Book Review
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1568134
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Book Information:
Author:
\break T.~Kilpel\"ainen J.~Heinonen, and O.~Martio
Title:
Nonlinear potential theory of degenerate elliptic equations
Additional book information:
Oxford University Press, London, 1993, v+363 pp., US$70.00. ISBN 0-19-853669-0.
Heinz Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin-New York, 1966 (German). Ausarbeitung einer im Sommersemester 1965 an der Universität Hamburg gehaltenen Vorlesung. MR 0210916
M. Brelot, Familles de Perron et problème de Dirichlet, Acta Litt. Sci. Szeged 9 (1939), 133–153 (French). MR 734
M. Brelot, Lectures on potential theory, Lectures on Mathematics, vol. 19, Tata Institute of Fundamental Research, Bombay, 1960. Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy. MR 0118980
[4] -, Sur le potentiel et les suites de fonctions sous-harmoniques, C. R. Acad. Sci. Paris Ser. I. Math. 207 (1938), 836-839.
[5] R. Caccioppoli, Sui teoremi d'esistenza di Riemann, Rend. Reale Accad. Sci. Fis. Mat. Napoli 4 (1934), 49-54.
Gianfranco Cimmino, Sulle equazioni lineari alle derivate parziali del secondo ordine di tipo ellittico sopra una superficie chiusa, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 7 (1938), no. 1, 73–96 (Italian). MR 1556798
Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer. MR 0419799
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Nicolaas du Plessis, An introduction to potential theory, University Mathematical Monographs, No. 7, Hafner Publishing Co., Darien, Conn.; Oliver and Boyd, Edinburgh, 1970. MR 0435422
Alexandre Eremenko and John L. Lewis, Uniform limits of certain $A$-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 361–375. MR 1139803, DOI 10.5186/aasfm.1991.1609
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Bent Fuglede, Finely harmonic functions, Lecture Notes in Mathematics, Vol. 289, Springer-Verlag, Berlin-New York, 1972. MR 0450590
F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
[17] O. D. Kellogg, Foundations of potential theory, Dover, New York, 1954.
T. Kilpeläinen, Potential theory for supersolutions of degenerate elliptic equations, Indiana Univ. Math. J. 38 (1989), no. 2, 253–275. MR 997383, DOI 10.1512/iumj.1989.38.38013
Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137–161. MR 1264000, DOI 10.1007/BF02392793
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171. MR 806413, DOI 10.1007/BF02392541
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
[23] V. G Maz'ya, On the continuity at a boundary point of solutions of quasilinear elliptic equations, Vestnik Leningrad Univ. Math. 3 (1976), 225-242.
Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
Oskar Perron, Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u=0$, Math. Z. 18 (1923), no. 1, 42–54 (German). MR 1544619, DOI 10.1007/BF01192395
Seppo Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), no. 3-4, 195–242. MR 781587, DOI 10.1007/BF02392472
Frédéric Riesz, Sur les Fonctions Subharmoniques et Leur Rapport à la Théorie du Potentiel, Acta Math. 48 (1926), no. 3-4, 329–343 (French). MR 1555229, DOI 10.1007/BF02565338
James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
[30] S. L. Sobolev, On a theorem in functional analysis, Mat. Sb. 46 (1938), 471-497. (Russian)
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Hermann Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444. MR 3331
[33] N. Wiener, Certain notions in potential theory, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 24-51.
[34] -, The Dirichlet problem, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 127-146.
[35] -, Note on a paper by O. Perron, J. Math. Phys. Mass. Inst. Tech. 4 (1925), 21-32.
V. A. Zorič, M. A. Lavrent′ev’s theorem on quasiconformal space maps, Mat. Sb. (N.S.) 74 (116) (1967), 417–433 (Russian). MR 0223569
- [1]
- H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math., vol. 22, Springer-Verlag, Berlin, 1966. MR 0210916 (35:1801)
- [2]
- M. Brelot, Familles de Perron et problème de Dirichlet, Acta Sci. Math. (Szeged) IX (1939), 133-153. MR 0000734 (1:121d)
- [3]
- -, Lectures on potential theory, Tata Inst. Fund. Res. Lectures on Math. and Phys., no. 19, Tata Inst. Fund. Res., Bombay, 1960. MR 0118980 (22:9749)
- [4]
- -, Sur le potentiel et les suites de fonctions sous-harmoniques, C. R. Acad. Sci. Paris Ser. I. Math. 207 (1938), 836-839.
- [5]
- R. Caccioppoli, Sui teoremi d'esistenza di Riemann, Rend. Reale Accad. Sci. Fis. Mat. Napoli 4 (1934), 49-54.
- [6]
- G. Cimmino, Sulle equazioni lineari alle derivate parziali del secondo ordine di tipo ellittico sopra una superficie chiusa, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1938), 73-96. MR 1556798
- [7]
- C. Costantinescu and A. Cornea, Potential theory of harmonic spaces, Springer-Verlag, Berlin, 1972. MR 0419799 (54:7817)
- [8]
- E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Reale Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3 (1957), 25-43. MR 0093649 (20:172)
- [9]
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren Math. Wiss., vol. 262, Springer-Verlag, New York, 1984. MR 731258 (85k:31001)
- [10]
- N. Du Plessis, An introduction to potential theory, Oliver and Boyd, Edinburgh, 1970. MR 0435422 (55:8382)
- [11]
- A. Eremenko and J. L. Lewis, Uniform limits of certain
-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn., Ser. A I. Math. 16 (1991), 361-375. MR 1139803 (93b:35039)
- [12]
- E. B. Fabes, C. E. Kenig, and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116. MR 643158 (84i:35070)
- [13]
- B. Fuglede, Finely harmonic functions, Lecture Notes in Math., vol. 289, Springer-Verlag, Berlin, 1972. MR 0450590 (56:8883)
- [14]
- F. Gehring, The
-integrability of the partial derivatives of quasiconformal mappings, Acta Math. 130 (1973), 265-277. MR 0402038 (53:5861)
- [15]
- L. L. Helms, Introduction to potential theory, Wiley-Interscience, New York, 1969. MR 0261018 (41:5638)
- [16]
- D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982), 80-147. MR 676988 (84d:31005b)
- [17]
- O. D. Kellogg, Foundations of potential theory, Dover, New York, 1954.
- [18]
- T. Kilpeläinen, Potential theory for supersolutions of degenerate elliptic equations, Indiana Univ. Math. J. 38 (1989), 253-275. MR 997383 (90e:35061)
- [19]
- T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161. MR 1264000 (95a:35050)
- [20]
- N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, Berlin, 1972. MR 0350027 (50:2520)
- [21]
- P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171. MR 806413 (87g:35074)
- [22]
- W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43-77. MR 0161019 (28:4228)
- [23]
- V. G Maz'ya, On the continuity at a boundary point of solutions of quasilinear elliptic equations, Vestnik Leningrad Univ. Math. 3 (1976), 225-242.
- [24]
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. MR 0159138 (28:2356)
- [25]
- J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954. MR 0100158 (20:6592)
- [26]
- O. Perron, Eine neue Behandlung der ersten Randwertaufgabe fur
, Math. Z. 18 (1923), 42-54. MR 1544619
- [27]
- S. Rickman, The analogue of Picard's theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), 195-242. MR 781587 (86h:30039)
- [28]
- F. Riesz, Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel. I, Acta Math. 48 (1926), 329-343. MR 1555229
- [29]
- J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302. MR 0170096 (30:337)
- [30]
- S. L. Sobolev, On a theorem in functional analysis, Mat. Sb. 46 (1938), 471-497. (Russian)
- [31]
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894 (22:5712)
- [32]
- H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411-444. MR 0003331 (2:202a)
- [33]
- N. Wiener, Certain notions in potential theory, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 24-51.
- [34]
- -, The Dirichlet problem, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 127-146.
- [35]
- -, Note on a paper by O. Perron, J. Math. Phys. Mass. Inst. Tech. 4 (1925), 21-32.
- [36]
- V. A. Zorich, The theorem of M. A. Lavrent'ev on quasiconformal mappings in space, Mat. Sb. 74 (1967), 417-433. MR 0223569 (36:6617)
Review Information:
Reviewer:
Nicola Garofalo
Journal:
Bull. Amer. Math. Soc.
31 (1994), 318-327
DOI:
https://doi.org/10.1090/S0273-0979-1994-00543-9