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Book Review

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Book Information:

Author: Carlos E. Kenig
Title: Harmonic analysis techniques for second order elliptic boundary value problems
Additional book information: CBMS Regional Conf. Series in Math., no. 83, Amer. Math. Soc., Providence, RI, 1994, xii + 146 pp., ISBN 0-8218-0309-3, $30.00

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  • [AC] I. Athanasopoulos and L. Caffarelli, A theorem of real analysis and its application to a free boundary problem, Comm. Pure Appl. Math. 38 (1985), 499--502. MR 86j:49062
  • [AgCS] N. Aguilera, L. Caffarelli and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa 14 (1987), 355--387. MR 89h:49016
  • [B] R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 339--379. MR 90d:35118
  • [C1] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part 1: Lipschitz free boundaries are $C^{1,\alpha}$, Revista Mat. Ib. 3 (1987), 139--162. MR 90d:35036
  • [C2] ------, A Harnack inequality approach to the regularity of free boundaries, Comm. Pure Appl. Math. 42 (1989), 55--78. MR 90b:35246
  • [Ca] A. P. Calderón, Cauchy integrals in Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324--1327. MR 57:6445
  • [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241--250. MR 50:10670
  • [CDM] R. Coifman, D. Deng, and Y. Meyer, Domaine de la racine carrée de certains operateurs differentiels accrétifs, Ann. Inst. Fourier 33 (1983), 123--134. MR 84h:35040
  • [CFK] L. Caffarelli, E. Fabes, and C. Kenig, Completely singular elliptic-harmonic measures, Indiana U. Math. J. 30 (1981), 189--213. MR 83a:35033
  • [CFMS] L. Cafarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of non-negative solutions of elliptic operators in divergence form, Ind. U. Math. J. 30 (1981), 621--640. MR 83c:35040
  • [CMcM] R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361--387. MR 84m:42027
  • [CZ] A. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171--225. MR 25:310
  • [D1] B. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech. Anal. 56 (1977), 272--288. MR 57:6470
  • [D2] ------, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119--1138. MR 88i:35061
  • [De] E. De. Giorgi, Sulla differenziabilita e analiticita delle estremali degli integrali multipli regolari, Mem. Acad. Sci. Torino 3 (1957), 25--43. MR 20:172
  • [DK] B. E. J. Dahlberg and C. Kenig, Hardy spaces and the $L^p$-Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math. 125 (1987), 437--465. MR 88d:35044
  • [Fa] E. Fabes, The initial value problem for parabolic equations with data in $L^p(\mathbf R^n)$, Studia Math. 44 (1972), 89--109. MR 48:6698
  • [FJK] E. Fabes, D. Jerison, and C. Kenig, Multilinear Littlewood-Paley estimates with applications to partial differential equations, Proc. Nat. Acad. Sci. USA 79 (1982), 5746--5750. MR 83k:47035
  • [FJR] E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary value problems on $C^1$ domains, Acta Math. 141 (1978), 165--186. MR 80b:31006
  • [Fe] R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), 127--135. MR 90b:35068
  • [FeKP] R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65--124. MR 93h:31010
  • [FSW] E. Fabes, S. Sroka, and K. Widman, Littlewood-Paley estimates for parabolic equations with sub-Dini continuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 305--334. MR 80h:35072
  • [HW] R. Hunt and R. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307--322. MR 37:1634
  • [JK] D. Jerison and C. Kenig, The Dirichlet problem on non-smooth domains, Ann. of Math. 113 (1981), 367--382. MR 84j:35076
  • [JN] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415--426. MR 24:A1348
  • [KP1] C. Kenig and J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Math. 113 (1993), 447--509. MR 95b:35046
  • [KP2] ------, The Neumann problem for elliptic equations with non-smooth coefficients, Part 2, Duke J. Math. (to appear).
  • [L] N. Lim, The Dirichlet problem for elliptic equations with data in $L^p$, J. Funct. Anal. (to appear).
  • [Na] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931--954. MR 20:6592
  • [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal functions, Trans. Amer. Math. Soc. 165 (1972), 207--226. MR 45:2461
  • [Mo] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577--591. MR 28:2356
  • [PV1] J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and $C^1$ domains, Comm. Math. Helv. 68 (1993), 385--414. MR 94j:35030
  • [PV2] ------, Maximum principles for the polyharmonic equation on Lipschitz domains, J. Potential Anal. (to appear).
  • [V] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation, J. Funct. Anal. 59 (1984), 572--611. MR 86e:35038
  • [WZ] M. Weiss and A. Zygmund, A note on smooth functions, Indag. Math. 62 (1959), 52--58. MR 21:5849
  • [Z] S. Zaremba, Sur le principe de Dirichlet, Acta Math. 34 (1911), 293--316.

Review Information:

Reviewer: Jill C. Pipher
Affiliation: Brown University
Journal: Bull. Amer. Math. Soc. 33 (1996), 229-236
Review copyright: © Copyright 1996 American Mathematical Society
American Mathematical Society