Some refinements of the Hodge decomposition and applications

K e y w o r d s H o d g e decomposition, Quasilinear equation, Measures. 1. I N T R O D U C T I O N We give some slight new extensions of the Hodge decomposition due to Iwaniec [2] (see, also [3-5]) by proving essentially that for v E W l ' n e n ( ~ ) , ~ C R n the vector field Dv(iDv] + e)-~n can be written as D~e + K~ with ~be E W l'r~ (~), re > n and K~ is a free divergence vector and is 1-en C~2--¢n. small, i.e., [[Ke[[r~ ~< c¢][Dv][~_e,~ + In [1], the functions do vanish on the boundary and they decompose the vector field, DvlDv[ -en. The main motivations of such decomposition is the study of equivalence problems for nonlinear boundary value problems with measured data. Many authors studied the existence and uniqueness of the solutions of such problems and different formulations have been introduced (see [1,3,6-11]). In particular, in [1], the authors consider the homogeneous Dirichlet problem for the equation div ('~(z, u, Du)) + b(u) = #, in D' (~) , (1.1) where ~ is an open bounded subset of ]R n, ~ : ~ x R x R n --* R n is a Carath~odory function with usual upper growth and such that ~(x, rl, ~)~ ~> a]~l n for a.e., x c ~t, for all (~, ~) E N x N n and b(u) c L 1 (gt). They prove that it is equivalent to find solutions in one of the three spaces 0893-9659/00/$ see front matter @ 2000 Elsevier Science Ltd. All rights reserved. Typeset by A~+5-TEX PII: S0893-9659 (00)00115-4