A characterization of first order nonlinear partial differential operators on Sobolev spaces

Abstract

Nonlinear partial differential operators G: W 1,p(Ω) → L q(Ω) (1 ⩽ p, q ∞) having the form G( u) = g( u, D 1 u,…, D N u), with g ϵ C( R × R N ), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, W 1,∞(Ω) , and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of W 1,p(Ω) and L q(Ω) .