A Singular Perturbation Result with a Fractional Norm
Adriana Garroni, Giampiero Palatucci · Birkhäuser Basel eBooks · 2006
Let I be an open bounded interval of ℝ and W a non-negative continuous function vanishing only at α, β ∈ ℝ. We investigate the asymptotic behavior in terms of Γ-convergence of the following functional $$ G_\varepsilon (u): = \varepsilon ^{p - 2} \iint_{I \times I} {\left| {\frac{{u(x) - u(y)}} {{x - y}}} \right|^p dxdy + \frac{1} {\varepsilon }\int_I {W(u)dx (p > 2),} } $$ , as ɛ → 0.