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Singular Integral Operators on Lipschitz Surfaces

Numer of credits: 8 hp

Examiner: Vladimir Kozlov

Course literature: See below. Also various articles.

[1] B.E.J. Dahlberg, Harmonic Analysis and Partial Differential Equations, Technological Report, Chalmers University of Technology, 1985; references made to digital version available at http://www.math.chalmers.se/Math/Research/GeometryAnalysis/Lecturenotes/HAPE.ps [2009-01-07].

[2] G. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University Press, Princeton, N.J. 1995.

[3] G.C. Hsiao and W.L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, Springer-Verlag, Berlin 2008.

[4] V. Maz'ya, Boundary Integral Equations; in V. Maz'ya and S.M. Nikol'skii, Encyclopaedia of Mathematical Sciences, Vol. 27, Springer-Verlag, Berlin 1991.

[5] Y. Meyer and R. Coifman, Wavelets: Calderón-Zygmund and multilinear operators, Cambridge University Press, Cambridge 1997.

[6] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J. 1970.

[7] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, N.J. 1993.

[8] E. M. Stein, Introduction to Fourier Analysis on Euclidian Spaces, Princeton University Press, 1975.

[9] M.E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, Providence, R.I. 2000.

Course contents: Singular integral operators. Riesz transforms. Operators of Calderón-Zygmund type. Pseudodifferential operators. Boundary Integral Methods and Layer Potentials. Riesz potentials. Lp-boundedness of the Cauchy integral operator on Lipschitz curves: different approaches. Layer Potentials on Lipschitz domains. BMO and the T1 theorem. Lp-results for Laplace equaion in Lipschitz domains.


Examination: Written and oral.

Prerequisites: MAI0001 Distributionsteori, MAI0063 Komplex analys, NM1002 Fourieranalys, MAI0067 Integrationsteori, MAI0065 Funktionalanalys (or equivalent courses).

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Last updated: 2014-04-29