Bulletin of the American Mathematical Society

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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, $30.00$

Ioannis Athanasopoulos and Luis A. Caffarelli, A theorem of real analysis and its application to free boundary problems, Comm. Pure Appl. Math. 38 (1985), no. 5, 499–502. MR 803243, DOI 10.1002/cpa.3160380503

N. E. Aguilera, L. A. Caffarelli, and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 3, 355–387 (1988). MR 951225

Russell M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339–379. MR 987761, DOI 10.2307/2374513

Sergio Campanato, A bound for the solutions of a basic elliptic system with nonlinearity $q\geq 2$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 80 (1986), no. 3, 81–88 (1987) (English, with Italian summary). MR 976693

Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. MR 973745, DOI 10.1002/cpa.3160420105

A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 466568, DOI 10.1073/pnas.74.4.1324

R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250

R. R. Coifman, D. G. Deng, and Y. Meyer, Domaine de la racine carrée de certains opérateurs différentiels accrétifs, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, x, 123–134 (French, with English summary). MR 699490

Luis A. Caffarelli, Eugene B. Fabes, and Carlos E. Kenig, Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30 (1981), no. 6, 917–924. MR 632860, DOI 10.1512/iumj.1981.30.30067

L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621–640. MR 620271, DOI 10.1512/iumj.1981.30.30049

R. 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. (2) 116 (1982), no. 2, 361–387 (French). MR 672839, DOI 10.2307/2007065

A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225

Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445

Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119–1138. MR 859772, DOI 10.2307/2374598

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

Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI 10.2307/1971407

Eugene Fabes, The initial value problem for parabolic equations with data in $L^{p}(R^{n})$, Studia Math. 44 (1972), 389–409. MR 328356, DOI 10.4064/sm-44-4-389-409

Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig, Multilinear Littlewood-Paley estimates with applications to partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), no. 18, 5746–5750. MR 674919, DOI 10.1073/pnas.79.18.5746

E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), no. 3-4, 165–186. MR 501367, DOI 10.1007/BF02545747

R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), no. 1, 127–135. MR 955604, DOI 10.1090/S0894-0347-1989-0955604-8

R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65–124. MR 1114608, DOI 10.2307/2944333

E. Fabes, S. Sroka, and K.-O. Widman, Littlewood-Paley a priori estimates for parabolic equations with sub-Dini continuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 2, 305–334. MR 541451

Richard A. Hunt and Richard L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 226044, DOI 10.1090/S0002-9947-1968-0226044-7

David S. Jerison and Carlos E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367–382. MR 607897, DOI 10.2307/2006988

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317

Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447–509. MR 1231834, DOI 10.1007/BF01244315

[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 , J. Funct. Anal. (to appear).

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841

Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6

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. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and $C^1$ domains, Comment. Math. Helv. 68 (1993), no. 3, 385–414. MR 1236761, DOI 10.1007/BF02565827

[PV2]

------, Maximum principles for the polyharmonic equation on Lipschitz domains, J. Potential Anal. (to appear).

Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1

Mary Weiss and Antoni Zygmund, A note on smooth functions, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 52–58. MR 0107122

[Z]

S. Zaremba, Sur le principe de Dirichlet, Acta Math. 34 (1911), 293--316.

[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 0803243

[AgCS]

N. Aguilera, L. Caffarelli and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa 14 (1987), 355--387. MR 0951225

[B]

R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 339--379. MR 0987761

[C1]

L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, Part 1: Lipschitz free boundaries are , Revista Mat. Ib. 3 (1987), 139--162. MR 0976693

[C2]

------, A Harnack inequality approach to the regularity of free boundaries, Comm. Pure Appl. Math. 42 (1989), 55--78. MR 0973745

[Ca]

A. P. Calderón, Cauchy integrals in Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324--1327. MR 0466568

[CF]

R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241--250. MR 0358205

[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 0699490

[CFK]

L. Caffarelli, E. Fabes, and C. Kenig, Completely singular elliptic-harmonic measures, Indiana U. Math. J. 30 (1981), 189--213. MR 0632860

[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 0620271

[CMcM]

R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361--387. MR 0672839

[CZ]

A. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171--225. MR 0136849

[D1]

B. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech. Anal. 56 (1977), 272--288. MR 0466593

[D2]

------, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119--1138. MR 0859772

[De]

E. De. Giorgi, Sulla differenziabilita e analiticita delle estremali degli integrali multipli regolari, Mem. Acad. Sci. Torino 3 (1957), 25--43. MR 0093649

[DK]

B. E. J. Dahlberg and C. Kenig, Hardy spaces and the -Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math. 125 (1987), 437--465. MR 0890159

[Fa]

E. Fabes, The initial value problem for parabolic equations with data in , Studia Math. 44 (1972), 89--109. MR 0328356

[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 0674919

[FJR]

E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary value problems on domains, Acta Math. 141 (1978), 165--186. MR 0501367

[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 0955604

[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 1114608

[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 0541451

[HW]

R. Hunt and R. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307--322. MR 0226044

[JK]

D. Jerison and C. Kenig, The Dirichlet problem on non-smooth domains, Ann. of Math. 113 (1981), 367--382. MR 0607897

[JN]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415--426. MR 0131498

[KP1]

C. Kenig and J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Math. 113 (1993), 447--509. MR 1231834

[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 , J. Funct. Anal. (to appear).

[Na]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931--954. MR 0100158

[M]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal functions, Trans. Amer. Math. Soc. 165 (1972), 207--226. MR 0293384

[Mo]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577--591. MR 0159138

[PV1]

J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and domains, Comm. Math. Helv. 68 (1993), 385--414. MR 1236761

[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 0769382

[WZ]

M. Weiss and A. Zygmund, A note on smooth functions, Indag. Math. 62 (1959), 52--58. MR 0107122

[Z]

S. Zaremba, Sur le principe de Dirichlet, Acta Math. 34 (1911), 293--316.