Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces

Abstract:

An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace $S = {\mathring {W}}_k^p[a,b]$ of the kth order Sobolev space $W_k^p[a,b], 1 \leqslant k, 1 \leqslant p \leqslant \infty$. It is shown that every mapping $T:S \to {L^1}[a,b]$ which is local, continuous and ${D^k}$-disjointly additive has the form $(Tu)(t) = H(t,{D^k}u(t))$, where $H:[a,b] \times R \to R$ is a function obeying Carathéodory conditions as well as $(\ast )H( \cdot ,0) = 0$. Here ${D^k}$-disjoint additivity means $T(u + v) = Tu + Tv$ whenever $({D^k}u)({D^k}v) = 0$. Likewise, every real functional N on S which is continuous and ${D^k}$-disjointly additive has the form $N(u) = \smallint Tu$, with T as above. Liapunov’s theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.