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Espaces biharmoniques. (Biharmonic spaces). (French) Zbl 0567.31006

Théorie du potentiel, Proc. Colloq. J. Deny, Orsay/France 1983, Lect. Notes Math. 1096, 116-148 (1984).
[For the entire collection see Zbl 0543.00004.]
This paper reduces the theory of biharmonic spaces introduced by E. P. Smyrnelis [Ann. Inst. Fourier 25, No.1, 35-97 (1975; Zbl 0295.31006), ibid. 26, No.3, 1-47 (1976; Zbl 0325.31020)] to a coupling of two harmonic spaces: (X,\({\mathcal H})\) is a biharmonic space if and only if there exist two harmonic spaces (X,\({\mathcal H}_ 1)\) and (X,\({\mathcal H}_ 2)\) having a common base \({\mathcal U}_ r\) of regular sets and a compatible family \((p_ V)_{V\in {\mathcal U}_ r}\) of potentials for (X,\({\mathcal H}_ 1)\) such that for every open U in X the space \({\mathcal H}(U)\) is the set of all ordered pairs \((h_ 1,h_ 2)\in {\mathcal C}(\bar U)\times {\mathcal C}(\bar U)\) satisfying \(h_ 1=H^ 1_ V h_ 1+K^ 1_ V h_ 2,\quad h_ 2=H^ 2_ V h_ 2\) for every \(V\in {\mathcal U}_ r\) with \(\bar V\subset U\) (where \(H^ i_ V\) denotes the harmonic kernel associated with V for (X,\({\mathcal H}_ i)\) and \(K^ 1_ V\) denotes the potential kernel associated with \(p_ V\) for (X,\({\mathcal H}_ 1))\). This characterization (which can be extended to polyharmonic functions) yields a better understanding for these spaces. Using this description as starting point, the introduction of all further notions for biharmonic spaces become more natural and proofs are getting simpler. One section consists of an application to second order differential operators.
Reviewer: W.Hansen

MSC:

31D05 Axiomatic potential theory
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J45 Systems of elliptic equations, general (MSC2000)