Evolution Equations in Sup-Norm Context And in L2 Variational Context
G. Lumer · Birkhäuser Basel eBooks · 1978
We consider evolution equations in sup-norm context for arbitrary open subsets of some locally compact space Ω (with 0 boundary values, and initial values in an appropriate domain), associated to a local operator A, as treated in [5]. We compare these evolution problems with the corresponding L2 variational problems for “the same” operator, (at least when the open sets are relatively compact), showing that under rather general assumptions the “variational operator” AV associated to a given relatively compact open set V, indeed extends the “sup-norm operator” AV associated to the same set V, (both operators corresponding of course to the same underlying local operator A on Ω). This relies on maximum principle considerations through the use of approximation techniques and results as given in [6], as well as on the extension of a variational argument given by R. Beals for the case of the ordinary laplacian (1).