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Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

“…Thanks to (84), we see that H(t) ∼ t −1/2 → 0 as t → +∞, while T (t) is a bounded function of t and, in fact, it converges to a finite limit as t → +∞: The proof of Lemma 13 builds upon classical compactness arguments for harmonic maps which are due to Luckhaus [22], and is based on the following result, which is an adaptation of Lemma 1 in [22]. In contrast with the case considered in [22], we are dealing here with 2-dimensional domains only; on the other hand, we have to include in our analysis the Ginzburg-Landau potential (1 − |u| 2 ) 2 , which is not present in [22].…”

“…The proof of Lemma 13 builds upon classical compactness arguments for harmonic maps which are due to Luckhaus [22], and is based on the following result, which is an adaptation of Lemma 1 in [22]. In contrast with the case considered in [22], we are dealing here with 2-dimensional domains only; on the other hand, we have to include in our analysis the Ginzburg-Landau potential (1 − |u| 2 ) 2 , which is not present in [22]. Similar results, for quadratic energies on three-dimensional domains, have been proven in [6].…”

Quantitative regularity for $p$-harmonic maps

“…By compactness, u i (sub)converges weakly in the W 1,p sense to a function u. According to [HL87, Corollary 2.8], since u i are p-minimizers the convergence is also strong W 1,p sense, and it is a minimizer by [Luc88] (see also [Sim96, section 2.9]). Moreover, by passing to a subsequence if necessary, we have lim i→∞ x (i) j = x j , and span(x j ) k+1 j=0 is a k + 1 dimensional subspace.…”

“…The strong convergence of T i and the fact that T is a Euclidean p-minimizer can be proved by a simple adaptation of [SU82, Proposition 4.7 and Proposition 5.2]. Alternatively, one can use the technique of ǫ-almost minimizers developed in [Luc88] (see also [Sim96, section 2]).…”

“…Indeed, note that many arguments in the proofs of these results rely on some compactness properties enjoyed by the family of p-minimizers. That is, if a sequence u i of p-minimizers converges weakly in the W 1,p sense to some u, then the convergence is actually strong and u is a p-minimizer (see [Luc88] or [Sim96, section 2.9]). Stationary maps do not enjoy this compactness property, and thus are in general worse behaved than minimizing ones.…”