  We deal with boundary value problems for second-order nonlinear elliptic
equations in divergence form, which emerge as Euler-Lagrange equations of
integral functionals of the Calculus of Variations built upon possibly
anisotropic norms of the gradient of trial functions. Integrands with non
polynomial growth are included in our discussion. The $W^{1,2}$-regularity of
the stress-field associated with solutions, namely the nonlinear expression of
the gradient subject to the divergence operator, is established under the
weakest possible assumption that the datum on the right-hand side of the
equation is a merely $L^2$-function. Global regularity estimates are offered in
domains enjoying minimal assumptions on the boundary. They depend on the weak
curvatures of the boundary via either their degree of integrability or an
isocapacitary inequality. By contrast, none of these assumptions is needed in
the case of convex domains. An explicit estimate for the constants appearing in
the relevant estimates is exhibited in terms of the Lipschitz characteristic of
the domains, when their boundary is endowed with H\"older continuous
curvatures.
