Second-order asymptotics of fractional Gagliardo seminorms as $$s\rightarrow 1^-$$ and convergence of the associated gradient flows

We study the second-order asymptotic expansion of the <i>s</i>-fractional Gagliardo seminorm as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi> <mml:mo>→</mml:mo> <mml:msup><mml:mn>1</mml:mn> <mml:mo>-</mml:mo></mml:msup> </mml:mrow> </mml:math> in terms of a higher-order nonlocal functional. We prove a Mosco-convergence result for the energy functionals and characterize the domain of the limit, which can be interpreted as the space of functions of differentiability order "logarithmically larger" than one. This is also motivated by the fact that the first variation of the limit is the formal derivative of the (scaled) <i>s</i>-fractional Laplacian as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi> <mml:mo>→</mml:mo> <mml:msup><mml:mn>1</mml:mn> <mml:mo>-</mml:mo></mml:msup> </mml:mrow> </mml:math> . In the second part of the paper, we show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:math> -gradient flows of the energies converge to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:math> -gradient flows of the Mosco-limit.