Existence of regular solutions for nonlinear Signorini's problems

Abstract

The aim of this paper is to prove the existence of a Lipschitz solution for nonlinear obstacle problems with quadratic growth in the gradient and Signorini’s boundary conditions. For the linear Signorini’s problem, existence and regularity results have been given by Brezis [S], and by Hanouzet and Joly [13]; in particular, the last two authors apply a dual estimation technique, consisting of estimating the conormal derivative of the solution as a measure on the boundary. For the nonlinear Signorini’s problem of the type considered in this paper, no existence results for weak solutions are known; the only results available are due to Frehse [lo], and refer to the interior regularity of arbitrary bounded weak solutions. To obtain the existence of a (Lipschitz) weak solution in the present nonlinear quadratic case, global estimates up to the boundary are needed. Let us remark incidentally that C’ regularity up to the boundary of the weak solution was previously obtained by da Veiga [2], da Veiga and Conti [3], by Giaquinta and Modica [15], using a direct variational approach; however, they only allow a sublinear growth in the gradient. In this case the existence of weak solutions can be obtained by standard methods. Our proof of the existence of a weak solution, in the case of quadratic growth in the gradient, relies on an a priori estimate up to the boundary of the Lipschitz norm for C’ n Hz arbitrary solution u, which is obtained in Section 3. We at first estimate the derivatives of u in the interior of the domain and its tangential derivatives on the boundary, adapting the differential quotient technique of Frehse to the present situation in which