Minimal lifting index and loss of regularity for $S^1$-valued Sobolev maps in dimension $n\ge3$, and lifting in Bessel potential spaces

When does an $S^1$-valued Sobolev map have a phase, and how much regularity must be sacrificed to obtain one? Bourgain, Brezis and Mironescu (2000) classified the pairs $(s,p)$ for which every $u\in W^{s,p}(\Omega;S^1)$, with $\Omega\subset\mathbb{R}^n$ smooth bounded simply connected, admits a phase $\varphi$ with $u=e^{i\varphi}$ in the same space $W^{s,p}$, and posed the quantitative refinement: determine the minimal lifting index $t^*(s,p,n)$, the supremum of all $t\in[0,s]$ such that every $u\in W^{s,p}(\Omega;S^1)$ has a phase in $W^{t,p}(\Omega)$ — the precise loss of regularity in lifting, stated as open for $n\ge3$, with the exemplar $n=3$, $p=4$: does some $u\in W^{s,4}$ for all $s<3/4$ have no phase even in $W^{1/4,4}$? A second open component concerns the Bessel potential (Triebel-Lizorkin) spaces $H^{s,p}=F^s_{p,2}$, where lifting is known only for $sp<1$ with $p\ge2$, and for $p=2$ (where $H^{s,2}=W^{s,2}$) by the Sobolev classification. The surrounding theory is settled — sum-space decompositions, density, optimal phase estimates, the Besov-scale and covering-space analogues, and the $d=1$ optimality of the 2000 estimate, settled in 2002 — but $t^*$ is determined only on the topological range $1\le sp<2$, and the remaining Bessel-potential ranges are open; both components remain open.

Background

Let $\Omega\subset\mathbb{R}^n$ be smooth, bounded and simply connected, and let $S^1=\{z\in\mathbb{C}:|z|=1\}$ be the unit circle. Maps $u:\Omega\to S^1$ are the natural state variables of models with a phase (Ginzburg-Landau vortices, liquid crystals), and one repeatedly needs a real-valued phase $\varphi$ with $u=e^{i\varphi}$ almost everywhere. For smooth $u$ a phase exists locally wherever $u$ has no winding; for rough $u$ the question is which function space the phase can occupy.

For $0<s<1$ and $1\le p<\infty$, the Sobolev-Slobodeckij space $W^{s,p}(\Omega)$ consists of the $L^p$ functions with finite Gagliardo seminorm $(\int_\Omega\int_\Omega|u(x)-u(y)|^p/|x-y|^{n+sp}\,dx\,dy)^{1/p}$; for integer $s\ge1$ it is the usual Sobolev space with $s$ derivatives in $L^p$. By $W^{s,p}(\Omega;S^1)$ one denotes the maps with values in $S^1$ a.e. A map is liftable in $W^{s,p}$ if it has a phase in $W^{s,p}(\Omega;\mathbb{R})$.

Bourgain, Brezis and Mironescu [1] classified liftability without loss of regularity: every $u\in W^{s,p}(\Omega;S^1)$ has a phase in $W^{s,p}$ exactly when (i) $n=1$; or (ii) $0<s<1$ with $sp<1$ or $sp\ge n$; or (iii) $s\ge1$ with $sp\ge2$. In the remaining negative ranges two explicit obstructions exist. The topological example $u(x)=(x_1,x_2)/|(x_1,x_2)|$ (defined off the codimension-2 plane $\{x_1=x_2=0\}$, extended arbitrarily on it) belongs to $W^{s,p}$ exactly when $sp<2$; it winds once around almost every circle linking the singular set. Off a branch cut any phase $\varphi$ differs from the smooth angular branch by a $2\pi\mathbb{Z}$-valued function, which is a.e. constant by the constancy lemma (a $\mathbb{Z}$-valued function in $W^{r,q}$ with $rq\ge1$ is a.e. constant on connected sets); hence every phase inherits the $2\pi$ jump of the branch across the cut, and such a jump is incompatible with membership in $W^{t,p}$ when $tp\ge1$. So no phase exists even in $W^{1/p,p}$. The analytical example uses, when $0<s<1$ and $1\le sp<n$, a phase $\psi\in W^{1,sp}\setminus W^{s,p}$ with $e^{i\psi}\in W^{s,p}$ (the latter by the Gagliardo-Nirenberg embedding $W^{1,sp}\cap L^\infty\hookrightarrow W^{s,p}$); every phase of $e^{i\psi}$ differs from $\psi$ by a $2\pi\mathbb{Z}$-valued function, and the same constancy lemma forces $\psi\in W^{s,p}$, a contradiction.

In the negative ranges one therefore asks how much regularity must be sacrificed. Define the minimal lifting index $t^*(s,p,n)$ as the supremum of all $t\in[0,s]$ such that every $u\in W^{s,p}(\Omega;S^1)$ has a phase in $W^{t,p}(\Omega)$. The construction of [1] always produces phases in $W^{t,p}$ once $tp<1$, and the topological example blocks $t\ge1/p$ whenever $sp<2$; hence on the range $1\le sp<2$ the index equals $1/p$ by the results of [1] itself. For $n\ge3$, however, the negative range also contains the region $0<s<1$ with $2\le sp<n$, where the topological example is absent, and [1] states verbatim: "When n ≥ 3 the precise loss of regularity in lifting is not fully understood." For the exemplar $n=3$, $p=4$, [1] notes that any $u\in W^{s,4}$ has a phase in $W^{s,4}$ if $s<1/4$ or $s\ge3/4$, that some maps in $W^{s,4}$ with $1/4\le s<3/4$ have no such phase, and asks whether there exists $u\in W^{s,4}$ for all $s<3/4$ which has no phase even in $W^{1/4,4}$.

The same paper raised the question for Bessel potential spaces $H^{s,p}(\Omega)=(I-\Delta)^{-s/2}L^p(\Omega)$, the Triebel-Lizorkin scale $F^s_{p,2}$: these coincide with $W^{s,2}$ for $p=2$ but differ from $W^{s,p}$ for $p\ne2$ (which is close to the Besov scale $B^s_{p,p}$). Appendix F of [1] proves lifting in $H^{s,p}$ when $sp<1$ and $p\ge2$, and states that the remaining cases — $p<2$, or $sp\ge1$ — "have not been investigated"; for $p=2$, where $H^{s,2}=W^{s,2}$, the full Sobolev classification of [1] of course applies.

Subsequent work settled the surrounding structure theory: phases in sums of spaces $W^{s,p}+W^{1,sp}$ [4, 5], the structure of $W^{s,p}(\mathbb{S}^n;S^1)$ through singular sets [7], factorizations $u=ve^{i\varphi}$ [6], density of smooth maps [8], optimal phase estimates [2, 9], the Besov-scale classification [10], lifting for compact covering spaces [11], the monograph synthesis [12], and a recent expository account [13].

Problem Statement

Let $n\ge3$, let $\Omega\subset\mathbb{R}^n$ be smooth, bounded and simply connected, and let $1<p<\infty$. First, for every $s>0$ such that $(s,p)$ lies in the negative ranges of the lifting classification — (i) $0<s<1$ with $1\le sp<n$, or (ii) $s\ge1$ with $1\le sp<2$ — determine the minimal lifting index $t^*(s,p,n)$, defined as the supremum of all $t\in[0,s]$ such that every $u\in W^{s,p}(\Omega;S^1)$ admits a phase $\varphi\in W^{t,p}(\Omega)$ with $u=e^{i\varphi}$ a.e., together with the endpoint behavior (whether lifting holds at $t=t^*$ itself). Of these, the range $1\le sp<2$ has the known value $t^*=1/p$ from the primary source [1], and $sp=n$ already gives $t^*=s$ by the classification, so the determination reduces to the still-open part: the region $0<s<1$ with $2\le sp<n$, which is nonempty exactly when $n\ge3$, together with the remaining borderline cases $sp=2$, the endpoint $t=1/p$, and the borderline index $t=1-n/(sp)+n/p$ when it exceeds $1/p$ (where the embedding $W^{1,sp}\hookrightarrow W^{t,p}$ is critical). Answer, as a separate question not implied by the per-$(s,p)$ determination of $t^*$ (which does not by itself supply a single map lying in $W^{s,4}$ for all $s<3/4$ at once), the exemplar of [1] for $n=3$, $p=4$: does there exist a map $u\in W^{s,4}(\Omega;S^1)$ for every $s<3/4$ which has no phase in $W^{1/4,4}(\Omega)$? The index is fixed by two matching bounds: every $u\in W^{s,p}(\Omega;S^1)$ has a phase in $W^{t,p}$ for all $t$ up to the index, while some $u\in W^{s,p}(\Omega;S^1)$ has no phase in $W^{t,p}$ for any $t$ above the index. Second, for the Bessel potential spaces $H^{s,p}(\Omega)=F^s_{p,2}(\Omega)$: determine for which $(s,p)$ every $u\in H^{s,p}(\Omega;S^1)$ has a phase in $H^{s,p}(\Omega)$. Known: yes for $sp<1$, $p\ge2$ (Corollary F.1 of [1]), and for $p=2$ by the classification of [1] since $H^{s,2}=W^{s,2}$; open: $p<2$ with arbitrary $s>0$, and $sp\ge1$ with $p>2$.

References

1. J. Bourgain, H. Brezis, P. Mironescu, "Lifting in Sobolev spaces", Journal d'Analyse Mathématique 80 (2000), 37-86. DOI: 10.1007/BF02791533. Author manuscript: https://hal.science/hal-00747691.
2. J. Bourgain, H. Brezis, P. Mironescu, "Limiting embedding theorems for $W^{s,p}$ when $s\uparrow1$ and applications", Journal d'Analyse Mathématique 87 (2002), 77-101. DOI: 10.1007/BF02868470. https://doi.org/10.1007/BF02868470. [Its Theorem 2 proves the one-dimensional optimality.]
3. P. Mironescu, "Sobolev maps on manifolds: degree, approximation, lifting", in: Perspectives in Nonlinear Partial Differential Equations (in honor of Haïm Brezis), Contemporary Mathematics 446, American Mathematical Society, 2007, 413-436. DOI: 10.1090/conm/446/08642. https://hal.science/hal-00747679.
4. P. Mironescu, "Lifting default for $S^1$-valued maps", Comptes Rendus Mathématique 346 (2008), 1039-1044. DOI: 10.1016/j.crma.2008.08.001. https://hal.science/hal-00747676.
5. P. Mironescu, "Lifting of $S^1$-valued maps in sums of Sobolev spaces", preprint, June 2008. HAL: hal-00747663. https://hal.science/hal-00747663. [Circulated preprint with detailed proofs of the announcement [4].]
6. P. Mironescu, "Decomposition of $S^1$-valued maps in Sobolev spaces", Comptes Rendus Mathématique 348 (2010), 743-746. DOI: 10.1016/j.crma.2010.06.020. https://hal.science/hal-00747677.
7. P. Mironescu, "$S^1$-valued Sobolev maps", Journal of Mathematical Sciences 170 (2010), 340-355. DOI: 10.1007/s10958-010-0090-z. https://hal.science/hal-00356333.
8. H. Brezis, P. Mironescu, "Density in $W^{s,p}(\Omega;N)$", Journal of Functional Analysis 269 (2015), 2045-2109. DOI: 10.1016/j.jfa.2015.04.005. https://hal.science/hal-01101299.
9. P. Mironescu, I. Molnar, "Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces", Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire 32 (2015), 965-1013. DOI: 10.1016/j.anihpc.2014.04.005. https://hal.science/hal-00960317.
10. P. Mironescu, E. Russ, Y. Sire, "Lifting in Besov spaces", Nonlinear Analysis 193 (2020), 111489. DOI: 10.1016/j.na.2019.03.012. arXiv:1705.04271. https://arxiv.org/abs/1705.04271.
11. P. Mironescu, J. Van Schaftingen, "Lifting in compact covering spaces for fractional Sobolev mappings", Analysis & PDE 14 (2021), 1851-1871. DOI: 10.2140/apde.2021.14.1851. arXiv:1907.01373. https://hal.science/hal-02171533.
12. H. Brezis, P. Mironescu, "Sobolev Maps to the Circle: From the Perspective of Analysis, Geometry, and Topology", Progress in Nonlinear Differential Equations and their Applications 96, Birkhäuser, New York, 2021. DOI: 10.1007/978-1-0716-1512-6. https://link.springer.com/book/10.1007/978-1-0716-1512-6.
13. P. Mironescu, "Sobolev maps to manifolds", graduate course notes, in: Function Spaces and Applications XII, Matfyz Press, Prague, 2023, pp. 1-74. ISBN: 978-80-7378-485-0. HAL: hal-04101540. https://hal.science/hal-04101540.