Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

Hyder, Ali; Segatti, Antonio; Sire, Yannick; Wang, Changyou

doi:10.1080/03605302.2022.2091453

Cited by 9 publications

(10 citation statements)

References 44 publications

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“…Moser Millot andSire [2015] contributed important results to the study of half-harmonic maps by exploiting the fact that for any smooth u : S 1 → ‫ޒ‬ n we can represent the half-Laplacian classically in the form…”

Section: Background and Resultsmentioning

confidence: 99%

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Untitled

2024

Analysis & PDE

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Section: Background and Resultsmentioning

confidence: 99%

“…Moreover, ū is conformal. For such mappings, smooth regularity on B was shown by Grüter, Hildebrandt, and Nitsche [Grüter et al 1981]; thus condition (1-6) holds everywhere on 1434 MICHAEL STRUWE ∂ B in the pointwise sense, and ū parametrizes a minimal surface of finite area supported by N which meets N orthogonally along its boundary.…”

Section: Appendixmentioning

confidence: 98%

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2024

Analysis & PDE

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“…Key features of the proof and related flow equations. In our approach, in a similar vein as [Lenzmann and Schikorra 2020], we uncover and exploit surprising regularity properties of the normal component dπ ⊥ N (u)∂ r u for the harmonic extension of u, likely related to the fractional commutator estimates for the normal projection in the work of Da Lio and Rivière [2011] or the regularity estimates of Da Lio and Pigati [2020], Mazowiecka and Schikorra [2018], and others.…”

Section: Background and Resultsmentioning

confidence: 99%

“…A different heat flow associated with half-harmonic maps, using the half-heat operator (∂ t − ) 1/2 instead of (1-1), was suggested by Hyder et al [2022], and they obtained global existence of partially regular, but possibly nonunique, weak solutions for their flow, with a possibly large singular set of measure zero.…”

Section: Background and Resultsmentioning

confidence: 99%

Plateau flow or the heat flow for half-harmonic maps

Struwe

2024

Analysis & PDE

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Using the interpretation of the half-Laplacian on S 1 as the Dirichlet-to-Neumann operator for the Laplace equation on the ball B, we devise a classical approach to the heat flow for half-harmonic maps from S 1 to a closed target manifold N ⊂ ‫ޒ‬ n , recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When N is a smoothly embedded, oriented closed curve ⊂ ‫ޒ‬ n , the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.I thank Amélie Loher and the anonymous referee for careful reading of the manuscript and useful suggestions.

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“…In particular, he used a Ginzburg-Landau approximation in the interior, hence keeping the same nonlinear boundary condition. Another approach was considered in [22], where the approximation is on the boundary.…”

Section: Introductionmentioning

confidence: 99%

Nematic liquid crystal flow with partially free boundary

Lin¹,

Sire²,

Wei³

et al. 2022

Preprint

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We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain. Contents Introduction Main resultsRoadmap to the construction 2. Reflection and symmetry of the full system 3. Notations and preliminaries 4. Approximation and improvement 5. Gluing system and derivation of the dynamics for parameters 6. Linear theories for linearized harmonic map heat flow and Stokes system 7. Solving the partially free boundary system 7.1. Couplings in the full system 7.2. Inner-outer gluing system and reduced problems 7.3. Weighted topologies and fixed point argument Appendix A. Derivation of the partially free boundary system Appendix B. Multiple bubbles: analysis of the interactions Appendix C. Analysis of the Stokes operator

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Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

Cited by 9 publications

References 44 publications

Untitled

Untitled

Plateau flow or the heat flow for half-harmonic maps

Nematic liquid crystal flow with partially free boundary

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