Regularity for systems

Abstract

In the last section of the previous chapter we presented De Giorgi’s regularity result for solutions u ∈ H¹ (Ω;ℝ) of the scalar elliptic problem

$$\sum\limits_{\alpha ,\beta } {{\partial _{{\chi _\alpha }}}({A^{\alpha \beta }}(x){\partial _{{\chi _\beta }}}u(x)) = 0} $$

with bounded Borel coefficients A^αß satisfying λI ≤ A ≤ ΛI: we proved that in fact $u \in C_{{\text{loc}}}^{0,\upsilon }(\Omega ;\mathbb{R})$, with ϑ = ϑ(n, λ, Λ).