Math. Ann. 262, 549-561 (1983) maen 9 Spdnger-Verlag 1983 Boundary Regularity for Minima of Certain Quadratic Functionals J. Jost and M. Meier* Mathematisches Institut der Universit~it, Wegelerstrasse 10. D-5300 Bonn, Federal Republic of Germany 0. Introduction Recently, Giaquinta and Giusti [4, 5] obtained several important results concern- ing the interior partial regularity for minima of quadratic variational integrals. It was shown in particular that such minima are H61der continuous in the interior of the underlying domain 12C~", except for a relatively closed set 1; whose Hausdorff dimension is less than n-2. Under an additional splitting condition on the coefficients of the functional, Giaquinta and Giusti also proved that the dimension of Z does not even exceed n-3. Furthermore, complete interior regularity has been established in the case when a certain one-sided condition holds (cf. [4]; it was pointed out in [7] how this condition can be generalized and suitably interpreted in a geometrical context). The above-mentioned results apply in particular to harmonic maps the images of which are covered by a single coordinate chart in the target manifold. On the other hand, taking a slightly different point of view, Schoen and Uhlenbeck [10] proved similar results for energy minimizing harmonic mappings between Riemannian manifolds, even without the coordinate chart restriction (but under stronger regularity conditions on the coefficients of the integral). Their method could also be modified to show complete boundary regularity for energy minimizing harmonic mappings with prescribed Dirichlet boundary values (cf. [i I]). The purpose of our paper is to establish boundary regularity for (bounded) minima of quadratic functionals of the type considered by Giaquinta and Giusti in [5]. In the first section, we will extend various interior results of [4, 5] to the case of boundary points. A combination with an argument due to Wood [12] yields the following theorem, the proof of which can be found in Sect. 2. (The corresponding result in [11] mentioned before has been obtained independently.) Theorem. Let 12 be a bounded open domain in F,", n > 2, with boundary 012 of class C 1. Assume that A~(x, v)= G~(x)gij(x,v) are bounded continuous coefficients on • IR N (~, fl = 1 ..... n" i,j = 1,..., N) satisfying the conditions * The authors have been supported by the Sonderforschungsbereich 72 at the University of Bonn