Classical boundary value problem (BVP) of the theory of elasticity requires one of the following surface conditions to be known on the whole boundary of a domain:(i) stress vector; (ii) displacement vector; or (iii) combinations of (i) and (ii). In all these cases the BVP is well-posed and hence possesses a unique solution. However one can consider another type of the BVP: overdetermined on a part of the oundary and underdetermined on the rest of it. This BVP originates from the problem of the stress state determination inside an elastic body by measuring the displacements over a part of the boundary. The other part of the boundary is loaded by unknown stresses and direct measurements on it are not available (it can be an internal boundary as well). I is shown how the problem can be reduced to a to the similar problem for holomorphic function. For the case of wedge-like domain the solution of the problem is found by the applying the Mellin transform. For the case when displacements are known at a number of discrete points, a numerical method has been suggested. The method is based on the conversion of the discrete data into the continuous data by using the orthogonal Laguerre polynomials. This method is discussed in the connection with an example of the problem of the in-situ stress determination in a rock mass on the basis of data of the displacement monitoring in an excavation.
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