Stress potentials on \(C^ 1\) domains (Q1096443)

The paper is a continuation of the authors' earlier work [Ark. Mat. 23, 217-240 (1985; Zbl 0602.35028) and Indiana Univ. Math. J. 32, 635-685 (1983; Zbl 0534.31003)] relating to the Dirichlet problem and adjoint boundary value problems for the biharmonic equation on \(C^ 1\)-domains in the plane. The concept of modified lower order potential introduced by the authors is used for determining the stress function from the boundary stresses. The results obtained provide a point-wise solution to the Neumann-type boundary value problem for a biharmonic equation. It is shown that the solution of the boundary value problem for a biharmonic equation can be obtained with boundary data in the sub-space \(L^ q\times L^ q(\Omega)\) and with the boundary values obtained nontangentially point-wise almost everywhere. As a matter of fact the classical elasto-static problem of obtaining interior stress from the prescribed stress on the boundary has been solved and the solution of biharmonic problem on \(C^ 1\) domain with data in \(L^ p\) for \(1<p<\infty\) has been obtained.