Nonsymmetric systems on nonsmooth planar domains

Verchota, G.; Vogel, A.

doi:10.1090/S0002-9947-97-02047-3

Nonsymmetric systems on nonsmooth planar domains
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by G. C. Verchota and A. L. Vogel PDF

Trans. Amer. Math. Soc. 349 (1997), 4501-4535 Request permission

Abstract:

We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
Heinrich Begehr and Robert P. Gilbert, Transformations, transmutations, and kernel functions. Vol. 2, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 59, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1242553
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
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
Martin Costabel and Monique Dauge, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems, Math. Nachr. 162 (1993), 209–237. MR 1239587, DOI 10.1002/mana.19931620117
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
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
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 Poisson integral for Lipschitz and $C^{1}$-domains, Studia Math. 66 (1979), no. 1, 13–24. MR 562447, DOI 10.4064/sm-66-1-13-24
B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota, Area integral estimates and maximum principles for higher order elliptic equations and systems, in preparation.
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
B. E. J. Dahlberg and C. E. Kenig, $L^p$ estimates for the three-dimensional systems of elastostatics on Lipschitz domains, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 621–634. MR 1044810
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135 (English, with French summary). MR 865663, DOI 10.5802/aif.1062
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 795–818. MR 975122, DOI 10.1215/S0012-7094-88-05735-3
L. Diomeda and B. Lisena, The Dirichlet problem for elliptic systems in piecewise $C^1$ plane domains, Indiana Univ. Math. J. 41 (1992), no. 3, 649–670. MR 1189905, DOI 10.1512/iumj.1992.41.41035
E. B. Fabes, Max Jodeit Jr., and J. E. Lewis, On the spectra of a Hardy kernel, J. Functional Analysis 21 (1976), no. 2, 187–194. MR 0394311, DOI 10.1016/0022-1236(76)90076-8
E. B. Fabes, Max Jodeit Jr., and Jeff E. Lewis, Double layer potentials for domains with corners and edges, Indiana Univ. Math. J. 26 (1977), no. 1, 95–114. MR 432899, DOI 10.1512/iumj.1977.26.26007
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
Eugene B. Fabes and Carlos E. Kenig, On the Hardy space $H^{1}$ of a $C^{1}$ domain, Ark. Mat. 19 (1981), no. 1, 1–22. MR 625534, DOI 10.1007/BF02384466
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Carlos E. Kenig, Weighted $H^{p}$ spaces on Lipschitz domains, Amer. J. Math. 102 (1980), no. 1, 129–163. MR 556889, DOI 10.2307/2374173
V. A. Kozlov, Singularities of solutions of the Dirichlet problem for elliptic equations in a neighborhood of corner points, Algebra i Analiz 1 (1989), no. 4, 161–177 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 4, 967–982. MR 1027465
Loo Keng Hua, Wei Lin, and Ci-Quian Wu, Second-order systems of partial differential equations in the plane, Research Notes in Mathematics, vol. 128, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 906654
Jeff E. Lewis and Cesare Parenti, Pseudodifferential operators and Hardy kernels on $L^{p}(\textbf {R}^{+})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 3, 481–503. MR 597548
Jeff E. Lewis and Cesare Parenti, Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations 8 (1983), no. 5, 477–544. MR 695401, DOI 10.1080/03605308308820276
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
Jill Pipher and Gregory C. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. of Math. (2) 142 (1995), no. 1, 1–38. MR 1338674, DOI 10.2307/2118610
J. Pipher and G. C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, Potential Anal. 4 (1995), no. 6, 615–636. MR 1361380, DOI 10.1007/BF02345828
Shia-kuai Ding, Kan-ting Wang, Ju-nien Ma, Chia-lo Shun, and Tong Chang, Definition of ellipticity of a system of second order partial differential equations with constant coefficients, Chinese Math. 1 (1960), 288–299. MR 152738
N. Th. Varopoulos, BMO functions and the $\overline \partial$-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 508035, DOI 10.2140/pjm.1977.71.221
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
Gregory Verchota, The Dirichlet problem for the biharmonic equation in $C^1$ domains, Indiana Univ. Math. J. 36 (1987), no. 4, 867–895. MR 916748, DOI 10.1512/iumj.1987.36.36048
G. C. Verchota, Potentials for the Dirichlet problem in Lipschitz domains, Potential Theory—ICPT 94 (Kouty, 1994), de Gruyter, Berlin, 1996, pp. 167–187.