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Green’s formulas and bilinear conservation laws. (English. Russian original) Zbl 0623.47055

Math. Notes 40, 776-780 (1986); translation from Mat. Zametki 40, No. 4, 478-483 (1986).
Let L and M be two linear differential operators of dimensions \(t\times g\) respectively \(t\times h\) and let \(L^*\) be the adjoint of L. One proves that a Green formula \[ (Lu,Mv)=\sum^{m}_{i=1}\partial_ iJ^ i[u,v] \] (where (Lu,Mv) and \(J^ i[u,v]\) are bilinear forms in u and v, u,v being regular vector functions) holds iff \(L^*M=0\). Different examples are shown.
Reviewer: C.Badea-Simionescu

MSC:

47F05 General theory of partial differential operators
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References:

[1] V. S. Vladimirov and V. V. Zharinov, ?Closed forms associated with linear differential operators,? Differents. Uravn.,16, No. 5, 845-867 (1980). · Zbl 0506.35015
[2] S. G. Mikhlin, Linear Partial Differential Equations [in Russian], Vysshaya Shkola, Moscow (1977). · Zbl 0378.45003
[3] N. G. Marchuk, ?A broad class of Green’s formulas and conservation laws,? Dokl. Akad. Nauk SSSR,285, No. 6, 1325-1328 (1985).
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