m-harmonic flow
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 24 (1997) no. 4, pp. 593-631.
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Hungerbühler, Norbert. $m$-harmonic flow. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 24 (1997) no. 4, pp. 593-631. http://archive.numdam.org/item/ASNSP_1997_4_24_4_593_0/

[1] F. Bethuel, The approximation problem for Sobolev maps between manifolds, Acta Math. 167 (1991),153-206. | MR | Zbl

[2] F. Bethuel - X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988), 60-75. | MR | Zbl

[3] K.-C. Chang - W.-Y. Ding - R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), 507-515. | MR | Zbl

[4] Y. Chen, The weak solutions to the evolution problem of harmonic maps, Math. Z. 201 (1989), 69-74. | MR | Zbl

[5] Y. Chen - M.-C. Hong - N. Hungerbühler, Heat flow of p-harmonic maps with values into spheres, Math. Zeit. 215 (1994), 25-35. | MR | Zbl

[6] Y. Chen - M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 83-103. | MR | Zbl

[7] H.J. Choe, Hölder continuity for solutions of certain degenerate parabolic systems, Nonlinear Anal. 18 (1992), 235-243. | MR | Zbl

[8] H.J. Choe, Hölder regularity for the gradient of solutions of certain singular parabolic equations, Comm. Partial Differential Equations 16 (1991), 1709-1732. | MR | Zbl

[9] J.-M. Coron, Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990), 335-344. | Numdam | MR | Zbl

[10] J.-M. Coron - R. Gulliver, Minimizing p-harmonic maps into spheres, J. Reine Angew. Math. 401 (1989), 82-100. | MR | Zbl

[ 11 ] E. Dibenedetto, C1,α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. | Zbl

[12] E. Dibenedetto, Degenerate parabolic equations, Universitext, Springer, Berlin, 1993. | MR | Zbl

[13] E. Dibenedetto - A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984), 83-128. | MR | Zbl

[14] E. Dibenedetto - A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1984), 1-22. | MR | Zbl

[15] F. Duzaar - M. Fuchs, Existence and regularity of functions which minimize certain energies in homotopy classes of mappings, Asymptotic Anal. 5 (1991), 129-144. | MR | Zbl

[16] J. Eells - L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. | MR | Zbl

[17] J. Eells - J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-169. | MR | Zbl

[18] J. L. ERICKSEN - D. KINDERLEHRER (ed.), Theory and applications of liquid crystals, IMA Vol. Math. Appl., vol. 5., Springer, New York, 1987. | MR | Zbl

[19] L.C. Evans, A new proof of local C 1, α regularity for solutions of certain degenerate elliptic P.D.E., J. Differential Equations 45 (1982), 356-373. | Zbl

[20] A. Freire, Uniqueness for the harmonic map flow in two dimensions, Calc. Var. Partial Differential Equations 3 (1995), 95-105. | MR | Zbl

[21 ] A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comment. Math. Helv. 70 (1995), no. 2, 310-338. | MR | Zbl

[22] A. Freire, Correction to: "Uniqueness for the harmonic map flow from surfaces to general targets" , Comment. Math. Helv. 71 (1996), no. 2, 330-337. | MR | Zbl

[23] A. Freire - S. Müller - M. Struwe, Weak convergence of wave maps from (1 + 2)-dimensional Minkowski space to Riemannian manifolds, Invent. Math., to appear. | MR | Zbl

[24] M. Fuchs, p-harmonische Hindernisprobleme, Habilitationsschrift Düsseldorf, Düsseldorf, 1987.

[25] M. Fuchs, Everywhere regularity theorems for mappings which minimize p-energy, Comment. Math. Univ. Carolin. 28 (1987), 673-677. | MR

[26] M. Fuchs, Some regularity theorems for mappings which are stationary points of the p-energy functional, Analysis 9 (1989), 127-143. | MR | Zbl

[27] M. Fuchs, p-harmonic obstacle problems, Part I: Partial regularity theory Ann. Mat. Pura Appl. 156 (1990), 127-158. | MR | Zbl

[28] M. Fuchs, p-harmonic obstacle problems, Part II: Extensions of Maps and Applications Manuscripta Math. 63 (1989), 381-419. | MR | Zbl

[29] M. Fuchs, p-harmonic obstacle problems, Part III: Boundary regularity. Ann. Mat. Pura Appl. 156 (1990), 159-180. | MR | Zbl

[30] M. Fuchs, p-harmonic obstacle problems, Part IV: Unbounded side conditions Analysis 13 (1993), 69-76 159-180. | MR | Zbl

[31] M. Fuchs - J. Hutchinson, Partial regularity for minimizers of certain functionals having non quadratic growth, Preprint, CMA, Canberra, 1985.

[32] M. Giaquinta - G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math. 57 (1986), 55-99. | MR | Zbl

[33] E. Giusti - M. Miranda, Un esempio di solutioni discontinue per un problema di minima relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. Ital. 2 (1968), 1-8. | MR | Zbl

[34] M. Günther, Zum Einbettungssatz von J. Nash, Math. Nachr.,144 (1989), 165-187. | MR | Zbl

[35] R.S. Hamilton, Harmonic maps of manifolds with boundary, Lect. Notes Math. 471, Springer, Berlin, 1975. | MR | Zbl

[36] R. Hardt - F.H. Lin, A remark on H1-mappings, Manuscripta Math. 56 (1986), 1-10. | MR | Zbl

[37] R. Hardt - F.H. Lin, Mappings minimizing the LP norm of the gradient, Comm. Pure and Appl. Math. 15 (1987), 555-588. | MR | Zbl

[38] F. Hélein, Regularité des applications faiblement harmoniques entre une surface et une variteé Riemannienne, C.R. Acad. Sci. Paris Ser. I Math. 312 (1991), 591-596. | MR | Zbl

[39] N. Hungerbühler, Non-uniqueness for the p-harmonic flow, Canad. Math. Bull. 40 (1997), 174-182. | MR | Zbl

[40] N. Hungerbühler, Global weak solutions of the p-harmonic flow into homogeneous spaces, Indiana Univ. Math. J. 45/1, (1996), 275-288. | MR | Zbl

[41] N. Hungerbühler, Compactness properties of the p-harmonic flow into homogeneous spaces, Nonlinear Anal. 28/5 (1997), 793-798. | MR | Zbl

[42] J.B. Keller - J. Rubinstein - P. Sternberg, Reaction-diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math. 49 (1989), 1722-1733. | MR | Zbl

[43] S. Klainerman - M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46, No. 9 (1993), 1221-1268. | MR | Zbl

[44] O.A. Lady - V.A. Solonnikov - N.N. Ural'Ceva, Linear and quasilinear equations of parabolic type, AMS, Providence R.I. 1968. | MR

[45] L. Lemaire, Applications harmoniques de surfaces riemannienne, J. Differential Geom.13 (1978),51-78. | MR | Zbl

[46] J. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 849-858. | MR | Zbl

[47] S. Luckhaus, Partial Holder continuity of energy minimizing p-harmonic maps between Riemannian manifolds, preprint, CMA, Canberra, 1986.

[48] A. Lunardi, On the local dynamical system associated to a fully nonlinear abstract parabolic equation, Nonlinear Analysis and Applications (ed. V. Laksmikantham, M. Decker), (1987), 319-326. | MR | Zbl

[49] C.B. Morrey, Multiple integrals in the calculus of variations, Grundlehren 130, Springer, Berlin, 1966. | Zbl

[50] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | MR | Zbl

[51 ] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740. | MR | Zbl

[52] S. Müller - M. Struwe, Global existence of wave maps in 1 + 2 dimensions for finite energy data, Top. Methods Nonlinear Analysis, to appear. | MR | Zbl

[53] J. Nash, The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63. | MR | Zbl

[54] J. Sacks - K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1-24. | MR | Zbl

[55] R.S. Schoen - K. Uhlenbeck, Approximation theorems for sobolev mappings, preprint.

[56] J. Shatah, Weak solutions and development of singularities in the SU(2) σ -model,, Comm. Pure Appl. Math. 41 (1988), 459-469. | Zbl

[57] M. Struwe, Weak compactness of harmonic maps from (2 + 1) -dimensional Minkowsky space to symmetric spaces, preprint, 1994.

[58] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Math. Helv. 60 (1985), 558-581. | MR | Zbl

[59] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28 (1988), 485-502. | MR | Zbl

[60] P. Tolksdorf, Everywhere regularity for some quasilinear systems with a lack of ellipticity, Ann. Math. Pura Appl. 134 (1983), 241-266. | MR | Zbl

[61 ] P. Tolksdorf, Regularity for a more general class ofquasilinear equations, J. of Differential Equations 51 (1984), 126-150. | MR | Zbl

[62] T. Toro - C. Wang, Compactness properties of weakly p-harmonic maps into homogeneous spaces, Indiana Univ. Math. J. 44 (1995), 87-113. | MR | Zbl

[63] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219-240. | MR | Zbl

[64] N.N. Ural'Ceva, Degenerate quasilinear elliptic systems, Zap. Naucn. Sem. Leningrad. Otel. Mat. Inst. Steklov 7 (1968), 184-222. | MR

[65] V. Vespri, Local existence, uniqueness and regularity for a class of degenerate parabolic systems arising in biological models, Quaderno/Dipartimento di Matematica "F.Enriques"', Universita degli Studi di Milano, no. 33/1988.