Linear operators satisfying the assumptions of some generalized Lax-Milgram theorems. (Q696145)

Let \(H\) be a Hilbert space and \(A: H\to H\) be an elliptic continuous linear operator, that is, there is a constant \(\alpha> 0\) such that \(\langle Ax,x\rangle\geq \alpha\| x\|^2\), \(x\in H\). Then \(A\) is invertible. That is, for every \(y\in H\), the equation \(Ax= y\) has a unique solution \(x= A^{-1}y\) and \(\| A^{-1}\|\leq {1\over\alpha}\). Recently, \textit{J. Saint Raymond} [Matematiche 52, 149--157 (1997; Zbl 0930.47005)] has extended the above theorem to many more cases. In this paper, the authors analyze the structure of two classes of linear operators satisfying the assumptions of two generalized Lax-Milgram theorems of Saint Raymond. The results of the present paper are very interesting and useful for studying boundary value problems and variational inequalities.