The four natural boundary problems for the weighted form Laplacians acting on polynomial differential forms in the -dimensional Euclidean ball are solved explicitly. Moreover, an algebraic algorithm for generating a solution from the boundary data is given in each case.
@article{ASNSP_2008_5_7_2_343_0, author = {Koz\l owski, Wojciech and Pierzchalski, Antoni}, title = {Natural boundary value problems for weighted form laplacians}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {343--367}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {2}, year = {2008}, zbl = {1178.35156}, mrnumber = {2437031}, language = {en}, url = {archive.numdam.org/item/ASNSP_2008_5_7_2_343_0/} }
Kozłowski, Wojciech; Pierzchalski, Antoni. Natural boundary value problems for weighted form laplacians. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, pp. 343-367. http://archive.numdam.org/item/ASNSP_2008_5_7_2_343_0/
[1] Non-Laplace type operators on manifolds with boundary, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ (2006), 107-140. | MR 2246767 | Zbl 1122.58013
,[2] I. G. Avramidi and T. P. Branson Heat kernel asymptotics of operators with non-Laplace principal part, Rev. Math. Phys. 13 (2001), 847-890. | MR 1843855 | Zbl 1031.58015
[3] Heat equation asymptotics of elliptic differential operators with non scalar leading symbol, Math. Nachr 166 (1994), 207-215. | MR 1273333 | Zbl 0831.35041
, and ,[4] Conditions for quasiconformal deformations in several variables, Contributions to Analysis, A collection of papers dedicated to L. Bers, Academic press, New York (1974), 19-25. | MR 367189 | Zbl 0304.30015
,[5] Invariant operators and integral representations in hyperbolic spaces, Math. Scand. 36 (1975), 27-43. | MR 402036 | Zbl 0313.31009
,[6] Quasiconformal deformations and mappings in , J. Anal. Math. 30 (1976), 74-97. | MR 492238 | Zbl 0338.30017
,[7] Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334-383. | MR 1491546 | Zbl 0904.58054
,[8] Natural boundary conditions for gradients, Łódź University, Faculty of Mathematics and Computer Science, preprint 2008.
and ,[9] Representations of compact groups and spherical harmonics, L'Enseignement Mathematique 14 (1969), 121-175. | MR 255877 | Zbl 0174.18902
and ,[10] “Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem”, Publish or Perish, Wilmington, Delaware, 1984. | MR 783634 | Zbl 0565.58035
,[11] Heat equation asymptotics of “nonminimal” operators on differential forms, J. Math. Phys. 32 (1991), 2089-2091. | MR 1123599 | Zbl 0778.58062
, and ,[12] The eta invariant for a class of elliptic boundary value problems, Comm. Pure Appl. Math. 98 (1976), 225-240. | MR 680084 | Zbl 0512.58035
and ,[13] DeWitt-Seeley-Gilkey coefficients for nonminimal operators in curved space, Fundam. Prikl. Math. 5 (1999), 649-674. | MR 1806847 | Zbl 1031.58014
and ,[14] Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), 87-114. | MR 2171224 | Zbl 1079.53026
,[15] Elliptic gradients and highest weights, Bull. Polon. Acad. Sci. Ser. Math. 44 (1996), 511-519. | MR 1420968 | Zbl 0899.22011
, , , and ,[16] Only one of generalized gradients can be elliptic, Ann. Pol. Math. 67 (1997), 111-120. | MR 1460594 | Zbl 0901.53017
, and ,[17] “Natural Operations in Differential Geometry”, Springer-Verlag Berlin Heidelberg, 1993. | MR 1202431 | Zbl 0782.53013
, and ,[18] Laplace type operators: Dirichlet problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2007), 53-80. | Numdam | MR 2341515 | Zbl 1185.35039
,[19] Boundary problems for the Ahlfors operator, (in Polish), Ph.D. Thesis, Łódź University (1996), 1-55.
,[20] “Seminar on the Atiyah-Singer Index Theorem”, Annals of Mathematics Studies, Vol. 57, Princeton University Press, Princeton, N.J., 1965. | MR 198494 | Zbl 0137.17002
,[21] The Ahlfors Laplacian on a Riemannian manifold with boundary, Michigan Math. J. 43 (1996), 99-122. | MR 1381602 | Zbl 0853.58108
and ,[22] “Geometry of Quasiconformal Deformations of Riemannian Manifolds”, Łódź University Press, 1997.
,[23] Ricci curvature and quasiconformal deformations of a Riemannian manifold”, Manuscripta Math. 66 (1989), 113-127. | MR 1027303 | Zbl 0698.53021
,[24] Rotation invariant differential equation for vector fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9 (1982), 160-174. | Numdam | MR 664106 | Zbl 0491.35027
,[25] Invariant system of differential operators, Proc. of a seminar held in Torino May-June 1982, Topics in modern harmonic analysis, Instituto di Alta Matematica, Roma, 1983. | MR 748882 | Zbl 0564.35017
,[26] Generalization of the Cauchy-Riemann equations and representation od the rotation group, Amer J. Math. 90 (1968), 163-196. | MR 223492 | Zbl 0157.18303
and ,[27] “Fourier Analysis on Euclidean Spaces”, Princeton University Press, 1971. | MR 304972 | Zbl 0232.42007
and ,[28] Das asymptotische Verteilungsgesetz der Eigenschingungen einer beliebig gestalten elastischen Koerpers, Rendiconti Cir. Mat. Palermo 39 (1915), 1-49. | JFM 45.1016.02
,