Regularity of solutions for arbitrary order variational inequalities with general convex sets (Q1375388)

The author considers systems of variational inequalities of arbitrary order of the form \[ u\in\mathbb{K}: \langle Au, v-u\rangle\geq 0,\quad\forall v\in \mathbb{K}, \] where \(A\) is a differential operator of order \(M\) satisfying certain growth and coercivity conditions, in particular, the pairing \(\langle Au,v\rangle\) is well-defined on the Sobolev space \(W^M_p\). The convex class \(\mathbb{K}\subset W^M_p\) is defined by boundary conditions and the additional requirement that \(T(v-\Psi)(x)\in X\) holds for all \(v\in\mathbb{K}\) and almost all \(x\). Here \(X\) denotes a closed convex set of some \(B\)-space, \(\Psi\) is a given function, and \(T\) denotes a continuous linear map. For various concrete cases, the author shows that the solution \(u\) actually has derivatives in the space \(L^{p+\varepsilon}_{\text{loc}}\) for some \(\varepsilon> 0\). This higher integrability result is established with the maximal function method.