Abstract
We give an overview of results concerning certain boundary value problems for elliptic Partial Differential Equations on nonsmooth domains, Clifford algebras, and wavelet techniques, and how these are related. In particular, we consider Laplace’s equation on nonsmooth domains and show how this leads to wavelet analysis of certain Clifford algebra valued singular integral operators. A natural context for stuying these operators is the theory of Hardy spaces of Clifford analytic functions, which we discuss in some detail. General elliptic operators are also considered.
The first author was partially supported by ONR Grant N0001-90-J-1343 and DARPA grant AFOSR 89-0455, and the second author partially by ONR Grant N0001-90-J-1343.
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Bibliography
L. Andersson, B. Jawerth, and M. Mitrea. The Cauchy singular integral operator and Clifford wavelets. Research Report 4, University of South Carolina, 1991.
F. Brackx, R. Delanghe, and F. Sommen. Clifford analysis. Pitman Research Notes in Math., 76, 1982.
A.P. Calderón. Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci. USA, 74:1324–1327, 1977.
A.P. Calderón. Boundary value problems for the Laplace equation in Lipschitz domains. In Recent Progress in Fourier Analysis, volume 111, pages 33–48. North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford, 1983.
R. Coifman, A. Mcintosh, and Y. Meyer. L’integrale de Cauchy definit un operateur borne sur L2 pour les courbes lipschitziennes. Annals of Mathematics, 116:361–387, 1982.
R.R. Coifman, P. Jones, and S. Semmes. Two elementary proofs of the L2 boundedness of the Cauchy integrals on Lipschitz curves. J. Amer. Math. Soc., 2:553–564, 1989.
B.J.E. Dahlberg. Harmonic functions in Lipschitz domains. Proc. of Symp. Pure Math., XXXV(1):313–322, 1979.
E.B. Dahlberg. On estimates of harmonic measure. Arch. Rat. Mech. and Analysis, 65:275–288, 1977.
J.E. Gilbert and M.A.M. Murray. Clifford algebras and Dirac operators in harmonic analysis. Cambridge Studies in Advanced Mathematics, 26, 1991.
K. Hoffman. Banach Spaces of Analytic Functions. Dover Publications, New York, 1962.
V. Iftimie. Fonctioneshypercomplexes. Bull. Math. de la Soc. Sci. Math. de la R. S. Roumanie, 4:279–332, 1965.
D.S. Jerison and C.E. Kenig. The Neumann problem on Lipschitz domains. Bull. AMS, 4:203–207, 1981.
J.-L. Journé. Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón. Lecture Notes in Mathematics, 994, 1983.
E.C. Kenig. Weighted Hp spaces on Lipschitz domains. Amer. J. Math., 102:129–163, 1980.
E.C. Kenig. Recent progress on boundary value problems on Lipschitz domains. Proc. of Symp. Pure Math., 43:175–205, 1985.
V.D. Kupradze. Potential methods in the theory of elasticity. Jerusalem, 1965.
C. Li, A. Mcintosh, and T. Qian. Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces. Macquarie Mathematics Report, 91-087, 1991.
Y. Meyer. Ondelettes et Opérateurs. Hermann, Paris, 1990.
M. Mitrea. Clifford algebras and boundary estimates for harmonic functions. Preprint, 1992.
M. Mitrea. Hardy spaces of harmonic functions in Lipschitz domains. Preprint, 1992.
M. Mitrea. Singular integrals, Hardy spaces and Clifford wavelets. Preprint, 1992.
M. Mitrea. Singular Integrals, Hardy Spaces and Clifford Wavelets. PhD thesis, Univ. of South Carolina, 1992.
G.C. Moisil and N. Teodorescu. Fonctions holomorphes dans l’espace. Mathematicae Cluij, 5:142–150, 1931.
J. Nečas. Les methodes directes en théorie des équations élliptiques. Academia, Prague, 1967.
I. Plemelj. Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend. Monatsch. Math. u. Physik, XIX:205–210, 1908.
E.M. Stein and G. Weiss. On the theory of harmonic functions of several variables, I. The theory of Hp spaces. Acta Math, 103:25–62, 1960.
G. Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Functional Analysis, 59(3):572–611, 1984.
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© 1994 Springer Science+Business Media Dordrecht
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Jawerth, B., Mitrea, M. (1994). Clifford wavelets, Hardy spaces, and elliptic boundary value problems. In: Byrnes, J.S., Byrnes, J.L., Hargreaves, K.A., Berry, K. (eds) Wavelets and Their Applications. NATO ASI Series, vol 442. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1028-0_11
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DOI: https://doi.org/10.1007/978-94-011-1028-0_11
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