Everywhere 𝒞 α -estimates for a class of nonlinear elliptic systems with critical growth
Miroslav Bulíček, Jens Frehse, Mark Steinhauer · Advances in Calculus of Variations · 2012
Abstract. We obtain everywhere 𝒞 α -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 ${\int F(u, abla u)\, dx}$ with quadratic growth in ∇ u ${ abla u}$ and which satisfies a generalized splitting condition that cover the case F ( u , ∇ u ) : = ∑ i Q i , $ {F(u, abla u):= \sum _i Q_i},\vspace*{-2.27621pt} $ where Q i : = ∑ α β A i α β ( u , ∇ u ) ∇ u α · ∇ u β ${Q_i := \sum _{\alpha \beta } A^{\alpha \beta }_i(u, abla u) abla u^{\alpha } \cdot abla u^{\beta }}$ , or the case F ( u , ∇ u ) : = ∏ i ( 1 + Q i ) θ i . $ {F(u, abla 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} $