Everywhere 𝒞α-estimates for a class of nonlinear elliptic systems with critical growth

Abstract

Abstract. We obtain everywhere 𝒞α-regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral ∫ F ( u , ∇ u ) d x ${\int F(u,\nabla u)\, dx}$ with quadratic growth in ∇ u ${\nabla u}$ and which satisfies a generalized splitting condition that cover the case F ( u , ∇ u ) : = ∑ i Q i , $ {F(u,\nabla u):= \sum _i Q_i},\vspace*{-2.27621pt} $ where Q i : = ∑ α β A i α β ( u , ∇ u ) ∇ u α · ∇ u β ${Q_i := \sum _{\alpha \beta } A^{\alpha \beta }_i(u,\nabla u) \nabla u^{\alpha } \cdot \nabla u^{\beta }}$ , or the case F ( u , ∇ u ) : = ∏ i ( 1 + Q i ) θ i . $ {F(u,\nabla u) := \prod _i (1+Q_i)^{\theta _i}}.\vspace*{-2.13394pt} $ A crucial assumption is the one-sided condition F u ( u , η ) · u ≥ - K $ F_u (u,\eta ) \cdot u \ge -K\vspace*{-0.62596pt} $ and related generalizations. In the elliptic case we obtain existence of 𝒞α-solutions. If the leading operator is not necessarily elliptic but coercive, possible minima are everywhere Hölder continuous and the same holds also for Noether solutions, i.e., extremals which are also stationary with respect to inner variations. In particular if A α β ( u , ∇ u ) = A α β ( u ) ${A^{\alpha \beta }(u,\nabla u)=A^{\alpha \beta } (u)}$ , our result generalizes a result of Giaquinta and Giusti. The technique of our proof (using weighted norms and inhomogeneous hole-filling method) does not rely on L ∞ ${L^{\infty }}$ -a priori estimates for the solution.