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

Haïm Brezis (1, 2) , Petru Mironescu (3)

1 Department of Mathematics - Rutgers School of Arts and Sciences
2 TECHNION - Department of Mathematics
3 EDPA - Équations aux dérivées partielles, analyse

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.

Mots clés: Sobolev spaces; singularities; circle-valued maps; Jacobian

Domaines: Analyse classique [math.CA]

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⟩