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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)].

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