Boundary Regularity for Quasiminima

Abstract

In [GG], Giaquinta & Giusti introduced the notion of quasiminima, a concept which is very useful in unifying the C ^0,α-regularity theory of elliptic equations in divergence form. In particular, their treatment provides regularity results for minima of nondifferentiable variational integrals. They also show that weak solutions of nonlinear elliptic equations in divergence form, under natural and general structural conditions, are quasiminima as are solutions to a wide class of obstacle problems. Because of the great interplay between quasiminima and the Hölder theory of partial differential equations, it is important to develop the subject fully. Indeed, significant progress has already been made. Recently, for example, Harnack inequalities have been established in [DBT] for sub and super-quasiminima which run parallel to the results obtained by Trudinger [T] for weak sub and supersolutions of quasilinear equations in divergence form. In particular, the results in [DBT] imply that quasiminima are Hölder continuous, a result first established by Giaquinta & Giusti, [GG].