On a fractional harmonic replacement
- Dipierro, Serena
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 31A05 35R11 46E35
- Harmonic replacement, fractional Sobolev spaces, energy estimates
Given $s ∈(0,1)$, we consider the problem of minimizing the 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.
- Discrete Contin. Dyn. Syst., 35 (2015) pp. 3377--3392.