no. 1
Laplace type operators: Dirichlet problem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 53-80.

We investigate Laplace type operators in the Euclidean space. We give a purely algebraic proof of the theorem on existence and uniqueness (in the space of polynomial forms) of the Dirichlet boundary problem for a Laplace type operator and give a method of determining the exact solution to that problem. Moreover, we give a decomposition of the kernel of a Laplace type operator into 𝖲𝖮(n)-irreducible subspaces.

Classification : 35J25, 34K10, 35J67 
@article{ASNSP_2007_5_6_1_53_0,
     author = {Koz{\l}, Wojciech},
     title = {Laplace type operators: {Dirichlet} problem},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {53--80},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {1},
     year = {2007},
     mrnumber = {2341515},
     zbl = {1185.35039},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2007_5_6_1_53_0/}
}
TY  - JOUR
AU  - Kozł, Wojciech
TI  - Laplace type operators: Dirichlet problem
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2007
SP  - 53
EP  - 80
VL  - 6
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2007_5_6_1_53_0/
LA  - en
ID  - ASNSP_2007_5_6_1_53_0
ER  - 
%0 Journal Article
%A Kozł, Wojciech
%T Laplace type operators: Dirichlet problem
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2007
%P 53-80
%V 6
%N 1
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2007_5_6_1_53_0/
%G en
%F ASNSP_2007_5_6_1_53_0
Kozł, Wojciech. Laplace type operators: Dirichlet problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 53-80. http://archive.numdam.org/item/ASNSP_2007_5_6_1_53_0/

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

[2] S. Axler, P. Bourdon and W. Ramey, “Harmonic Function Theory”, Springer-Verlag, New York, 2001. | MR | Zbl

[3] R. R. Coifman and G. Weiss, Representations of compact groups and spherical harmonics, Enseign. Math. 14 (1969), 121-175. | MR | Zbl

[4] G. B. Folland Harmonic analysis of the de Rham complex on the sphere, J. Reine Angew. Math. 398 (1989), 130-143. | MR | Zbl

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

[6] A. Korányi and S. Vági, Group theoretic remarks on Riesz system on balls, Proc. Amer. Math. Soc. 85 (1982), 200-205. | MR | Zbl

[7] N. V. Krylov, “Lectures on Elliptic and Parabolic Equations in Hölder Spaces”, Graduate Studies in Mathematics, Vol. 12, American Mathematical Society, Providence, RI, 1996. | MR | Zbl

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

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

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

[11] H. M. Reimann, Rotation invariant differential equation for vector fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9, (1982), 160-174. | Numdam | MR | Zbl

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

[13] A. Strasburger, Differential operators of gradient type associated with spherical harmonics, Ann. Polon. Math. 53 (1991), 161-183. | MR | Zbl

[14] H. Weyl, Eigenschwingungen eines beliebig gestatleten elastischen Korpers, Rend. Circ. Mat. Palermo 39 (1915), 1-50. | JFM

[15] K. Yano, “Integral Formulas in Riemannian Geometry”, Marcel Dekker INC, New York, 1970. | MR | Zbl