Boundary partial regularity for minimizers of discontinuous quasiconvex integrals with general growth (Q2099236)

The authors prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \[ \mathcal{F}(u):=\int_{\Omega}f(x, u, Du) dx, \] where f satisfies a uniform VMO condition with respect to the x-variable, is continuous with respect to u and has a general growth with respect to the gradient variable. c. The regularity theory for functionals of this form and related partial differential equations are one of the classical topics in the calculus of variations. The study of functionals/equations with general growth has been initiated by Marcellini. Partial regularity results for quasi-convex functionals with p-growth was studied by Mingione and collaborators. Here, the authors prove that if the boundary and the boundary datum are of class \(C^1\) then the minimizer u of the functional satisfying the general growth condition assumed in this paper is locally Hölder continuous for every Hölder exponent \(\alpha\in (0, 1)\) at any boundary point that is Lebesgue type, in some sense, with respect to Du. Moreover, the authors assume that the functional is non-degenerate, and this allows them to convert the original functional with \(C^1\) boundary datum to a functional with the zero boundary value.