Supremum principles for the higher-order derivatives of solutions to the Dirichlet problem for parabolic equations without compatibility conditions between the data

Abstract

The Dirichlet problem is considered for the heat equation u t = a u x x , a > 0 a constant, for ( x , t ) ∈ [ 0 , 1 ] × [ 0 , T ] , without assuming any compatibility condition between initial and boundary data at the corner points ( 0 , 0 ) and ( 1 , 0 ) . Under some smoothness restrictions on the data (stricter than those required by the classical maximum principle), weak and strong supremum and infimum principles are established for the higher-order derivatives, u t and u x x , of the bounded classical solutions. When compatibility conditions of zero order are satisfied (i.e., initial and boundary data coincide at the corner points), these principles allow to estimate the higher-order derivatives of classical solutions uniformly from below and above on the entire domain, except that at the two corner points. When compatibility conditions of the second order are satisfied (i.e., classical solutions belong to C x , t 2 , 1 on the closed domain), the results of the paper are a direct consequence of the classical maximum and minimum principles applied to the higher-order derivatives. The classical principles for the solutions to the Dirichlet problem with compatibility conditions are generalized to the case of the same problem without any compatibility condition. The Dirichlet problem without compatibility conditions is then considered for general linear one-dimensional parabolic equations. The previous results as well as some new properties of the corresponding Green functions derived here allow to establish uniform L 1 -estimates for the higher-order derivatives of the bounded classical solutions to the general problem.