Minimizers of the $W^{1,1}$-energy of $\mathbb S^1$-valued maps with prescribed singularities. Do they exist? - Archive ouverte HAL Access content directly
Journal Articles Nonlinear Analysis: Theory, Methods and Applications Year : 2018

Minimizers of the $W^{1,1}$-energy of $\mathbb S^1$-valued maps with prescribed singularities. Do they exist?

Abstract

The paper is concerned with the least $W^{1,1}$-energy required to produce maps from a domain $\Omega\subset{\mathbb R}^2$ with values into ${\mathbb S}^1$ having prescribed singularities $(a_1)_{1\le i\le k}$. The value of infimum has been known for a long time and corresponds to the length of minimal configurations connecting the points $(a_i)$ between themselves and/or to the boundary. We tackle here the question whether the infimum of this $W^{1,1}$-energy is achieved. This natural topic turns out to be delicate and we have a complete answer only when $k=1$. The bottom line for $k\ge 1$ is that the infimum is ``rarely'' achieved. As a ``substitute'', we give a full description of the asymptotic behavior of all minimizing sequences and show that they ``concentrate'' along ``convex combinations'' of minimal configurations.
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Dates and versions

hal-01766732 , version 1 (13-04-2018)
hal-01766732 , version 2 (16-04-2018)
hal-01766732 , version 3 (28-04-2018)

Identifiers

  • HAL Id : hal-01766732 , version 3

Cite

Haïm Brezis, Petru Mironescu. Minimizers of the $W^{1,1}$-energy of $\mathbb S^1$-valued maps with prescribed singularities. Do they exist?. Nonlinear Analysis: Theory, Methods and Applications, 2018, 177 part A, pp.105-134. ⟨hal-01766732v3⟩
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