352 D. В. Silin

Ъ ) if a strongly convex function (p{t^ x), continuously differentiable in a neighbourhood of (to, xq), satisfies

( p { t , x)'^u{t, x) and ip(tQ, xo) = u{to, xq), (42)

then

М|^ + я(^^(|^)<0. (43)

The definition above differs from the one from [3, 4] by the requirements of strong convexity and, respectively, strong concavity of test functions (f in (40)- (43). Let us prove that both definitions are equivalent. Indeed, if there exists a continuously differentiable function (fi{t, x) satisfying (40)-(41), then by virtue of Lemma 4 there exists a strongly concave continuously differentiable function (p{t, x) such that

( p { t , x)^(pi (r, x) for all (/, x) and (p{tQ, xq) = ^i{tQ, xq). Hence

d ( p { to , xo ) _d(fi{to,xo) dipjto, xp) _ dipijto, xp) dt dt ^ dx dx ^

and (40)-(41) are valid for the strongly concave continuously differentiable function (/?, too. A similar argument verifies the equivalence of the definitions for (42H43). For

/ ( r , p ) =r H{s,')ds{p). (44)

J / o , 0

denote

u { t , x ) =f { t , p ) , (45)

where the conjugate transform is carried out only in p.

Lemma 9. The function (45) is continuous in {t^x) at every (r, jc) € intdomM(-, •).

Proof , Choose arbitrarily {tp, xp) G int dom м(-, •) and let T > tp. Consider a sequence of partitions ^k of the interval [0, T] with diam ^k = 2/A: such that tp-l/k and tp + l/k are the nodes of ^k with no other nodes of â^k in between. By Theorem 1

\f { t , p ) - I { 0^k' . t ) { p ) \^Tu ; i2 / k ) forevery (r, p) G [0, Г] x dom w*(-).

( 46 )

Denote

u , it , x ) =Ii^ , ; tfix ) . (47)

From (6) and (19)

vt { t , x ) = sup {{x, p) -1 (^t; to - l/k) -{t-to- \/k))H{to - \/k, p)) p