A note on the regularity of the solutions to two variational inequalities involving a pseudodifferential operator (Q925417)

The author considers the equation \[ -\Delta u+ k^2u= 0\quad\text{for }x_{n+1}> 0 \] in \(\mathbb{R}^{n+1}\) with obstacle boundary conditions at \(x_{n+1}= 0\). The problem is reduced to a pseudodifferential equation on the boundary \(x_{n+1}= 0\). Namely, the author obtains a variational inequality involving the first-order pseudodifferential operator with symbol \(A(\eta)= (k^2+ |\eta|^2)^{1/2}\). Existence and uniqueness of the solution follow then from \textit{J. L. Lions} and \textit{G. Stampacchia} [Commun. Pure Appl. Math. 20, 493--519 (1967; Zbl 0152.34601)]. Global regularity and interior regularity of the solution are further deduced.