The case when f is of bounded variation and u Holder continuous
2] > 0, x [member of] [a, b], while the function u: [a, b] [right arrow] R satisfies some local Holder continuous, namely,
The case when f is absolutely continuous and u Holder continuous
T] (S) we denote the spaces of Holder continuous
functions and Holder continuous
tangential fields (0 < [alpha] < 1), respectively.
The authors show that solutions of large classes of subelliptic equations with bounded measurable coefficients are Holder continuous
X], which is Holder continuous with exponent [alpha] [member of] (0,1].
Then f is Holder continuous on [0,1] with Holder exponent 1/2:
7 The latter Lemma cannot be substantially improved, as in t = 1/4, the density f (t) is not Holder continuous with Holder exponent 1/2 + [member of] for any [member of] > 0.
Topics include a review of preliminaries such as continuous and Holder continuous
functions, Sobolev spaces and convex analysis; classical methods such as Euler-Lagrange equations; direct methods such as the Dirichlet integral; regularity, such as the one-dimensinal case; minimal surfaces such as in the Douglas- Courant-Tonelli method; and isoperimetric inequality.