Abstract
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.
Highlights
Given a smooth function φ : Rn → Rn and a subset Tε of Rn, we consider vε(x) solution of the following obstacle problem: vε(x)≥ φ(x) for x ∈ Tε (−∆)svε (−∆)svε ≥ =0 0 for x ∈ Rn for x ∈ Rn \ Tε and for x ∈ Tε if vε(x) > φ(x). (1)The operator (−∆)s denotes the fractional Laplace operator of order s, where s is a real number between 0 and 1
We will see that this system of equations can be stated as a boundary obstacle problem for elliptic degenerate equations
In the remainder of this section, we briefly motivate the problem and we introduce the extension problem for the fractional Laplace operators, which allows us to rewrite (1) as a boundary obstacle problem for a local elliptic operator
Summary
Given a smooth function φ : Rn → Rn and a subset Tε of Rn, we consider vε(x) solution of the following obstacle problem: vε(x). In the case of the regular Laplace operator, this problem was first studied for periodic Tε by L. In the remainder of this section, we briefly motivate the problem and we introduce the extension problem for the fractional Laplace operators, which allows us to rewrite (1) as a boundary obstacle problem for a local (degenerate) elliptic operator. When s = 1/2, (1) naturally arises as a boundary obstacle problem for the regular Laplace operator ( know as Signorini problem): We consider the following problem set in the upper-half space Rn++1 = {(x, y) ∈ Rn × R ; y ≥ 0}:. The concentration inside the cell is given by the solution u(x, y) of (4)
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