  We present a general blow-up technique to obtain local regularity estimates
for solutions, and their derivatives, of second order elliptic equations in
divergence form in H\"older spaces with variable exponent. The procedure allows
to extend the estimates up to a portion of the boundary where Dirichlet or
Neumann boundary conditions are prescribed and produces a Schauder theory for
partial derivatives of solutions of any order $k\in\mathbb{N}$. The strategy
relies on the construction of a class of suitable regularizing problems and an
approximation argument. The estimates we obtain are sharp with respect to the
regularity or integrability conditions on variable coefficients, boundaries,
boundary data and right hand sides respectively in H\"older and Lebesgue
spaces, both with variable exponent
