Boundary Value Problems for Higher Order Operators in Lipschitz and $C^1$ Domains

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1. [A] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, 1965. Zbl0142.37401MR178246
2. [ADN1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math.12 (1959), 623 - 727. Zbl0093.10401MR125307
3. [ADN2] S. Agmon, A. Douglis and L. Nirenberg, ibid II, Comm. Pure Appl. Math.22 (1964), 35 - 92. Zbl0123.28706MR162050
4. [B] F. Browder, On the regularity of properties of solutions of elliptic boundary value problems, Comm. Pure Appl. Math.9 (1956), 351 - 361. Zbl0070.09601MR90740
5. [C] J. Cohen, BMO estimates for biharmonic multiple layer potentials, Studia Math.XCI (1988), 109 - 123. Zbl0682.31003MR985078
6. [CGI] J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a Cl domain in the plane, Ind. U. Math. J.32 (1983), 635 - 685. Zbl0534.31003MR711860
7. [CG2] J. Cohen and J. Gosselin, Stress potentials on Cl domains, Jour. Math. Anal. Appl125 (1987), no. 1, 22 - 46. Zbl0633.73005MR891347
8. [CMM] R. Coifman, A. McIntRosh and Y. Meyer, L'integrale de Cauchy definit un Operateur borne sur L2 pour les courbes lipschitziennes, Ann. of Math.116 (1982), 361 - 387. Zbl0497.42012MR672839
9. [Da] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Math.1341, Springer-Verlag, 1988. Zbl0668.35001MR961439
10. [D] B. E. J. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech, Anal.65 (1977), 272 - 288. Zbl0406.28009MR466593
11. [DK] B. E. J. Dahlberg and C. Kenig, Hardy spaces and the Lp - Neumann problem for Laplace's equation in a Lipschitz domain, Annals, of Math.125 (1987), 437 - 465. Zbl0658.35027MR890159
12. [DKV] B. Dahlberg, C. Kenig and G. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. de l'Inst. Fourier36 (1986), 109 - 134. Zbl0589.35040MR865663
13. [G] P. Garabedian, A partial differential equation arising in conformal mapping., Pac. J. Math.1 (1951), 485 - 524. Zbl0045.05102MR46440
14. [Gr] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Publishing, 1985. Zbl0695.35060MR775683
15. [JK1] D. Jerison and C. Kenig, The Dirichlet problem in non-smooth domains, Ann. of Math.113 (1981), 367 - 382. Zbl0434.35027MR607897
16. [JK2] D. Jerison and C. Kenig, Boundary value problems on Lipschitz domains, MAA Studies in Math.23 (1982). Zbl0529.31007MR716504
17. [KO] V. Kondratiev and O. Oleinik, Estimates near the boundary for 2nd order derivatives of solutions of the Dirichlet problem for the biharmonic equation, Att. Accad. Naz. Lincei Rend. Cl. Sei. Fis. Mat. Natur.80 (8) (1986), 525529. Zbl0675.35028MR976945
18. [Ko1] V. Kozlov, The strong zero theorem for an elliptic boundary value problem in a corner, Dokl. Akad. nauk. SSSR309 (6) (1989), 1299 - 1301. Zbl0715.35010MR1045320
19. [Ko2] V. Kozlov, The Dirichlet problem for elliptic equations in domains with conical points, Diff. Eq.26 (1990), 739 - 747. Zbl0712.35033MR1073239
20. [KoM1] V. Kozlov and V. Maz'ya, Estimates of the Lp means and asymptotic behavior of solutions of elliptic boundary value problems in a cone, I, Seminar analysis, Akad. Wiss. DDR, Berlin (1986), 55- 91. Zbl0631.35023MR890526
21. [KoM2] V. Kozlov and V. Maz'ya, ibid, II, Math. Nachr.137 (1988), 113 - 139. Zbl0736.35035MR968991
22. [KoM3] V. Kozlov and V. Maz'ya, Spectral properties of operator pencils generated by elliptic boundary value problems in a cone, Funct. Anal. Appl.22 (1988), 114 - 121. Zbl0672.35050MR947604
23. [KrM] G. Kresin and V. Maz'ya, A sharp constant in a Miranda-Agmon type inequality for solutions to elliptic equations, Soviet Math.32 (1988), 49 -59. Zbl0672.35019MR960965
24. [MN] V. Maz'ya and S. Nazarov, The apex of a cone can be irregular in Wiener's sense for a fourth-order elliptic equation, Mat. Zametki39 (1986), 24 - 28. Zbl0604.35016MR830840
25. [MNP] V. Maz'ya, S. Nazarov and B. Plamenevskii, On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone, Math. Sb.50 (1980), 415 - 437. Zbl0599.35056
26. [MP] V. G. Maz'ya and B. A. Plamenevskii, Estimates in Lp and in Holder classes and the Mir and a-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Amer. Math. Soc. Transl. (2) 123 (1984), 1 - 56; Transl. from Math. Nachr.81 (1978), 25 - 82. Zbl0554.35035MR492821
27. [MNP1] V. Maz'ya, S. Nazarov and B. lamenevskii, Asymptotische Theorie Elliptischer Randwertaufgaben in singular gestorten Gebieten I, Mathematische Lehrbrucher und Monographien, Akademie- Verlag, Berlin, 1991. MR1101139
28. [MR] V. Maz'ya and J. Rossman, On the Miranda-Agmon maximum principle for solutions of elliptic equations in polyhedral and polygonal domains, preprint. Zbl0753.35013MR1143406
29. [O] S. Osher, On Green's function for the biharmonic equation in a right angle wedge, J. Math. Anal. Appl.43 (1973), 705-716. Zbl0263.31005MR324209
30. [PV1] J. Pipher and G. Verchota, The Dirichlet problem in Lp for the biharmonic operator on Lipschitz domains, Amer. J. Math.114 (1992), 923 -972. Zbl0768.31006MR1183527
31. [PV2] J. Pipher and G. Verchota, Area integral estimates for the biharmonic operator in Lipschitz domains, Trans. Amer. Math. Soc.327 (2) (1991), 903-917. Zbl0774.35022MR1024776
32. [PV3] J. Pipher and G. Verchota, A maximum principle for biharmonic functions in non-smooth domains, Comm. Math. Helv., to appear. Zbl0794.31005MR1236761
33. [PV4] J. Pipher and G. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, preprint. Zbl0844.35013MR1361380
34. [PV5] J. Pipher and G. Verchota, Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators, preprint. Zbl0878.35035MR1338674
35. [S] V. Slobodin, Homogeneous boundary value problems for a polyharmonic operator with boundary conditions on thin sets, Izv. Vyssh. Vchebn. Zaved. Mat.10 (1989), 57 - 63. Zbl0697.31007MR1044480
36. [V1] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation, J.of Punct. Anal.59 (1984), 572 - 611. Zbl0589.31005MR769382
37. [V2] G. Verchota, the Dirichlet problem for the biharmonic equation in Cl domains, Ind. U. Math. J.36 (1987), 867 - 895. Zbl0644.35039MR916748
38. [V3] G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Ind. U. Math. J.39 (1990), 671702. Zbl0724.31005MR1078734