Continuity of operators of the type \(T_ f(u)=f(x,u,Du)\). (Italian. English summary) Zbl 0785.47031
The author gives a continuity condition for the operator \(T_ f\) defined by
\[
(T_ f u)(x)= f(x,u(x),Du(x))
\]
between \(W^{1,q}(\Omega,\mathbb{R}^ m)\) and \(L^ r(\Omega,\mathbb{R})\). This complements and generalizes well- known results by M. Marcus and V. J. Mizel [J. Funct. Anal. 33, 217-229 (1979; Zbl 0418.46024)] and L. Ambrosio, G. Buttazzo and A. Leaci [Boll. Unione Mat. Ital. VII. Ser. B 2, No. 1, 99-108 (1988; Zbl 0639.47051)].
Reviewer: J.Appell (Würzburg)
MSC:
47F05 | General theory of partial differential operators |
47J05 | Equations involving nonlinear operators (general) |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |