Nonlinear Elliptic Equations

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 117)


Methods of the calculus of variations applied to problems in geometry and classical continuum mechanics often lead to elliptic PDE that are not linear. We discuss a number of examples and some of the developments that have arisen to treat such problems.


Riemannian Manifold Minimal Surface Dirichlet Problem Nonlinear Elliptic Equation Morrey Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ber]
    S. Bernstein, Sur les equations du calcul des variations, Ann. Sci. Ecole Norm.Sup. 29(1912), 431–485.MathSciNetzbMATHGoogle Scholar
  2. [BJS]
    L. Bers, F. John, and M. Schechter, Partial Differential Equations, Wiley, New York, 1964.zbMATHGoogle Scholar
  3. [BN]
    L. Bers and L. Nirenberg, On linear and non-linear elliptic boundary value problems in the plane, pp. 141–167 in Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste. Edizioni Cremonese, Rome, 1955.Google Scholar
  4. [Bet]
    F. Bethuel, On the singularity set of stationary maps, Preprint, 1993.Google Scholar
  5. [Bom]
    E. Bombieri (ed.), Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, N. J., 1983.zbMATHGoogle Scholar
  6. [Br]
    F. Browder, A priori estimates for elliptic and parabolic equations, Proc. Symp. Pure Math. 4(1961), 73–81.MathSciNetGoogle Scholar
  7. [Br2]
    F. Browder, Non-linear elliptic boundary value problems, Bull. AMS 69(1963), 862–874.MathSciNetzbMATHGoogle Scholar
  8. [Br3]
    F. Browder, Existence theorems for nonlinear partial differential equations, Proc.Symp. Pure Math. 16(1970), 1–60.Google Scholar
  9. [Cal]
    L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat Fis. Milano 58(1988), 253–284.MathSciNetzbMATHGoogle Scholar
  10. [Ca2]
    L. Caffarelli, Interior a priori estimates for solutions of fully non linear equations, Ann. of Math. 130(1989), 189–213.MathSciNetzbMATHGoogle Scholar
  11. [Ca3]
    L. Caffarelli, A priori estimates and the geometry of the Monge-Ampere equation, pp. 7–63 in [HW].Google Scholar
  12. [CKNS]
    L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations II. Complex Monge-Ampere, and uniformly elliptic, equations. CPAM 38(1985), 209–252.MathSciNetzbMATHGoogle Scholar
  13. [CNS]
    L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Ampere equations, CPAM 37(1984), 369–402.MathSciNetzbMATHGoogle Scholar
  14. [Caml]
    S. Campanato, Equazioni ellittiche del IF ordine e spazi C 2,x, Ann. Math. Pura Appl. 69(1965), 321–381.MathSciNetzbMATHGoogle Scholar
  15. [Cam2]
    S. Campanato, Sistemi ellittici in forma divergenza—Regolarita all’ interno, Quaderni della Sc. Norm. Sup. di Pisa, 1980.zbMATHGoogle Scholar
  16. [Cam3]
    S. Campanato, Non variational basic elliptic systems of second order, Rendi. Sem. Mate Fis. Pisa 8(1990), 113–131.MathSciNetGoogle Scholar
  17. [CY]
    S. Cheng and S.-T. Yau, On the regularity of the Monge-Ampere equation det(δ 2 u/δxiδxj) = F(x, u) , CPAM 30(1977), 41–68.MathSciNetzbMATHGoogle Scholar
  18. [Cher]
    S.S. Chern, Minimal Submanifolds in a Riemannian Manifold, Technical Report #19, Univ. of Kansas, 1968.Google Scholar
  19. [Cher2]
    S. S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations, MSRI Publ. #2, Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  20. [Chow]
    B. Chow, The Ricci flow on the 2-sphere, J. Diff Geom. 33(1991), 325–334.zbMATHGoogle Scholar
  21. [CK]
    D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of theMinkowski Space, Princeton Univ. Press, Princeton, N. J., 1993.zbMATHGoogle Scholar
  22. [CT]
    R. Cohen and M. Taylor, Weak stability for the map x|x for liquid crystal functionals, Comm. PDE 15(1990), 675–692.MathSciNetzbMATHGoogle Scholar
  23. [CF]
    P. Conçus and R. Finn (eds.), Variational Methods for Free Surface Interfaces, Springer-Verlag, New York, 1987.zbMATHGoogle Scholar
  24. [Cor]
    H. Cordes, über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann. 131(1956), 278–312.MathSciNetzbMATHGoogle Scholar
  25. [Cou]
    R. Courant, Dirichlet’s Principle, Conformai Mapping, and Minimal Surfaces, Interscience, New York, 1950.Google Scholar
  26. [Cou2]
    R. Courant, The existence of minimal surfaces of given topological type, ActaMath. 72(1940), 51–98.MathSciNetGoogle Scholar
  27. [CH]
    R. Courant and D. Hilbert, Methods of Mathematical Physics II, J. Wiley, New York, 1966.Google Scholar
  28. [CIL]
    M. Crandall, H. Ishii, and R Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. AMS 27(1992), 1–67.MathSciNetzbMATHGoogle Scholar
  29. [Dae]
    B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, New York, 1989.zbMATHGoogle Scholar
  30. [DeG]
    E. DeGiorgi, Sulla differenziabilita e 1 ‘analiticita degli integrali multipli regolari, Accad. Sci. Torino Cl Fis. Mat. Natur. 3(1957), 25–43.MathSciNetGoogle Scholar
  31. [DeG2]
    E. DeGiorgi, Frontiere orientate di misura minima, Quaderni Sc. Norm. Sup.Pisa (1960–61).Google Scholar
  32. [Deim]
    K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  33. [DT]
    D. DeTurck and D. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. Ecole Norm. Sup. 14(1980), 249–260.MathSciNetGoogle Scholar
  34. [DHKW]
    U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal Surfaces, Vols. 1 and 2, Springer-Verlag, Berlin, 1992.Google Scholar
  35. [Dou]
    J. Douglas, Solution of the problem of Plateau, Trans. AMS 33(1931), 263–321.Google Scholar
  36. [Dou2]
    J. Douglas, Minimal surfaces of higher topological structure, Ann. Math. 40 (1939), 205–298.Google Scholar
  37. [Eis]
    G. Eisen, A counterexample for some lower semicontinuity results, Math. Zeit. 162(1978), 241–243.MathSciNetzbMATHGoogle Scholar
  38. [Ev]
    L. C. Evans, Classical solutions of fully nonlinear convex second order elliptic equations, CPAM 35(1982), 333–363.zbMATHGoogle Scholar
  39. [Ev2]
    L. C. Evans, Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators, Trans. AMS 275(1983), 245–255.Google Scholar
  40. [Ev3]
    L. C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rat. Mech. Anal. 95(1986), 227–252.zbMATHGoogle Scholar
  41. [Ev4]
    L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch.Rat. Math. Anal. 116(1991), 101–113.zbMATHGoogle Scholar
  42. [EG]
    L. C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Fla., 1992.zbMATHGoogle Scholar
  43. [Fed]
    H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.zbMATHGoogle Scholar
  44. [Fef]
    C. Fefferman, Monge-Ampere equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 103(1976), 395–416.MathSciNetzbMATHGoogle Scholar
  45. [Fl]
    W. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo 11(1962), 69–90.MathSciNetzbMATHGoogle Scholar
  46. [Fol]
    G. Folland, Real Analysis: Modern Techniques and Applications, Wiley-Interscience, New York, 1984.zbMATHGoogle Scholar
  47. [Fom]
    A. Fomenko, The Plateau Problem, 2 vols., Gordon and Breach, New York, 1990.zbMATHGoogle Scholar
  48. [Freh]
    J. Frehse, A discontinuous solution of a mildly nonlinear elliptic system, Math.Zeit. 134(1973), 229–230.MathSciNetzbMATHGoogle Scholar
  49. [Fri]
    K. Friedrichs, On the differentiability of the solutions of linear elliptic equations, CPAM 6(1953), 299–326.MathSciNetzbMATHGoogle Scholar
  50. [FuH]
    N. Fusco and J. Hutchinson, Partial regularity in problems motivated by nonlinear elasticity, SIAMJ. Math. Anal 22(1991), 1516–1551.MathSciNetzbMATHGoogle Scholar
  51. [Ga]
    P. Garabedian, Partial Differential Equations, Wiley, New York, 1964.zbMATHGoogle Scholar
  52. [Geh]
    F. Gehring, The Lp-integrability of the partial derivatives of a quasi conformai mapping, Acta Math. 130(1973), 265–277.MathSciNetzbMATHGoogle Scholar
  53. [Gia]
    M. Giaquinta, Multiple Integrals in the Calculus of Variations and NonlinearElliptic Systems, Princeton Univ. Press, Princeton, N. J., 1983.zbMATHGoogle Scholar
  54. [Gia2]
    M. Giaquinta (ed.), Topics in Calculus of Variations, LNM #1365, Springer-Verlag, New York, 1989.zbMATHGoogle Scholar
  55. [GiaM]
    M. Giaquinta and G. Modica, Partial regularity of minimizers of quasieonvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3(1986), 185–208.MathSciNetzbMATHGoogle Scholar
  56. [GiaS]
    M. Giaquinta and J. Soucek, Cacciopoli’s inequality and Legendre-Hadamard condition, Math. Annalen 270(1985), 105–107.MathSciNetzbMATHGoogle Scholar
  57. [GT]
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of SecondOrder, 2nd ed., Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  58. [Giu]
    E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984.zbMATHGoogle Scholar
  59. [GiuM]
    E. Giusti and M. Miranda, Sulla regolarita delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Rat. Mech. Anal. 31(1968), 173–184.MathSciNetzbMATHGoogle Scholar
  60. [GS]
    B. Guan and J. Spruck, Boundary-value problems on S” for surfaces of constant Gauss curvature, Annn, of Math. 138(1993), 601–624.MathSciNetzbMATHGoogle Scholar
  61. [Gui]
    M. Günther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Ann. Global Anal. Geom. 7(1989), 69–77.MathSciNetzbMATHGoogle Scholar
  62. [Gu2]
    M. Günther, Zum Einbettungssatz von J.Nash, Math. Nachr. 144(1989), 165–187.MathSciNetzbMATHGoogle Scholar
  63. [Gu3]
    M. Günther, Isometric embeddings of Riemannian manifolds, Proc. Intern.Congr. Math. Kyoto, 1990, pp. 1137–1143.Google Scholar
  64. [Ham]
    R. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71(1988), 237–262.MathSciNetGoogle Scholar
  65. [HKL]
    R. Hardt, D. Kinderlehrer, and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105(1986), 547–570.MathSciNetzbMATHGoogle Scholar
  66. [HW]
    R. Hardt and M. Wolf (eds.), Nonlinear Partial Differential Equations in Differential Geometry, IAS/Park City Math. Ser., Vol. 2, AMS, Providence, R. L, 1995.Google Scholar
  67. [Hei]
    E. Heinz, On elliptic Monge-Ampere equations and Weyl’s embedding problem, Analyse Math. 7(1959), 1–52.MathSciNetzbMATHGoogle Scholar
  68. [HH]
    E. Heinz and S. Hildebrandt, Some remarks on minimal surfaces in Riemannian manifolds, CPAM 23(1970), 371–377.MathSciNetzbMATHGoogle Scholar
  69. [Hel]
    F. Helein, Minima de la fonctionelle énergie libre des cristaux liquides, CRAS Paris 305(1987), 565–568.MathSciNetzbMATHGoogle Scholar
  70. [Hel2]
    F. Helein, Régularité des applications faiblement harmonique entre une surface et une variété riemannienne, CR Acad. Sci. Paris 312(1991), 591–596.MathSciNetzbMATHGoogle Scholar
  71. [Hild]
    S. Hildebrandt, Boundary regularity of minimal surfaces, Arch. Rat. Mech. Anal. 35(1969), 47–82.MathSciNetzbMATHGoogle Scholar
  72. [HW]
    S. Hildebrandt and K. Widman, Some regularity results for quasilinear systems of second order, Math. Zeit. 142(1975), 67–80.MathSciNetzbMATHGoogle Scholar
  73. [HM1]
    D. Hoffman and W. Meeks, A complete embedded minimal surface in E3 with genus one and three ends, J. Diff. Geom. 21(1985), 109–127.MathSciNetzbMATHGoogle Scholar
  74. [HM2]
    D. Hoffman and W. Meeks, Properties of properly imbedded minimal surfaces of finite topology, Bull. AMS 17(1987), 296–300.MathSciNetzbMATHGoogle Scholar
  75. [HRS]
    D. Hoffman, H. Rosenberg, and J. Spruck, Boundary value problems for surfaces of constant Gauss curvature, CPAM 45(1992), 1051–1062.MathSciNetzbMATHGoogle Scholar
  76. [IL]
    H. Ishii and P. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Diff. Eqs. 83(1990), 26–78.MathSciNetzbMATHGoogle Scholar
  77. [JS]
    H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, 7. Reine Angew. Math. 229(1968), 170–187.MathSciNetzbMATHGoogle Scholar
  78. [Jo]
    F. John, Partial Differential Equations, Springer-Verlag, New York, 1975.zbMATHGoogle Scholar
  79. [Jos]
    J. Jost, Conformai mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reine Angew. Math. 359(1985), 37–54.MathSciNetzbMATHGoogle Scholar
  80. [Kaz]
    J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Reg. Conf. Sen Math. #57, AMS, Providence, R. L, 1985.Google Scholar
  81. [KaW]
    J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99(1974), 14–47.MathSciNetzbMATHGoogle Scholar
  82. [KS]
    D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.zbMATHGoogle Scholar
  83. [Kryl]
    N. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math.USSR Izv. 20(1983), 459–492.zbMATHGoogle Scholar
  84. [Kry2]
    N. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Math. USSR Izv. 22(1984), 67–97.zbMATHGoogle Scholar
  85. [Kry3]
    N. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, D. Reidel, Boston, 1987.zbMATHGoogle Scholar
  86. [KrS]
    N. Krylov and M. Safonov, An estimate of the probability that a diffusion process hits a set of positive measure, Soviet Math. Dokl. 20(1979), 253–255.zbMATHGoogle Scholar
  87. [LU]
    O. Ladyzhenskaya and N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.zbMATHGoogle Scholar
  88. [Law]
    H. B. Lawson, Lectures on Minimal Submanifolds, Publish or Perish, Berkeley, Calif., 1980.zbMATHGoogle Scholar
  89. [Law2]
    H. B. Lawson, Minimal Varieties in Real and Complex Geometry, Univ. of Montreal Press, 1974.zbMATHGoogle Scholar
  90. [LO]
    H. B. Lawson and R. Osserman, Non-existence, non-uniqueness, and irregularity of solutions to the minimal surface equation, Acta Math. 139(1977), 1–17.MathSciNetzbMATHGoogle Scholar
  91. [LS]
    J. Leray and J. Schauder, Topologie et équations fonctionelles, Ann. Sel EcoleNorm. Sup. 51(1934), 45–78.MathSciNetGoogle Scholar
  92. [LM]
    J. Lions and E. Magenes, Non-homogeneous Boundary Problems and Applications I, II, Springer-Verlag, New York, 1972.Google Scholar
  93. [LiPl]
    P. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rat. Mech.Anal. 74(1980), 335–353.zbMATHGoogle Scholar
  94. [LiP2]
    P. Lions, Sur les équations de Monge-Ampere, I, Manuscripta Math. 41(1983), 1–43;MathSciNetzbMATHGoogle Scholar
  95. [LiP2a]
    P. Lions, Sur les équations de Monge-Ampere, II, Arch. Rat. Mech. Anal. 89(1985), 93–122.zbMATHGoogle Scholar
  96. [MM]
    U. Massari and M. Miranda, Minimal Surfaces of Codimension One, North-Holland, Amsterdam, 1984.zbMATHGoogle Scholar
  97. [MY]
    W. Meeks and S.-T. Yau, The classical Plateau problem and the topology of three dimensional manifolds, Topology 4(1982), 409–442.MathSciNetGoogle Scholar
  98. [Mey]
    N. Meyers, An L p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa 17(1980), 189–206.MathSciNetGoogle Scholar
  99. [Min]
    G. Minty, On the solvability of non-linear functional equations of “monotonie” type, Pacific J. Math. 14(1964), 249–255.MathSciNetzbMATHGoogle Scholar
  100. [Mir]
    C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970.zbMATHGoogle Scholar
  101. [Morg]
    F. Morgan, Geometric Measure Theory: A Beginner’s Guide, Academic Press, New York, 1988.zbMATHGoogle Scholar
  102. [Mori]
    C. B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. Math. 49(1948), 807–851.MathSciNetzbMATHGoogle Scholar
  103. [Mor2]
    C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.zbMATHGoogle Scholar
  104. [Mor3]
    C. B. Morrey, Partial regularity results for nonlinear elliptic systems, J. Math. and Mech. 17(1968), 649–670.MathSciNetzbMATHGoogle Scholar
  105. [Mo 1]
    J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations, Ann. Scoula Norm. Sup. Pisa 20(1966), 265–315.zbMATHGoogle Scholar
  106. [Mo2]
    J. Moser, A new proof of DeGiorgi’s theorem concerning the regularity problem for elliptic differential equations, CPAM 13(1960), 457–468.zbMATHGoogle Scholar
  107. [Mo3]
    J. Moser, On Harnack’s theorem for elliptic differential equations, CPAM 14 (1961), 577–591.zbMATHGoogle Scholar
  108. [MW]
    T. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math, and Phys. 31(1952), 253–259.MathSciNetGoogle Scholar
  109. [Nal]
    J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20–63.MathSciNetzbMATHGoogle Scholar
  110. [Na2]
    J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J.Math. 80(1958), 931–954.MathSciNetzbMATHGoogle Scholar
  111. [Nee]
    J. Necas, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and concilions for regularity, in Theory of Non Linear Operators, Abh. Akad. der Wissen, der DDR, 1977.Google Scholar
  112. [Nil]
    L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, CPAM 6(1953), 103–156.MathSciNetzbMATHGoogle Scholar
  113. [Ni2]
    L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, CPAM 6(1953), 337–394.MathSciNetzbMATHGoogle Scholar
  114. [Ni3]
    L. Nirenberg, Estimates and existence of solutions of elliptic equations, CPAM 9(1956), 509–530.MathSciNetzbMATHGoogle Scholar
  115. [Ni4]
    L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa 13(1959), 116–162.MathSciNetGoogle Scholar
  116. [Ni5]
    L. Nirenberg, Lectures on Linear Partial Differential Equations, Reg. Conf. Ser. in Math., #17, AMS, Providence, R. L, 1972.Google Scholar
  117. [Ni6]
    L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute Lecture Notes, New York, 1974.zbMATHGoogle Scholar
  118. [Ni7]
    L. Nirenberg, Variational and topological methods in nonlinear problems, Bull.AMS 4(1981), 267–302.MathSciNetzbMATHGoogle Scholar
  119. [Nit1]
    J. Nitsche, Vorlesungen über Minimalflächen, Springer-Verlag, Berlin, 1975.zbMATHGoogle Scholar
  120. [Nit2]
    J. Nitsche, Lectures on Minimal Surfaces, Vol. 1, Cambridge Univ. Press, 1989.zbMATHGoogle Scholar
  121. [Oss1]
    R. Osserman, A Survey of Minimal Surfaces, van Nostrand, New York, 1969.zbMATHGoogle Scholar
  122. [Oss2]
    R. Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem, Ann. of Math. 9(1970), 550–569.MathSciNetGoogle Scholar
  123. [P]
    J. Peetre, On the theory of CP,λspaces, J. Funct. Anal. 4(1969), 71–87.MathSciNetzbMATHGoogle Scholar
  124. [Pi]
    J. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Princeton Univ. Press, Princeton, N. J., 1981.Google Scholar
  125. [Po]
    A. Pogorelov, On convex surfaces with a regular metric, Dokl. Akad. Nauk SSSR 67(1949), 791–794.MathSciNetzbMATHGoogle Scholar
  126. [Po2]
    A. Pogorelov, Monge-Ampere Equations of Elliptic Type, Noordhoff, Groningen, 1964.zbMATHGoogle Scholar
  127. [PrW]
    M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  128. [Radi]
    T. Rado, On Plateau’s problem, Ann. of Math. 31(1930), 457–469.MathSciNetzbMATHGoogle Scholar
  129. [Rad2]
    T. Rado, On the Problem of Plateau, Springer-Verlag, New York, 1933.Google Scholar
  130. [RT]
    J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampere equation, Rocky Mountain J. Math. 7(1977), 345–364.MathSciNetzbMATHGoogle Scholar
  131. [Reif]
    E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104(1960), 1–92.MathSciNetzbMATHGoogle Scholar
  132. [Riv]
    T. Riviere, Everywhere discontinuous maps into spheres. Preprint, 1993.Google Scholar
  133. [SU]
    J. Sachs and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113(1981), 1–24.MathSciNetGoogle Scholar
  134. [Saf]
    M. Safonov, Hamack inequalities for elliptic equations and Hölder continuity of their solutions, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 96(1980), 272–287.MathSciNetzbMATHGoogle Scholar
  135. [Sch]
    R. Schoen, Conformai deformation of a Riemannian metric to constant scalar curvature,/. Diff. Geom. 20(1984), 479–495.MathSciNetzbMATHGoogle Scholar
  136. [Sch2]
    R. Schoen, Analytic aspects of the harmonic map problem, pp. 321–358 in [Cher2].Google Scholar
  137. [ScU]
    R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff.Geom. 17(1982), 307–335; 18(1983), 329.MathSciNetGoogle Scholar
  138. [SY]
    R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with non-negative scalar curvature, Ann. Math. 110(1979), 127–142.MathSciNetzbMATHGoogle Scholar
  139. [Schw]
    J. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.zbMATHGoogle Scholar
  140. [Sel]
    J. Serrin, On a fundamental theorem in the calculus of variations, Acta Math. 102(1959), 1–32.MathSciNetzbMATHGoogle Scholar
  141. [Se2]
    J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. Royal Soc. London Ser. A 264(1969), 413–496.MathSciNetzbMATHGoogle Scholar
  142. [Si]
    L. Simon, Survey lectures on minimal submanifolds, pp. 3–52 in [Bom].Google Scholar
  143. [Si2]
    L. Simon, Singularities of geometrical variational problems, pp. 187–256 in [HW].Google Scholar
  144. [So]
    S. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1964.zbMATHGoogle Scholar
  145. [Spi]
    M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. 1–5, Publish or Perish Press, Berkeley, Calif., 1979.Google Scholar
  146. [Sto]
    J. J. Stoker, Differential Geometry, Wiley-Interscience, New York, 1969.zbMATHGoogle Scholar
  147. [Strl]
    M. Struwe, Plateau’s Problem and the Calculus of Variations, Princeton Univ. Press, Princeton, N. J., 1988.Google Scholar
  148. [Str2]
    M. Struwe, Variational Methods, Springer-Verlag, New York, 1990.zbMATHGoogle Scholar
  149. [T]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.zbMATHGoogle Scholar
  150. [T2]
    M. Taylor, Microlocal analysis on Morrey spaces, IMA Preprint #1322, 1995.Google Scholar
  151. [ToT]
    F. Tomi and A. Tromba, Existence Theorems for Minimal Surfaces of Non-zeroGenus Spanning a Contour, Memoirs AMS #382, Providence, R. I., 1988.Google Scholar
  152. [Tro]
    G. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum, New York, 1987.zbMATHGoogle Scholar
  153. [Trul]
    N. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61(1980), 67–79.MathSciNetzbMATHGoogle Scholar
  154. [Tru2]
    N. Trudinger, Elliptic equations in nondivergence form, Proc. Miniconf. on Partial Differential Equations, Canberra, 1981, pp. 1–16.Google Scholar
  155. [Tru3]
    N. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. AMS 278(1983), 751–769.MathSciNetzbMATHGoogle Scholar
  156. [Tru4]
    N. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh 108(1988), 57–65.MathSciNetzbMATHGoogle Scholar
  157. [TU]
    N. Trudinger and J. Urbas, On the Dirichlet problem for the prescribed Gauss curvature equation, Bull. Austral. Math. Soc. 28(1983), 217–231.MathSciNetzbMATHGoogle Scholar
  158. [U]
    K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 38(1977), 219–240.MathSciNetGoogle Scholar
  159. [Wen]
    H. Wente, Large solutions to the volume constrained Plateau problem, Arch. Rat. Mech. Anal. 75(1980), 59–77.MathSciNetzbMATHGoogle Scholar
  160. [Wid]
    K. Widman, On the Hölder continuity of solutions of elliptic partial differential equations in two variables with coefficients in L. Comm. Pure Appl. Math. 22(1969), 669–682.MathSciNetzbMATHGoogle Scholar
  161. [Yau 1]
    S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation I, CPAM 31(1979), 339–411.Google Scholar
  162. [Yau2]
    S.-T. Yau (ed.), Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N. J., 1982.Google Scholar
  163. [Yau3]
    S.-T. Yau, Survey on partial differential equations in differential geometry, pp. 3–72 in [Yau2].Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

Personalised recommendations