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We study the -norm of the function 1 on tubular neighbourhoods of curves in ${ℝ}^2$. We take the limit of small thickness, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit → 0, containing contributions from the length of the curve (at order ), the ends ( ), and the curvature ( ). The second result is a Γ-convergence result, in which the central curve may vary along the sequence...

We study the -norm of the function 1 on tubular neighbourhoods of curves in ${ℝ}^2$. We take the limit of small thickness , and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit → 0, containing contributions from the length of the curve (at order ), the ends ( ), and the curvature ( ). The second result is a Γ-convergence result, in which the central curve may vary along the...

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}\left({ℝ}^n\right)$. This criterion consists of comparing the value of a functional $\int f\left(u\right)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization...