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, p. 343-367
The four natural boundary problems for the weighted form Laplacians L=adδ+bδd,a,b>0 acting on polynomial differential forms in the n-dimensional Euclidean ball are solved explicitly. Moreover, an algebraic algorithm for generating a solution from the boundary data is given in each case.
Classification:  35J67,  35J25,  34K10
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {2},
     year = {2008},
     pages = {343-367},
     zbl = {1178.35156},
     mrnumber = {2437031},
     language = {en},
     url = {http://www.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://www.numdam.org/item/ASNSP_2008_5_7_2_343_0/

[1] I. G. Avramidi, 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] T. P. Branson, P. B. Gilkey and A. Pierzchalski, Heat equation asymptotics of elliptic differential operators with non scalar leading symbol, Math. Nachr 166 (1994), 207-215. | MR 1273333 | Zbl 0831.35041

[4] L. V. Ahlfors, 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] L. V. Ahlfors, Invariant operators and integral representations in hyperbolic spaces, Math. Scand. 36 (1975), 27-43. | MR 402036 | Zbl 0313.31009

[6] L. V. Ahlfors, Quasiconformal deformations and mappings in n , J. Anal. Math. 30 (1976), 74-97. | MR 492238 | Zbl 0338.30017

[7] T. P. Branson, Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334-383. | MR 1491546 | Zbl 0904.58054

[8] T. P. Branson and A. Pierzchalski, Natural boundary conditions for gradients, Łódź University, Faculty of Mathematics and Computer Science, preprint 2008.

[9] R. R. Coifman and G. Weiss, Representations of compact groups and spherical harmonics, L'Enseignement Mathematique 14 (1969), 121-175. | MR 255877 | Zbl 0174.18902

[10] P. B. Gilkey, “Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem”, Publish or Perish, Wilmington, Delaware, 1984. | MR 783634 | Zbl 0565.58035

[11] P. B. Gilkey, T. P. Branson and S. A. Fulling, Heat equation asymptotics of “nonminimal” operators on differential forms, J. Math. Phys. 32 (1991), 2089-2091. | MR 1123599 | Zbl 0778.58062

[12] P. B. Gilkey and L. Smith, The eta invariant for a class of elliptic boundary value problems, Comm. Pure Appl. Math. 98 (1976), 225-240. | MR 680084 | Zbl 0512.58035

[13] V. P. Gusynin and V. V. Kornyak, DeWitt-Seeley-Gilkey coefficients for nonminimal operators in curved space, Fundam. Prikl. Math. 5 (1999), 649-674. | MR 1806847 | Zbl 1031.58014

[14] Y. Homma, Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), 87-114. | MR 2171224 | Zbl 1079.53026

[15] J. Kalina, B. Ørsted, A. Pierzchalski, P. Walczak and G. Zhang, Elliptic gradients and highest weights, Bull. Polon. Acad. Sci. Ser. Math. 44 (1996), 511-519. | MR 1420968 | Zbl 0899.22011

[16] J. Kalina, A. Pierzchalski and P. Walczak, Only one of generalized gradients can be elliptic, Ann. Pol. Math. 67 (1997), 111-120. | MR 1460594 | Zbl 0901.53017

[17] I. Kolář, P. W. Michor and J. Slovák, “Natural Operations in Differential Geometry”, Springer-Verlag Berlin Heidelberg, 1993. | MR 1202431 | Zbl 0782.53013

[18] W. Kozłowski, Laplace type operators: Dirichlet problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2007), 53-80. | Numdam | MR 2341515 | Zbl 1185.35039

[19] A. Lipowski, Boundary problems for the Ahlfors operator, (in Polish), Ph.D. Thesis, Łódź University (1996), 1-55.

[20] R. Palais, “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] B. Ørsted and A. Pierzchalski, The Ahlfors Laplacian on a Riemannian manifold with boundary, Michigan Math. J. 43 (1996), 99-122. | MR 1381602 | Zbl 0853.58108

[22] A. Pierzchalski, “Geometry of Quasiconformal Deformations of Riemannian Manifolds”, Łódź University Press, 1997.

[23] A. Pierzchalski, Ricci curvature and quasiconformal deformations of a Riemannian manifold”, Manuscripta Math. 66 (1989), 113-127. | MR 1027303 | Zbl 0698.53021

[24] H. M. Reimann, 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] H. M. Reimann, 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] E. M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representation od the rotation group, Amer J. Math. 90 (1968), 163-196. | MR 223492 | Zbl 0157.18303

[27] E. M. Stein and G. Weiss, “Fourier Analysis on Euclidean Spaces”, Princeton University Press, 1971. | MR 304972 | Zbl 0232.42007

[28] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschingungen einer beliebig gestalten elastischen Koerpers, Rendiconti Cir. Mat. Palermo 39 (1915), 1-49. | JFM 45.1016.02