The regularity of solutions of vector variational inequalities has a rather recent history. Frehese proved for the first time the continuity of the first derivatives of the solutions. Subsequently Kinderlehrer was able to prove the \(C^{1,\alpha}\)-continuity of solutions for systems of two-dimensional elasticity. A further progress in studying more general systems was attempted by Vralčeva and Archipora, but their results were confined to systems diagol with respect to their principal part, thus excluding the system of elasticity.
Here the question is tackled in an apparently artificial way, but which has the advantage of transforming the vector variational inequality to a scalar inequality. The artifice is first illustrated in the fully scalar case. Let us suppose we have to solve a variational inequality involving the Laplace operator in a domain \(\Omega\) for a scalar function \(u\) constrained by condition of the type \(u\geq 0\) on a portion \(\Gamma\) of the boundary \(\partial \Omega\), a condition called of “thin obstacle”.
Thus it is possible to prove that the problem is perfectly equivalent to solving a variational inequality formulated on \(\partial \Omega\) and not on \(\Omega\), provided that we re-define the differential operator and the convex set where solutions are sought. The modified operator is a pseudodifferential operator associated to the Laplace operator, that is the operator \(\partial /\partial n\), \(n\) being the exterior normal to \(\partial \Omega\); the convex set is the set of functions \(\phi\) of the \(W^{,2}(\partial \Omega)\) space such that \(\phi\geq \psi\) a.e. on \(\Gamma\). For the new scalar problem it is proved that, if data are regular, the solution is of class \(C^{\alpha}_{loc}(\Gamma)\); and this regularity can be further improved to \(C_{loc}^{1+\alpha}(\Gamma).\)
The main idea of the scalar case extends to Signorini’s problem, since, in this situation too, it is possible to define an equivalent boundary variational inequality involving the pseudodifferential operator associated to elasticity operator. Therefore, it is simple to prove that the solution is of class \(C_{loc}^{1+\alpha}(\Gamma)\), but the precise value of \(\alpha\in (0,1)\) cannot be determined.