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Behaviour of superharmonic functions and fine Riquier problem at the biharmonic Martin boundary. (Comportement des fonctions bisurharmoniques et problème de Riquier fin a la frontière de Martin biharmonique.) (French) Zbl 1148.31007
The behaviour of biharmonic functions is an old care of the authors who continue it in the present paper with new results. One of these new results is the extension of K. Gowrisankaran’s studies [Ann. Inst. Fourier 13, No. 2, 307–356 (1963; Zbl 0134.09503)], respecting harmonic functions and boundary value problems. They extend the results to the case of biharmonic functions in a biharmonic hard space \((\Omega,{\mathcal K})\), where \(\Omega\) is a Green’s space.
Another result of the paper is the continuation of the results obtained by the first author [Positivity 6, No. 2, 129–145 (2002; Zbl 0998.31004)] concerning the biharmonic Martin’s boundary. Connected to this problem, they propose a minimum principle with the fine filters for the study of the Riquier generalised problem.
All the obtained results in the paper can be extended to a hard polyharmonic space of arbitrary order. The restriction to biharmonic case is made for simplicity reasons.
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
31C35 Martin boundary theory