Higher regularity of weak solutions of strongly nonlinear elliptic equations
Sobolev’s original definition of his spaces is revisited. It only assumed that is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions with respect to appropriate norms, and equivalence of these norms is proved.
We discuss how the choice of the functional setting and the definition of the weak solution affect the existence and uniqueness of the solution to the equation where is a very general domain in , including the case .
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