On Lipschitz Continuity of Nonlinear Differential Operators

In connection with approximations for nonlinear evolution equations, it is standard to assume that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that f = f(x, ∇u(x)) is Lipschitz continuous from the subspace W 1,1 ⊂ L2 into W 1,2, and maps W 2,1 into W 1,1, if and only if f is affine withW 1,1 coefficients. In fact, a local version of this claim is proved. This paper follows efforts to sharply estimate the convergence of some fully discrete approx- imations for semilinear parabolic partial differential equations (3). At a certain point in the analysis, it is tempting to postulate that the semilinearity, viewed as a nonlinear operator, is Lipschitz continuous in a sense described below. However, it is proved here that this condition can hold if and only if the function in question is actually affine with respect to the argument for which Lipschitz continuity is assumed. Hence, while Lipschitz assumptions are standard in proving convergence of schemes for nonlinear evolution equations, generalizing them even very weakly to a function space setting may amount to linearizing the equation. To establish some notation, suppose that is a bounded domain in RN. For 1 ≤ p ≤ ∞ and integers m ≥ 0, let W m,p() represent the well-known Sobolev spaces consisting of functions with distributional derivatives of order ≤ m in Lp(). Also, kk W m,p() denotes the usual norm. Next, let C ∞ 0 () consist of infinitely differentiable functions with support compactly contained in . Completing the latter with respect to kk W m,p() produces the spaces W m,p 0 (). Then, for 1 ≤ p < ∞, p−1+ q−1 = 1 and integers m ≥ 1, define W −m,q() ≡ W m,p 0 () ∗ equipped with the norm: k vk W m,q() ≡ sup u∈W m,p 0 () |(u, v)|/k uk W m,p() (u, v) ≡ Z u(x)v(x)dx.