Regularity of Weak Solutions to Some Anisotropic Elliptic Equations

We consider boundary value problem of the form i£± i£² i£³ Pn i=1 Di(ai(x, Du(x))) = f, x ∈ Ω, u(x) = u∗(x), x ∈ ∂Ω. We show that regularity of boundary datum u∗ forces u to have regularity as well. A similar result is obtained for obstacle problem.