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For a simly connected region, the solution of the first problem of plane elasticity can be reduced - roughly speaking - to the solution of a biharmonic problem. This problem can then be solved approximately by the method of least squares on the boundary, developed by K. Rektorys and V. Zahradník in Apl. mat. 19 (1974), 101-131. The present paper gives a generalization of this method for multiply connected regions. Two fundamental questions which arise in this case are answered, namely: (i) How to formulate the problem in order that it correspond to the reality. (ii) How to modify the method and prove the convergence.
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