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically equal to zero.