Conformally invariant variational integrals
HTML articles powered by AMS MathViewer
- by S. Granlund, P. Lindqvist and O. Martio PDF
- Trans. Amer. Math. Soc. 277 (1983), 43-73 Request permission
Abstract:
Let $f:G \to {R^n}$ be quasiregular and $I = \int {F(x,\nabla u) dm}$ a conformally invariant variational integral. Hölder-continuity, Harnack’s inequality and principle are proved for the extremals of $I$. Obstacle problems and their connection to subextremals are studied. If $u$ is an extremal or a subextremal of $I$, then $u \circ f$ is again an extremal or a subextremal if an appropriate change in $F$ is made.References
- Ronald Gariepy and William P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), no. 1, 25–39. MR 492836, DOI 10.1007/BF00280825
- F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 139735, DOI 10.1090/S0002-9947-1962-0139735-8
- Seppo Granlund, Harnack’s inequality in the borderline case, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 159–163. MR 595186, DOI 10.5186/aasfm.1980.0507
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672 H. Lebesgue, Sur le problème de Dirichlet, Rend. Circ. Mat. Palermo 24 (1907), 371-402. P. Lindqvist, A new proof of the lower-semicontinuity of certain convex variational integrals in Sobolev spaces, Report HTKK-MAT-A97 (1977), 1-10.
- Peter Lindqvist, On the Hölder continuity of monotone extremals in the “borderline case”, Ark. Mat. 19 (1981), no. 1, 117–122. MR 625540, DOI 10.1007/BF02384472
- O. Martio, A capacity inequality for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 474 (1970), 18. MR 291449
- O. Martio, Equicontinuity theorem with an application to variational integrals, Duke Math. J. 42 (1975), no. 3, 569–581. MR 380599
- O. Martio, Reflection principle for solutions of elliptic partial differential equations and quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 1, 179–187. MR 639975, DOI 10.5186/aasfm.1981.0610
- O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 448 (1969), 40. MR 0259114
- O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970), 13. MR 0267093
- O. Martio and J. Sarvas, Density conditions in the $n$-capacity, Indiana Univ. Math. J. 26 (1977), no. 4, 761–776. MR 477038, DOI 10.1512/iumj.1977.26.26059
- V. G. Maz′ja, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ. 25 (1970), no. 13, 42–55 (Russian, with English summary). MR 0274948
- Norman G. Meyers and Alan Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J. 42 (1975), 121–136. MR 417568
- 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
- G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383
- Ju. G. Rešetnjak, Spatial mappings with bounded distortion, Sibirsk. Mat. Ž. 8 (1967), 629–658 (Russian). MR 0215990
- Ju. G. Rešetnjak, General theorems on semicontinuity and convergence with functionals, Sibirsk. Mat. Ž. 8 (1967), 1051–1069 (Russian). MR 0220127
- Ju. G. Rešetnjak, Mappings with bounded distortion as extremals of integrals of Dirichlet type, Sibirsk. Mat. Ž. 9 (1968), 652–666 (Russian). MR 0230900
- Ju. G. Rešetnjak, Extremal properties of mappings with bounded distortion, Sibirsk. Mat. Ž. 10 (1969), 1300–1310 (Russian). MR 0276465 R. T. Rockafeller, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1972.
- T. Rado and P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. MR 0079620
- James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 43-73
- MSC: Primary 30C70; Secondary 49A21
- DOI: https://doi.org/10.1090/S0002-9947-1983-0690040-4
- MathSciNet review: 690040