On a strong extremum principle for a weakly parabolically second-order connected operator

Abstract

The following strong extremum principle holds for the heat conduction equation ~2n/ds~-~uidt=0, considered in a rectangle G with sides parallel to the coordinate axes: the solution u (x, t), which takes the greatest or least value inside G at the point (x0, to), is equal to a constant in G for t < to. (Hence, as distinct from the strong extremum principle for Laplace’s equation, the present principle for the heat conduction equation is anisotropic.) A strong anisotropic extremum principle was proved in [I] for a second-order uniformly parabolic equation. A strong anisotropic principle for general second-order equations with non-negative characteristic form was first studied in [2]. We may also mention [3] (cf. also [4, 5]), in which sufficient conditions were obtained for a second-order operator, whose characteristic form is a sum of squares, to be parabolically connected in Aleksandrov’s sense. The results obtained in [3] (cf. also [4]) are a particular case of the results in [2] (see Note 17 of the present paper). The present paper is a continuation of [6], and extends the theory of [2] (for second-order parabolically connected equations), which was based on the use of Giraud exact theorems (on the signs of the directional and time derivatives) for the solutions of second-order parabolic equations with non-negative characteristic form. The strong anisotropic extremum principle obtained is applicable to various topics of mathematical physics, e.g. when proving the uniqueness of the solution of certain boundary value problems (see [7]). We state our main results in Section 1, and prove them in Section 2. Some definitions and results of [6, 71 are employed.