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On the traction problem for the Stokes system. (English) Zbl 1163.35463

Summary: The purpose of this paper is to prove the existence and uniqueness of classical solutions to the traction problem associated with the Stokes system in a bounded as well as in an exterior domain of the $n$-dimensional space $\left(n\ge 2\right)$. This problem is of some interest in the theory of viscous fluid dynamics and of importance in incompressible linear elastostatics. Despite its relevance in the applications, in its more general formulation (not connected boundaries) it has mainly been studied in the framework of the variational theory and not by the methods of the potential theory that lead to an integral expression of the solution by means of suitable densities and permit to show the existence of a regular solution under weaker (classical) hypotheses, for instance the sole continuity of the datum.

We consider domains of class ${C}^{1,\alpha }$ and are able to prove that the classical solution to our problem can be written as the sum of a single layer potential and a double layer potential whose density belongs to the cokernel of the Fredholm-Riesz operator associated with the boundary value problem and is (in principle) known. Moreover, we derive a maximum modulus theorem for the traction, field in the direction of the normal to the boundary.

The problem of finding an explicit expression of a basis of both the kernel and cokernel of the Fredholm-Riesz operator associated with the boundary value problem is of practical interest, because their knowledge gives the analytic form oi the solution to the traction problem in an exterior domain vanishing at infinity and corresponding to a uniform pressure on the boundary. As far as the particular but interesting case of the ellipsoid is concerned, we furnish here its form by using a very simple technique.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory 35C15 Integral representations of solutions of PDE 74B05 Classical linear elasticity 76D07 Stokes and related (Oseen, etc.) flows