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On a fractional harmonic replacement

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  • Given $s\in(0,1)$, we consider the problem of minimizing the fractional Gagliardo seminorm in $H^s$ 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.
    Mathematics Subject Classification: 31A05, 35R11, 46E35.


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