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\(L^p\) polyharmonic Dirichlet problems in regular domains. I: The unit disc. (English) Zbl 1281.31004

The paper is devoted to the study of the Dirichlet boundary value problem for polyharmanic equation in the \(L^{p}\)-setting on the unit disc. It continues the investigation of the group by Prof. H. Begehr and his followers.
New forms of the higher order Poisson kernels are found and integral representations of the solutions to the considred problems are obtained on their base.

MSC:

31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31A10 Integral representations, integral operators, integral equations methods in two dimensions
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[1] Aksoy Ü, Adv. Dyn. Syst. Appl. 5 (2) pp 133– (2010)
[2] Begehr H, Acta Math. Vietnamica 36 pp 169– (2011)
[3] DOI: 10.1007/s10231-007-0050-5 · Zbl 1223.30016 · doi:10.1007/s10231-007-0050-5
[4] Begehr H, Oper. Theory Adv. Appl. 205 pp 101– (2009)
[5] Begehr H, Georgian Math. J. 14 pp 33– (2007)
[6] DOI: 10.4171/ZAA/1244 · Zbl 1089.30040 · doi:10.4171/ZAA/1244
[7] DOI: 10.1080/0278107031000103412 · Zbl 1146.30307 · doi:10.1080/0278107031000103412
[8] Du Z, Doctoral Dissertation (2008)
[9] DOI: 10.1016/j.jmaa.2009.07.048 · Zbl 1183.31001 · doi:10.1016/j.jmaa.2009.07.048
[10] Gaertner E, Doctoral Dissertation (2006)
[11] Nicolesco M, Les Fonctions Polyharmoniques (1936)
[12] Riquier Ch, J. Math. 9 (5) pp 297– (1926)
[13] Stein EM, Introduction to Fourier Analysis on Euclidean Spaces (1971)
[14] Stein EM, Complex Analysis (2003)
[15] Hardy H, Inequalities (1934)
[16] Stein EM, Singular Integrals and Differentiability Properties of Functions (1970)
[17] Folland GB, Real Analysis: Modern Techniques and their Applications (1999) · Zbl 0924.28001
[18] Jost J, Partial Differential Equations, GTM 214 (2002)
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