A Strong Maximum Principle for the Laplace Equation with Mixed Boundary Condition

In this work we present a comparison result for two solutions of the Laplace equation in a smooth bounded domain, satisfying the same mixed boundary condition (zero Dirichlet data on part of the boundary and zero Neumann data on the rest). The result is in some sense a generalization of the Hopf lemma to the case of mixed boundary conditions, where the barrier function is not given explicitly, but as the solution of the Laplace equation with a constant right hand side and mixed boundary condition