×

Nonlinear Hodge theory on manifolds with boundary. (English) Zbl 0963.58003

Authors’ summary: The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, elliptic PDEs called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the \({\mathcal L}^p\)-theory for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated.

MSC:

58A14 Hodge theory in global analysis
58J05 Elliptic equations on manifolds, general theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahlfors, L. V.; Beurling, A., Conformal invariants and function theoretic null sets, Acta Math., 83, 101-129 (1950) · Zbl 0041.20301 · doi:10.1007/BF02392634
[2] Ahlfors, L. V., Lectures on quasiconformal mappings (1996), Princeton: Van Nostrand, Princeton · Zbl 0138.06002
[3] Astala, K., Area distortion and quasiconformal mappings, Acta Math., 173, 37-60 (1994) · Zbl 0815.30015 · doi:10.1007/BF02392568
[4] Bojarski, B.; Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in ℝ^n, Ann. Acad. Sci. Fenn., Ser. A.I., 8, 257-324 (1983) · Zbl 0548.30016
[5] Bojarski, B. V., Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk. SSSR, 102, 661-664 (1955) · Zbl 0068.06402
[6] Browder, F., Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69, 862-874 (1963) · Zbl 0127.31901 · doi:10.1090/S0002-9904-1963-11068-X
[7] Budney, L. R.; Iwaniec, T.; Stroffolini, B., Removability of Singularities of A-Harmonic functions, Differential and Integral Equations, 12, no. 2, 261-274 (1999) · Zbl 1064.35505
[8] [BS96]L. R. Budney—C. Scott,A characterization of higher order Sobolev space, (preprint) (1996).
[9] Cartan, H., Differential forms (1970), Boston: Houghton Mifflin Co., Boston · Zbl 0213.37001
[10] [Con56]P. E. Conner,The Neumann’s problem for differential forms on Riemannian manifolds, Memoirs of the AMS,20 (1956). · Zbl 0070.31404
[11] Duff, G. F. D.; Spencer, D. C., Harmonic tensors on Riemannian manifolds with boundary, Ann. Math., 56, 2, 128-156 (1952) · Zbl 0049.18901 · doi:10.2307/1969771
[12] Donaldson, S. K.; Sullivan, D. P., Quasiconformal 4-manifolds, Acta Math., 163, 181-252 (1989) · Zbl 0704.57008 · doi:10.1007/BF02392736
[13] Duff, G. F. D., Differential forms on manifolds with boundary, Annals of Math., 56, 115-127 (1952) · Zbl 0049.18804 · doi:10.2307/1969770
[14] Friedrichs, K., The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc., 55, 132-151 (1944) · Zbl 0061.26201 · doi:10.2307/1990143
[15] Fefferman, C.; Stein, E. M., H^p-spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215
[16] [Gaf54]M. Gaffney,The heat equation method of Milgram and Rosenbloom for open Riemannian manifolds, Ann. Math.,60 (1954). · Zbl 0057.07501
[17] Gehring, F. W., Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103, 353-393 (1962) · Zbl 0113.05805 · doi:10.2307/1993834
[18] Gehring, F. W., The ℒ^p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021 · doi:10.1007/BF02392268
[19] Giachetti, D.; Leonetti, F.; Schianchi, R., On the regularity of very weak minima, Proc. Royal Soc. Edinburgh, 126A, 287-296 (1996) · Zbl 0851.49026
[20] Hamburger, C., Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math., 431, 7-64 (1992) · Zbl 0776.35006
[21] [HKM93]J. Heinonen—T. Kilpeläinen—O. Martio,Nonlinear Potential Theory of Degenerate Elliptic Operators, Oxford University Press (1993). · Zbl 0780.31001
[22] Hodge, W. V. D., A dirichlet problem for a harmonic functional, Proc. London Math. Soc., 2, 257-303 (1933) · Zbl 0008.02203
[23] Hörmander, L., Weak and strong extensions of differential operators, Comm. Pure Appl. Math., 14, 371-379 (1941) · Zbl 0111.29202
[24] [IL93]T. Iwaniec—A. Lutoborski,Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. (1993), pp. 25-79. · Zbl 0793.58002
[25] Iwaniec, T.; Martin, G., Quasiregular mappings in even dimensions, Acta Math., 170, 29-81 (1993) · Zbl 0785.30008 · doi:10.1007/BF02392454
[26] Iwaiec, T.; Mitrea, M.; Scott, C., Boundary value estimates for harmonic fields, Proc. Amer. Math. Soc., 124, 1467-1471 (1995) · Zbl 0854.31002 · doi:10.1090/S0002-9939-96-03142-5
[27] [IS93]T. Iwaniec—C. Sbordone,Weak minima of variational integrals, J. Reine Angew. Math. (1993). · Zbl 0802.35016
[28] Iwaniec, T., Projections onto Gradient fields and ℒp-estimates for degenerate elliptic operators, Studia Math., 75, 293-312 (1983) · Zbl 0552.35034
[29] Iwaniec, T., p-harmonic tensors and quasiregular mappings, Ann. Math., 136, 589-624 (1992) · Zbl 0785.30009 · doi:10.2307/2946602
[30] Iwaniec, T., Integrability theory of the Jacobians, Lipshitz Lectures, no. 36 Sonderforschungsbereich, Bonn, 256, 1-68 (1995)
[31] Kodaira, K., Harmonic fields in Riemannian manifolds, Ann. Math., 50, 587-665 (1949) · Zbl 0034.20502 · doi:10.2307/1969552
[32] Lewis, J., On very weak solutions of certain elliptic systems, Comm. PDE, 18, 1515-1537 (1993) · Zbl 0796.35061 · doi:10.1080/03605309308820984
[33] Manfredi, J., Quasiregular mappings form the multilinear point of view, Fall School in Analysis, Jyväskylä 1994, Preprint, 68, 55-94 (1995) · Zbl 0843.30020
[34] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Berlin: Springer-Verlag, Berlin · Zbl 0142.38701
[35] Martio, O.; Rickman, S.; Väisälä, J., Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A.I., 448, 1-40 (1969) · Zbl 0189.09204
[36] Martio, O.; Rickman, S.; Väisälä, J., Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A.I., 465, 1-13 (1970) · Zbl 0197.05702
[37] Martio, O.; Rickman, S.; Väisälä, J., Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A.I., 488, 1-31 (1971) · Zbl 0223.30018
[38] Reshetnyak, Y. G., On extremal properties of mappings with bounded distortion, Siberian Math. J., 10, 1300-1310 (1969) · Zbl 0201.09802
[39] [Res89]Y. G. Reshetnyak,Space mappings with bounded distortion, Trans. Math. Monographs Amer. Math. Soc.,73 (1989). · Zbl 0667.30018
[40] [Ric93]S. Rickman,Quasiregular mappings, Springer-Verlag (1993). · Zbl 0816.30017
[41] Robbin, J. W.; Rogers, R. C.; Temple, B., On weak continuity and the Hodge decomposition, Trans. Amer. Math. Soc., 303, 405-417 (1988)
[42] Rochberg, R.; Weiss, G., Derivatives of analytic families of Banach spaces, Ann. Math., 118, 315-347 (1983) · Zbl 0539.46049 · doi:10.2307/2007031
[43] Scott, C., L^p theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347, 2075-2096 (1995) · Zbl 0849.58002 · doi:10.2307/2154923
[44] Stroffolini, B., On weakly A-harmonic tensors, Studia Math., 114, 3, 289-301 (1995) · Zbl 0868.35015
[45] [Str99]B. Stroffolini,Nonlinear Hodge Projections, preprint (1999).
[46] Tucker, A. W., A boundary value problem with a volume constraint, Bull. Amer. Math. Soc., 47, 714-714 (1941)
[47] Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems, Acta Math., 138, 219-250 (1977) · Zbl 0372.35030 · doi:10.1007/BF02392316
[48] [Vek62]I. N. Vekua,Generalized analytic functions, Pergamon Press (1962). · Zbl 0127.03505
[49] Zoretti, L., Sur les functions analytiques uniformes qui possedent un ensemble parfait discontinu de points singuliers, J. Math. Pures Appl., 1, 6, 1-51 (1905) · JFM 36.0451.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.