Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition

This paper is concerned with systems of nonlinear partial differential equations −Dαai (x, u,∇u) = bi(x, u,∇u) (i = 1, . . . , N) where the coefficients ai are assumed to satisfy the condition ai (x, u, ξ)− ai (y, v, ξ) ≤ ω |x− y|+ |u− v| (1 + |ξ|) for all {x, u}, {y, v} ∈ Ω × R and all ξ ∈ R , and where R 1 0 ω(t) t dt < +∞ while the functions ∂ai ∂ξ j β satisfy the standard boundedness and ellipticity conditions and the function ξ 7→ bi(x, u, ξ) may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions u to the above system provided the usual smallness condition on ‖u‖L∞(Ω) is fulfilled.