Stability for the Surface Diffusion Flow

Abstract We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the n –dimensional flat torus. More precisely, we show that if a smooth set is “close enough” to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a “translated” of the critical set. This result was obtained in dimension $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> by Acerbi, Fusco, Julin and Morini in [1] (extending previous results for spheres of Escher, Mayer and Simonett [15], Wheeler [37, 38] and Elliott and Garcke [14]). Our work generalizes such conclusion to any dimension $$n\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . For sake of clarity, we show all the details in dimension $$n=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and we list the necessary modifications to the quantities involved in the proof in the general n –dimensional case, in the last section.