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ISSN 1947-6221(e) ISSN 0065-9266(p)

-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians

Authors: Tadeusz Iwaniec and Jani Onninen
Journal: Memoirs of the AMS
MSC (2000): Primary 30C65, 30C75, 35J20
Posted: September 19, 2011
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Abstract: The central theme of this paper is the variational analysis of homeomorphisms $h \colon \mathbb{X} \onto \mathbb{Y}$ between two given domains $\mathbb{X} , \mathbb{Y} \subset \mathbb{R}^n$ . We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb{X},\mathbb{Y})$ which minimize the energy integral

$\displaystyle {\mathscr E}_h=\int_{{\mathbb{X}}} \vert\vert Dh(x) \vert\vert ^n \textrm{d}x.$

Because of the natural connections with quasiconformal mappings this

-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal

-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.

References

Bibliography

1. Stuart S. Antman, Fundamental mathematical problems in the theory of nonlinear elasticity, Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 35–54. MR 0468503 (57 #8335)
2. K. Astala, T. Iwaniec, and G. Martin, Deformations of annuli with smallest mean distortion, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 899–921. MR 2591976 (2011c:30059), http://dx.doi.org/10.1007/s00205-009-0231-z
3. Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875 (2010j:30040)
4. K. Astala, T. Iwaniec, G. J. Martin, and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 (2005), no. 3, 655–702. MR 2180459 (2006h:30016), http://dx.doi.org/10.1112/S0024611505015376
5. John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 0475169 (57 #14788)
6. J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), no. 2, 135–174. MR 615159 (83f:49031), http://dx.doi.org/10.1016/0022-1236(81)90085-9
7. B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in 𝑅ⁿ, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786 (85h:30023)
8. Eugenio Calabi and Philip Hartman, On the smoothness of isometries, Duke Math. J. 37 (1970), 741–750. MR 0283727 (44 #957)
9. Y. W. Chen, Discontinuity and representations of minimal surface solutions, Proceedings of the conference on differential equations (dedicated to A. Weinstein), University of Maryland Book Store, College Park, Md., 1956, pp. 115–138. MR 0086557 (19,203e)
10. Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420 (89e:73001)
11. Cristina, J., Iwaniec, T., Kovalev, L. V., and Onninen, J. The Hopf-Laplace equation, arXiv:1011.5934.
12. Marianna Csörnyei, Stanislav Hencl, and Jan Malý, Homeomorphisms in the Sobolev space 𝑊^{1,𝑛-1}, J. Reine Angew. Math. 644 (2010), 221–235. MR 2671780 (2011h:46048), http://dx.doi.org/10.1515/CRELLE.2010.057
13. Frank Duzaar and Giuseppe Mingione, The 𝑝-harmonic approximation and the regularity of 𝑝-harmonic maps, Calc. Var. Partial Differential Equations 20 (2004), no. 3, 235–256. MR 2062943 (2005d:35094), http://dx.doi.org/10.1007/s00526-003-0233-x
14. Peter Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. MR 2048384 (2005d:31001)
15. Dominic G. B. Edelen, The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal. 11 (1962), 117–121. MR 0150623 (27 #618)
16. F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 0139735 (25 #3166)
17. Zheng-Chao Han, Remarks on the geometric behavior of harmonic maps between surfaces, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, 1996, pp. 57–66. MR 1417948 (98a:58048)
18. Robert Hardt and Fang-Hua Lin, Mappings minimizing the 𝐿^{𝑝} norm of the gradient, Comm. Pure Appl. Math. 40 (1987), no. 5, 555–588. MR 896767 (88k:58026), http://dx.doi.org/10.1002/cpa.3160400503
19. Stanislav Hencl and Pekka Koskela, Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 75–95. MR 2211707 (2007e:30020), http://dx.doi.org/10.1007/s00205-005-0394-1
20. Stanislav Hencl, Pekka Koskela, and Jani Onninen, A note on extremal mappings of finite distortion, Math. Res. Lett. 12 (2005), no. 2-3, 231–237. MR 2150879 (2006a:30022)
21. Stefan Hildebrandt and Johannes C. C. Nitsche, A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles, Arch. Rational Mech. Anal. 79 (1982), no. 3, 189–218. MR 658387 (83k:49068), http://dx.doi.org/10.1007/BF00251903
22. Tadeusz Iwaniec, On the concept of the weak Jacobian and Hessian, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 181–205. MR 1886622 (2003f:30027)
23. Tadeusz Iwaniec, Pekka Koskela, and Jani Onninen, Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), no. 3, 507–531. MR 1833892 (2002c:30029), http://dx.doi.org/10.1007/s002220100130
24. Iwaniec, T., Koh, N.-T., Kovalev, L. V., and Onninen, J. Existence of energy-minimal diffeomorphisms between doubly connected domains, Invent. Math., to appear.
25. Tadeusz Iwaniec, Leonid V. Kovalev, and Jani Onninen, Harmonic mappings of an annulus, Nitsche conjecture and its generalizations, Amer. J. Math. 132 (2010), no. 5, 1397–1428. MR 2732352 (2011j:31002), http://dx.doi.org/10.1353/ajm.2010.0000
26. Iwaniec, T., Kovalev, L. V. and Onninen, J. The Nitsche conjecture, J. Amer. Math. Soc. 24 (2011), no. 2, 345-373.
27. Iwaniec, T., Kovalev, L. V. and Onninen, J. Harmonic mapping problem and affine capacity, Proc. Roy. Soc. Edinburgh Sect. A, to appear.
28. Iwaniec, T., Kovalev, L. V. and Onninen, J. Doubly connected minimal surfaces and extremal harmonic mappings, J. Geom. Anal., to appear.
29. Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2001. MR 1859913 (2003c:30001)
30. Tadeusz Iwaniec and Jani Onninen, Deformations of finite conformal energy: Boundary behavior and limit theorems, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5605–5648. MR 2817402, http://dx.doi.org/10.1090/S0002-9947-2011-05106-8
31. Tadeusz Iwaniec and Jani Onninen, Deformations of finite conformal energy: existence and removability of singularities, Proc. Lond. Math. Soc. (3) 100 (2010), no. 1, 1–23. MR 2578466 (2011a:30063), http://dx.doi.org/10.1112/plms/pdp016
32. Tadeusz Iwaniec and Vladimír Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), no. 1, 181–188. MR 1160301 (93k:30023), http://dx.doi.org/10.1090/S0002-9939-1993-1160301-5
33. Jürgen Jost and Richard Schoen, On the existence of harmonic diffeomorphisms, Invent. Math. 66 (1982), no. 2, 353–359. MR 656629 (83i:58033), http://dx.doi.org/10.1007/BF01389400
34. David Kalaj, On the Nitsche conjecture for harmonic mappings in ℝ² and ℝ³, Israel J. Math. 150 (2005), 241–251. MR 2255810 (2007i:31001), http://dx.doi.org/10.1007/BF02762382
35. Abdallah Lyzzaik, The modulus of the image annuli under univalent harmonic mappings and a conjecture of Nitsche, J. London Math. Soc. (2) 64 (2001), no. 2, 369–384. MR 1853457 (2002h:30023), http://dx.doi.org/10.1112/S0024610701002460
36. Giuseppe Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), no. 4, 355–426. MR 2291779 (2007k:35086), http://dx.doi.org/10.1007/s10778-006-0110-3
37. Charles B. Morrey Jr., The Topology of (Path) Surfaces, Amer. J. Math. 57 (1935), no. 1, 17–50. MR 1507053, http://dx.doi.org/10.2307/2372017
38. Charles B. Morrey Jr., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53. MR 0054865 (14,992a)
39. Charles B. Morrey Jr., A variational method in the theory of harmonic integrals. II, Amer. J. Math. 78 (1956), 137–170. MR 0087765 (19,408a)
40. Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511 (34 #2380)
41. Johannes C. C. Nitsche, Mathematical Notes: On the Module of Doubly-Connected Regions Under Harmonic Mappings, Amer. Math. Monthly 69 (1962), no. 8, 781–782. MR 1531840, http://dx.doi.org/10.2307/2310779
42. Kai Rajala, Distortion of quasiconformal and quasiregular mappings at extremal points, Michigan Math. J. 53 (2005), no. 2, 447–458. MR 2152710 (2006b:30046), http://dx.doi.org/10.1307/mmj/1123090778
43. Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644 (90d:30067)
44. Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941 (95g:30026)
45. Schottky, F.H. Über konforme Abbildung von mehrfach zusammenhängenden Fläche. J. für Math., 83, (1877).
46. Gudrun Turowski, Behaviour of doubly connected minimal surfaces at the edges of the support surface, Arch. Math. (Basel) 77 (2001), no. 3, 278–288. MR 1865870 (2002h:53016), http://dx.doi.org/10.1007/PL00000492
47. K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 0474389 (57 #14031)
48. N. N. Ural′ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222 (Russian). MR 0244628 (39 #5942)
49. Changyou Wang, A compactness theorem of 𝑛-harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 4, 509–519 (English, with English and French summaries). MR 2145723 (2006a:58022), http://dx.doi.org/10.1016/j.anihpc.2004.10.007
50. Allen Weitsman, Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche, Israel J. Math. 124 (2001), 327–331. MR 1856525 (2002f:30011), http://dx.doi.org/10.1007/BF02772628

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