Interior decay of solutions to elliptic equations with respect to frequencies at the boundary

We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency of the Dirichlet boundary datum. We show that, as the frequency at the boundary grows, the square of a suitable norm of the solution in a compact subset of the domain decays in an inversely proportional manner with respect to the corresponding frequency. Under Lipschitz regularity assumptions, these estimates are essentially optimal and have important consequences for the choice of optimal measurements for corresponding inverse boundary value problems.