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

Let be a portion of a C 1, 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 (T, T), 0 . We prove that u satis.es a three cylinder inequality near (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 , then u is identically equal to zero.