Evolution of spoon-shaped networks

Alessandra  Pluda; Alessandra  Pluda

doi:10.3934/nhm.2016007

Networks and Heterogeneous Media

2016, Volume 11, Issue 3: 509-526. doi: 10.3934/nhm.2016007

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Evolution of spoon-shaped networks

Alessandra Pluda ^{1
,}

1.
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127

Received: 01 March 2015 Revised: 01 September 2015
Primary: 53C44; Secondary: 53A04, 35K55.

We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain $\Omega$. The two curves meet only at one point, forming angles of $120$ degrees. The non-closed curve has a fixed end--point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.
- Curvature flow,
- networks,
- singularity formation,
- Brakke spoon,
- grain boundaries
Citation: Alessandra Pluda. Evolution of spoon-shaped networks[J]. Networks and Heterogeneous Media, 2016, 11(3): 509-526. doi: 10.3934/nhm.2016007

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Abstract

We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain $\Omega$. The two curves meet only at one point, forming angles of $120$ degrees. The non-closed curve has a fixed end--point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.

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