Geodesic
Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities
Sbornik. Mathematics, Tome 214 (2023) no. 4, pp. 550-566

Voir la notice de l'article provenant de la source Math-Net.Ru

The main aim of this paper is to study the geometric and metric properties of $B$- and $C$-capacities related to problems of uniform approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of Euclidean spaces. In the harmonic case this problem is well known, and it was studied in detail in the framework of classical potential theory in the first half of the 20th century. For a wide class of equations mentioned above, we obtain two-sided estimates between the corresponding $B_+$- and $C_+$-capacities (defined in terms of potentials of positive measures) and the harmonic capacity in the same dimension. Our research method is based on new simple explicit formulae obtained for the fundamental solutions of the equations under consideration. Bibliography: 12 titles.
Keywords: elliptic quadratic form, homogeneous second-order elliptic equation, fundamental solution, capacity, Calderon-Zygmund kernel
Mots-clés : Fourier transform. 
@article{SM_2023_214_4_a3,
     author = {P. V. Paramonov and K. Yu. Fedorovskiy},
     title = {Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities},
     journal = {Sbornik. Mathematics},
     pages = {550--566},
     publisher = {mathdoc},
     volume = {214},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2023_214_4_a3/}
}
TY  - JOUR
AU  - P. V. Paramonov
AU  - K. Yu. Fedorovskiy
TI  - Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities
JO  - Sbornik. Mathematics
PY  - 2023
SP  - 550
EP  - 566
VL  - 214
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SM_2023_214_4_a3/
LA  - en
ID  - SM_2023_214_4_a3
ER  - 
%0 Journal Article
%A P. V. Paramonov
%A K. Yu. Fedorovskiy
%T Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities
%J Sbornik. Mathematics
%D 2023
%P 550-566
%V 214
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SM_2023_214_4_a3/
%G en
%F SM_2023_214_4_a3
P. V. Paramonov; K. Yu. Fedorovskiy. Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities. Sbornik. Mathematics, Tome 214 (2023) no. 4, pp. 550-566. http://geodesic.mathdoc.fr/item/SM_2023_214_4_a3/