Schwartz-type boundary-value problems for canonical domains in a biharmonic plane

摘要

A commutative algebra Bdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathbbm{B} $$end{document} over the complex field with a basis {e1, e2} satisfying the conditions e12+e222=0,e12+e22≠0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {left({e}_1^2+{e}_2^2ight)}^2=0,{e}_1^2+{e}_2^2e 0 $$end{document} is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form.