WIAS Preprint No. 1928, (2014)

On a fractional harmonic replacement


  • Dipierro, Serena
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

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.

Appeared in

  • Discrete Contin. Dyn. Syst., 35 (2015) pp. 3377--3392.

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