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- ItemOn a fractional harmonic replacement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dipierro, Serena; Valdinoci, EnricoGiven s e (0, 1), we consider the problem of minimizing the 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.