Hölder Continuity of Local Minimizers

H�lder continuity of local minimizers of vectorial integral functionals

“…In this section we prove Theorem 1.2. The proof is similar to that of Theorem 3.1 in [3], where the scalar case m = 1 is considered. However, differently from the scalar case, when p < N and m > 1, we do not have a priori Hölder continuity of the local minimizer.…”

“…Hölder continuity of local minimizers of vectorial integral functionals 273 [3] is an up to the boundary higher integrability result for scalar-valued local minimizers. With obvious changes, the proof holds for vectorvalued Q-minimizers, too.…”

“…I. Fonseca and N. Fusco prove in [7] that if F = F (z) satisfies (1.2) and (v), then local minimizers are Lipschitz continuous. When F also depends on x and u, then (1.2), (iv) and (v) imply that local minimizers are Hölder continuous for any exponent less then 1, but they may not be Lipschitz continuous (see G. Cupini, N. Fusco and R. Petti [3]). …”

A regularity result for a convex functional and bounds for the singular set

“…In the first step we assume that f (·, ξ) ∈ C 2 but we are able to establish the estimates (1.6) and (1.9) independently of the C 2 norm of the integrand f , by adopting an argument first used in [14]. In the second step we remove the assumption f (·, ξ) ∈ C 2 using an approximation procedure introduced in [14] and developed in [7,10,15]. More precisely we approximate f by a sequence {f h } of C 2 functions which are strictly elliptic (and the ellipticity constant is precisely the ν appearing in (H2)).…”

“…We need now the following fundamental result that can be obtained with a procedure first introduced in [14,15] and then developed in [7,10], that plays a key role in the completion of the proof of our theorems. …”