On a fractional harmonic replacement

Given s ∈ (0, 1), we consider the problem of minimizing the fractional Gagliardo seminorm in Hs with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set K. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set A to K increases the energy of at most the measure of A (this may be seen as a perturbation result for small sets A). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.