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Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality
Published 13 Feb 2023 in math.AP and math.DG | (2302.06738v2)
Abstract: We study regularity of minimizing $p$-harmonic maps $u \colon B3 \to \mathbb{S}3$ for $p$ in the interval $[2,3]$. For a long time, regularity was known only for $p = 3$ (essentially due to Morrey) and $p = 2$ (Schoen-Uhlenbeck), but recently Gastel extended the latter result to $p \in [2,2+\frac{2}{15}]$ using a version of Kato inequality. Here, we establish regularity for a small interval $p\in [2.961,3]$ by combining Morrey's methods with Hardt and Lin's Extension Theorem. We also improve on the other result by obtaining regularity for $p \in [2,p_0]$ with $p_0 = \frac{3+\sqrt{3}}{2} \approx 2.366$. In relation to this, we address a question posed by Gastel and prove a sharp Kato inequality for $p$-harmonic maps in two-dimensional domains, which is of independent interest.
- Strong approximation of fractional Sobolev maps. J. Fixed Point Theory Appl., 15(1):133–153, 2014.
- Harmonic maps with defects. Comm. Math. Phys., 107(4):649–705, 1986.
- Liouville properties for p𝑝pitalic_p-harmonic maps with finite q𝑞qitalic_q-energy. Trans. Amer. Math. Soc., 368(2):787–825, 2016.
- J.-M. Coron and R. Gulliver. Minimizing p𝑝pitalic_p-harmonic maps into spheres. J. Reine Angew. Math., 401:82–100, 1989.
- Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109–160, 1964.
- H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960.
- A. Gastel. Regularity issues for Cosserat continua and p𝑝pitalic_p-harmonic maps. SIAM J. Math. Anal., 51(6):4287–4310, 2019.
- Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys., 105(4):547–570, 1986.
- Stable defects of minimizers of constrained variational principles. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5(4):297–322, 1988.
- The p𝑝pitalic_p-energy minimality of x/|x|𝑥𝑥x/|x|italic_x / | italic_x |. Comm. Anal. Geom., 6(1):141–152, 1998.
- R. Hardt and F.-H. Lin. Mappings minimizing the Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT norm of the gradient. Comm. Pure Appl. Math., 40(5):555–588, 1987.
- An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math., 138(1-2):1–16, 1977.
- M.-C. Hong. On the minimality of the p𝑝pitalic_p-harmonic map x|x|:Bn→Sn−1:𝑥𝑥→superscript𝐵𝑛superscript𝑆𝑛1\frac{x}{|x|}\colon B^{n}\to S^{n-1}divide start_ARG italic_x end_ARG start_ARG | italic_x | end_ARG : italic_B start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → italic_S start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT. Calc. Var. Partial Differential Equations, 13(4):459–468, 2001.
- M.-C. Hong. Partial regularity of stable p𝑝pitalic_p-harmonic maps into spheres. Bull. Austral. Math. Soc., 76(2):297–305, 2007.
- On the singular set of stable-stationary harmonic maps. Calc. Var. Partial Differential Equations, 9(2):141–156, 1999.
- C. P. Hopper. Partial regularity for holonomic minimisers of quasiconvex functionals. Arch. Ration. Mech. Anal., 222(1):91–141, 2016.
- N. Hungerbuehler. p-harmonic Flow. ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland).
- W. Jäger and H. Kaul. Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J. Reine Angew. Math., 343:146–161, 1983.
- Y. Li and C. Wang. Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity. Int. Math. Res. Not. IMRN, (6):4620–4658, 2022.
- F. Lin and C. Wang. The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
- F.-H. Lin. A remark on the map x/|x|𝑥𝑥x/|x|italic_x / | italic_x |. C. R. Acad. Sci. Paris Sér. I Math., 305(12):529–531, 1987.
- On the size of the singular set of minimizing harmonic maps. Mem. Amer. Math. Soc., to appear.
- P. Mironescu and J. Van Schaftingen. Trace theory for Sobolev mappings into a manifold. Ann. Fac. Sci. Toulouse Math. (6), 30(2):281–299, 2021.
- C. B. Morrey, Jr. The problem of Plateau on a Riemannian manifold. Ann. of Math. (2), 49:807–851, 1948.
- N. Nakauchi. Regularity of minimizing p𝑝pitalic_p-harmonic maps into the sphere. Boll. Un. Mat. Ital. A (7), 10(2):319–332, 1996.
- N. Nakauchi. Regularity of minimizing p𝑝pitalic_p-harmonic maps into the sphere. In Proceedings of the Third World Congress of Nonlinear Analysts, Part 2 (Catania, 2000), volume 47, pages 1051–1057, 2001.
- T. Okayasu. Regularity of minimizing harmonic maps into S4superscript𝑆4S^{4}italic_S start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, S5superscript𝑆5S^{5}italic_S start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT and symmetric spaces. Math. Ann., 298(2):193–205, 1994.
- R. Schoen and K. Uhlenbeck. A regularity theory for harmonic maps. J. Differential Geometry, 17(2):307–335, 1982.
- R. Schoen and K. Uhlenbeck. Regularity of minimizing harmonic maps into the sphere. Invent. Math., 78(1):89–100, 1984.
- D. Valtorta. Sharp estimate on the first eigenvalue of the p𝑝pitalic_p-Laplacian. Nonlinear Anal., 75(13):4974–4994, 2012.
- Regularity of p𝑝pitalic_p-harmonic maps into certain manifolds with positive sectional curvature. J. Reine Angew. Math., 466:1–17, 1995.
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