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A tour of the theory of absolutely minimizing functions

Authors: Gunnar Aronsson, Michael G. Crandall and Petri Juutinen
Journal: Bull. Amer. Math. Soc. 41 (2004), 439-505
MSC (2000): Primary 35J70, 49K20, 35B50
Published electronically: August 2, 2004
MathSciNet review: 2083637
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Abstract: These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

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  • 1. ALMANSA, A., Échantillonage, interpolation et détection. Applications en imagerie satellitaire, Ph.D. thesis, E.N.S. de Cachan (2002).
  • 2. ARONSSON, G., Hur kan en sandhög se ut? (What is the possible shape of a sandpile?) NORMAT, vol. 13 (1965), 41-44.
  • 3. ARONSSON, G., Minimization problems for the functional ${\rm sup}\sb{x}\,F(x,\,f(x),\,f\sp{\prime} (x))$, Ark. Mat. 6 (1965), 33-53. MR 0196551 (33:4738)
  • 4. ARONSSON, G., Minimization problems for the functional ${\rm sup}\sb{x}\,F(x, f(x),f\sp\prime (x))$. II., Ark. Mat. 6 (1966), 409-431. MR 0203541 (34:3391)
  • 5. ARONSSON, G., Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551-561. MR 0217665 (36:754)
  • 6. ARONSSON, G., On the partial differential equation $u\sb{x}{}\sp{2}u\sb{xx} +2u\sb{x}u\sb{y}u\sb{xy}+u\sb{y}{}\sp{2}u\sb{yy}=0$, Ark. Mat. 7 (1968), 395-425. MR 0237962 (38:6239)
  • 7. ARONSSON, G., Minimization problems for the functional ${\rm sup}\sb{x}\,F(x,\,f(x),\,f\sp{\prime} \,(x))$. III, Ark. Mat. 7 (1969), 509-512. MR 0240690 (39:2035)
  • 8. ARONSSON, G., On certain singular solutions of the partial differential equation $u\sp{2}\sb{x}u\sb{xx}+2u\sb{x}u\sb{y}u\sb{xy}+u\sp{2}\sb{y}u\sb{yy}=0$, Manuscripta Math. 47 (1984), no. 1-3, 133-151. MR 0744316 (85m:35011)
  • 9. ARONSSON, G., Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=\infty$, Manuscripta Math. 56 (1986), no. 2, 135-158. MR 0850366 (87j:35070)
  • 10. ARONSSON, G., EVANS, L. C., AND WU, Y., Fast/slow diffusion and growing sandpiles, J. Differential Equations 131 (1996), no. 2, 304-335. MR 1419017 (97i:35068)
  • 11. BANACH, S., Wstep do teorii funkcji rzeczywistych. (Polish) [Introduction to the theory of real functions], Monografie Matematyczne, Warszawa-Wroc\law, 1951. MR 0043161 (13:216a)
  • 12. BARLES, G., AND BUSCA, J. Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Diff. Equations 26 (2001), 2323-2337. MR 1876420 (2002k:35078)
  • 13. BARRON, E. N., JENSEN, R. R., AND WANG, C. Y., The Euler equation and absolute minimizers of $L\sp \infty$ functionals, Arch. Ration. Mech. Anal. 157 (2001), no. 4, 255-283. MR 1831173 (2002m:49006)
  • 14. BARRON, E. N., JENSEN, R. R., AND WANG, C. Y., Lower semicontinuity of $L\sp \infty$ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 4, 495-517. MR 1841130 (2002c:49020)
  • 15. BELLONI, M., AND KAWOHL, B., The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$, ESAIM Control Optim. Calc. Var. 10 (2004), 28-52.
  • 16. BHATTACHARYA, T., An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions, Electron. J. Differential Equations, No. 44 (2001), 8 pp. (electronic). MR 1836812 (2002b:35071)
  • 17. BHATTACHARYA, T., On the properties of $\infty$-harmonic functions and an application to capacitary convex rings, Electron. J. Differential Equations, No. 101 (2002), 22 pp. (electronic). MR 1938397 (2003j:35126)
  • 18. BHATTACHARYA, T., DIBENEDETTO, E., AND MANFREDI, J., Limits as $p\to\infty$ of $\Delta\sb pu\sb p=f$ and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 15-68 (1991). MR 1155453 (93a:35049)
  • 19. BIESKE, T., On $\infty$-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), no. 3-4, 727-761. MR 1900561 (2003g:35033)
  • 20. BIESKE, T., Viscosity solutions on Grushin-type planes, Illinois J. Math. 46 (2002), no. 3, 893-911. MR 1951247 (2003k:35037)
  • 21. BIESKE, T., Lipschitz extensions on generalized Grushin spaces, Michigan Math. J. (to appear).
  • 22. BIESKE, T., AND CAPOGNA, L., The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics, Trans. Amer. Math. Soc. (to appear).
  • 23. CAFFARELLI, L., AND CABRE, J., Fully Nonlinear Elliptic Equations, American Mathematical Society, Colloquium Publications vol. 43, Providence, 1995. MR 1351007 (96h:35046)
  • 24. CAO, F., Absolutely minimizing Lipschitz extension with discontinuous boundary data, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 6, 563-568. MR 1650611 (99h:49042)
  • 25. CASELLES, V., MOREL, J.-M., AND SBERT, C., An axiomatic approach to image interpolation, IEEE Trans. Image Process. 7 (1998), no. 3, 376-386. MR 1669524 (2000d:94001)
  • 26. CHAMPION, T., AND DE PASCALE, L., A principle of comparison with distance functions for absolute minimizers, preprint.
  • 27. CHAMPION, T., DE PASCALE, L., AND PRINARI, F.,$\Gamma$-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var. (to appear).
  • 28. CRANDALL, M. G., An efficient derivation of the Aronsson equation, Arch. Rational Mech. Anal. 167 (2003), 271-279. MR 1981858 (2004b:35053)
  • 29. CRANDALL, M. G., EVANS, L. C., AND GARIEPY, R. F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123-139. MR 1861094 (2002h:49048)
  • 30. CRANDALL, M. G., AND EVANS, L. C., A remark on infinity harmonic functions, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viã del Mar-Valparaiso, 2000), 123-129 (electronic), Electron. J. Differ. Equ. Conf., 6, Southwest Texas State Univ., San Marcos, TX, 2001. MR 1804769 (2001j:35076)
  • 31. CRANDALL, M. G., AND LIONS, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 0690039 (85g:35029)
  • 32. CRANDALL, M. G., ISHII, H., AND LIONS, P. L., User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67. MR 1118699 (92j:35050)
  • 33. CRANDALL, M. G., AND ZHANG, J., Another way to say harmonic, Trans. Amer. Math. Soc. 355 (2003), 241-263. MR 1928087 (2003k:35062)
  • 34. CZIPSZER, J., AND GEH´ER, L., Extension of functions satisfying a Lipschitz condition, Acta Math. Acad. Sci. Hungar. 6 (1955), 213-220. MR0071493 (17:136b)
  • 35. EVANS, L. C., Estimates for smooth absolutely minimizing Lipschitz extensions, Electron. J. Differential Equations 1993, No. 03, approx. 9 pp. (electronic only). MR 1241488 (94k:35117)
  • 36. EVANS, L. C., AND GARIEPY, R. F., Measure Theory and Fine Properties of Functions, 1999, CRC Press, Boca Raton, Florida. MR 1158660 (93f:28001)
  • 37. FEDERER, H., Geometric Measure Theory, Springer-Verlag, New York, 1969. MR 0257325 (41:1976)
  • 38. FRIEDMAN, A., Foundations of Modern Analysis, Dover Publications, Inc., New York, 1982. MR 0663003 (83h:46001)
  • 39. FUKAGAI, N., ITO, M., AND NARUKAWA, K., Limit as $p\to\infty$ of $p$-Laplace eigenvalue problems and $L\sp\infty$-inequality of the Poincaré type, Differential Integral Equations 12 (1999), no. 2, 183-206. MR 1672746 (99m:35058)
  • 40. GASPARI, T., The infinity Laplacian in infinite dimensions, Calc. Var. Partial Differential Equations (to appear).
  • 41. GILBARG, D., AND TRUDINGER, N. S., Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2001. MR 1814364 (2001k:35004)
  • 42. GIUSTI, E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933 (2004g:49003)
  • 43. HEINONEN, J., KILPELÄINEN, T., AND MARTIO, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, The Clarendon Press, Oxford University Press, New York, 1993. MR 1207810 (94e:31003)
  • 44. ISHIBASHI, T., AND KOIKE, S., On fully nonlinear PDEs derived from variational problems of $L\sp p$ norms, SIAM J. Math. Anal. 33 (2001), no. 3, 545-569 (electronic). MR 1871409 (2002j:49037)
  • 45. JANFALK, U., Behaviour in the limit, as $p\to\infty$, of minimizers of functionals involving $p$-Dirichlet integrals, SIAM J. Math. Anal. 27 (1996), no. 2, 341-360. MR 1377478 (96m:49023)
  • 46. JENSEN, R. R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), no. 1, 1-27. MR 0920674 (89a:35038)
  • 47. JENSEN, R. R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51-74. MR 1218686 (94g:35063)
  • 48. JUUTINEN, P., Minimization problems for Lipschitz functions via viscosity solutions, Ann. Acad. Sci. Fenn. Math. Diss. No. 115 (1998), 53 pp. MR 1632063 (2000a:49055)
  • 49. JUUTINEN, P., Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57-67. MR 1884349 (2002m:54020)
  • 50. JUUTINEN, P., LINDQVIST, P., AND MANFREDI, J., The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 89-105. MR 1716563 (2000g:35047)
  • 51. LINDQVIST, P., AND MANFREDI, J., The Harnack inequality for $\infty$-harmonic functions, Electron. J. Differential Equations (1995), No. 04, approx. 5 pp. MR 1322829 (96b:35025)
  • 52. LINDQVIST, P., AND MANFREDI, J., Note on $\infty$-superharmonic functions, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 471-480. MR 1605682 (98k:35043)
  • 53. LINDQVIST, P., MANFREDI, J., AND SAKSMAN, E., Superharmonicity of nonlinear ground states, Rev. Mat. Iberoamericana 16 (2000), no. 1, 17-28. MR 1768532 (2001f:35133)
  • 54. MANFREDI, J., PETROSYAN, A., AND SHAHGHOLIAN, H., A free boundary problem for $\infty$-Laplace equation, Calc. Var. Partial Differential Equations 14 (2002), no. 3, 359-384. MR 1899452 (2003a:35212)
  • 55. MCSHANE, E. J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
  • 56. MIL'MAN, V. A., Lipschitz extensions of linearly bounded functions (Russian), Mat. Sb. 189 (1998), no. 8, 67-92; translation in Sb. Math. 189 (1998), no. 7-8, 1179-1203. MR 1669631 (2000b:46043)
  • 57. MIL'MAN, V. A., Absolutely minimal extensions of functions on metric spaces (Russian), Mat. Sb. 190 (1999), no. 6, 83-110; translation in Sb. Math. 190 (1999), no. 5-6, 859-885. MR 1719573 (2000j:41041)
  • 58. REICH, S., Review of ``Geometry of Banach spaces, duality mappings and nonlinear problems" by Ioana Cioranescu, Bull. Amer. Math. Soc. 26 (N.S.) (1992), 367-370.
  • 59. ROSSET, E., A lower bound for the gradient of $\infty$-harmonic functions, Electron. J. Differential Equations, No. 02 (1996), approx. 6 pp. (electronic). MR 1371219 (96k:35066)
  • 60. ROSSET, E., Symmetry and convexity of level sets of solutions to the infinity Laplace's equation, Electron. J. Differential Equations, No. 34 (1998), 12 pp. (electronic). MR 1656591 (99j:35068)
  • 61. SAVIN, O.,$C^1$ regularity for infinity harmonic functions in two dimensions, preprint.
  • 62. WANG, C., The Aronsson equation for absolute minimizers of $L^\infty$-functionals associated with vector fields satisfying Hörmander's condition, preprint (2003).
  • 63. WHITNEY, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63-89. MR 1501735
  • 64. WU, Y., Absolute minimizers in Finsler metrics, Ph.D. dissertation, UC Berkeley, 1995.
  • 65. YU, Y., Sufficiency of Aronsson-Euler equations without zeroth order terms, preprint.

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Additional Information

Gunnar Aronsson
Affiliation: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden

Michael G. Crandall
Affiliation: Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106

Petri Juutinen
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland

Received by editor(s): July 18, 2003
Received by editor(s) in revised form: May 24, 2004
Published electronically: August 2, 2004
Additional Notes: “Absolutely minimizing" has other meanings besides the one herein. We might more properly say “absolutely minimizing in the Lipschitz sense" instead, but prefer to abbreviate.
The third author is supported by the Academy of Finland, project #80566.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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