Besides bounded mappings from a Sobolev space to its dual, there is an alternative understanding of differential operators as unbounded operators from a (typically dense) subset of a function space to itself. This calls for a generalization of a monotonicity concept for mappings D → X, with X a Banach space and D its subset. Moreover, X need not be reflexive because the weak-compactness arguments will be replaced by metric properties and completeness. The main benefit from this approach will be achieved for evolution problems in Chapter 9 but the method is of some interest in steady-state problems themselves.
KeywordsWeak Solution Duality Mapping Integral Identity Distributional Solution Uniform Convexity
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