Maximum principles for implicit parabolic equations
Norio Yoshida · Proceedings of the Japan Academy Series A Mathematical Sciences · 1973
In [6], Nirenberg derived a strong maximum principle for second order linear parabolic equations.This result was extended by Besala [2] to nonlinear parabolic equations of the form ( u (t, x, u, u, u), where x--(x, ,x), Ut--U/t, U--(U/3Xi)= and u=(3u/x3x).,=.On the other hand, Picone [7] and Krzyiafiski [4] established a maximum principle in unbounded domains which was particularly useful to the study of the Cauchy problem for second order linear parabolic equations.An extension of this principle to nonlinear equa- tions of the form (.) was given by Besala [1].The purpose of this paper is to generalize the above mentioned results of Besala to the implicit parabolic equation ( 1) F(t, x, u, ut, u, u) 0. Let D be a domain in the (n+ 1)-dimensional Euclidean space R / of points (t,x).For each fixed point (t ,x)eD we define S(t , x)[Sg(t , x)] to be the set of all points (t, x)e D which can be joined to (t , x) by a upward [downward] directed broken line contained in D, with (t , x) as initial point and (t, x) as endpoint.Consider a function F(t, x, z, p, Q, R) defined for all (t, x) e D, z, p, Q=(q)\l and R-(r),__.The function F(t, x, z, p, Q, R) is said to