Absolute continuity on tracks and mappings of Sobolev spaces

The present paper is concerned with the circumstances under which a function g(x,t ,...,t ) provides, via composition, a ^ 1 m mapping between Sobolev spaces. That is, we examine conditions which ensure that for every system of functions u_,...,u e W _ (Q) 1 m 1, q (where W, (Q) is the class of L functions with L summable i,q q q strong first derivatives on the domain Q, c: R )S the composite function v given by v(x) = g(x,un(x) , . . . ,u (x) ) belongs to Wn _(£})> with preassigned 1 <. P < oo. Our overall approach in this paper is patterned after a classical chain rule result of Vallee Poussin [8,p. 467] for real functions on a real interval. By introducing a (seemingly new) definition for absolute continuity of a function g(t ,...,t ) on the track of an absolutely continuous curve and exploring its properties, we have been able to attain an exact analogue of the above result of Vallee Poussin in the case of functions g(t_,...,t ) defined on R . This re1 m m suit is thereafter utilized in obtaining necessary and sufficient loc conditions in order that for given functions u_,...,u e W_ _(Q) 1 m 1,1 loc the composite function v = g(u_,...,u ) belong to Wn . (Q) . This 1 m 1,1 last result leads in a relatively straightforward manner to conResearch partially supported by the National Science Foundation under Grants GP 24339 and GP 28377. ditions for g to map Wn (ft) to Wn (ft) . We also obtain a l q 1 P different set of conditions on g under which g(t .t,,....t ) o 1 n takes W. (ft) into W_ (ft) via the composition 2,q l p v(x) = g(u(x),d u(x),...,S u(x)). On the other hand for functions g(x.t......t ) xeO, we have ^ 1 m obtained fully analogous results only when the function g satisf ies a local Lipschitz condition on fi x R . * m The entire approach relies heavily on a characterization of the spaces W (ft) due to Gagliardo [2]. 1 ABSOLUTE CONTINUITY ON TRACKS AND MAPPINGS OF SOBOLEV SPACES by M. Marcus and V. J. Mizel Introduction, The present paper is concerned with the circumstances under which a function gfx.t,,...,t ) provides, via composition, a ~ 1 m mapping between Sobolev spaces. That is, we examine conditions which ensure that for every system of functions u......u e W 1 m where W, (fl) is the class of L functions with L summable l q q q strong first derivatives on the domain Q c R 5 the composite func tion v given by v(x) = g(x,u_ (x) 9 • . . ,u (x) ) belongs to W_ (Q) 1 ~ m ~ l P with preassigned 1 £ p < 00. Our overall approach in this paper is patterned after a classical chain rule result of Vallee Poussin [8,p.467] for real functions on a real interval. He showed that when g and u are both absolutely continuous functions then the composite function g u will be absolutely continuous if and only if g (u(x))u(x) is summable (when the product is properly interpreted) , and that then the chain rule (c) -jg(u(x)) = g' (u(x))u'(x) ax is valid almost everywhere. In this direction Serrin has shown [5] loc that for g : R —> R locally absolutely continuous and ueW (H) , JL _L -L} 1 "I r\r+ lOC one has v(x) = g(u(x)) eW, ,(fl) if and only if g' (u(x) ) Vu(x) eL. (0) ~ ~ 1,1 ~ ~ l