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}, mrnumber = {2437031}, zbl = {1178.35156}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_2_343_0/} }
TY - JOUR AU - Kozłowski, Wojciech AU - Pierzchalski, Antoni TI - Natural boundary value problems for weighted form laplacians JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 SP - 343 EP - 367 VL - 7 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_2_343_0/ LA - en ID - ASNSP_2008_5_7_2_343_0 ER -
%0 Journal Article %A Kozłowski, Wojciech %A Pierzchalski, Antoni %T Natural boundary value problems for weighted form laplacians %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2008 %P 343-367 %V 7 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2008_5_7_2_343_0/ %G en %F 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, Serie 5, Volume 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 | Zbl
,[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 | Zbl
[3] Heat equation asymptotics of elliptic differential operators with non scalar leading symbol, Math. Nachr 166 (1994), 207-215. | MR | Zbl
, 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 | Zbl
,[5] Invariant operators and integral representations in hyperbolic spaces, Math. Scand. 36 (1975), 27-43. | MR | Zbl
,[6] Quasiconformal deformations and mappings in , J. Anal. Math. 30 (1976), 74-97. | MR | Zbl
,[7] Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334-383. | MR | Zbl
,[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 | Zbl
and ,[10] “Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem”, Publish or Perish, Wilmington, Delaware, 1984. | MR | Zbl
,[11] Heat equation asymptotics of “nonminimal” operators on differential forms, J. Math. Phys. 32 (1991), 2089-2091. | MR | Zbl
, and ,[12] The eta invariant for a class of elliptic boundary value problems, Comm. Pure Appl. Math. 98 (1976), 225-240. | MR | Zbl
and ,[13] DeWitt-Seeley-Gilkey coefficients for nonminimal operators in curved space, Fundam. Prikl. Math. 5 (1999), 649-674. | MR | Zbl
and ,[14] Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), 87-114. | MR | Zbl
,[15] Elliptic gradients and highest weights, Bull. Polon. Acad. Sci. Ser. Math. 44 (1996), 511-519. | MR | Zbl
, , , and ,[16] Only one of generalized gradients can be elliptic, Ann. Pol. Math. 67 (1997), 111-120. | MR | Zbl
, and ,[17] “Natural Operations in Differential Geometry”, Springer-Verlag Berlin Heidelberg, 1993. | MR | Zbl
, and ,[18] Laplace type operators: Dirichlet problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2007), 53-80. | Numdam | MR | Zbl
,[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 | Zbl
,[21] The Ahlfors Laplacian on a Riemannian manifold with boundary, Michigan Math. J. 43 (1996), 99-122. | MR | Zbl
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 | Zbl
,[24] Rotation invariant differential equation for vector fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9 (1982), 160-174. | Numdam | MR | Zbl
,[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 | Zbl
,[26] Generalization of the Cauchy-Riemann equations and representation od the rotation group, Amer J. Math. 90 (1968), 163-196. | MR | Zbl
and ,[27] “Fourier Analysis on Euclidean Spaces”, Princeton University Press, 1971. | MR | Zbl
and ,[28] Das asymptotische Verteilungsgesetz der Eigenschingungen einer beliebig gestalten elastischen Koerpers, Rendiconti Cir. Mat. Palermo 39 (1915), 1-49. | JFM
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