On the Dirichlet problem for the nonlinear equation of the vibrating string. I (Q1321795)

The aim of this paper is the solvability of the Dirichlet problem \[ u_{xx} -u_{yy} + f(x,y,u) = 0,\;(x,y) \in \Omega,\quad u(x,y) = 0,\;(x,y) \in \partial \Omega, \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ 2\), convex relative to the characteristics \(x \pm y = cte\). The existence and uniqueness of weak solutions of (1) is proved, for some class of domains \(\Omega\) when \(f\) is continuous, monotone with respect to \(u\) and satisfies an additional growth restriction related to the first negative and first positive eigenvalue of the operator \(A\), the closure in \(L^ 2(\Omega)\) of the symmetric operator \(A_ 0u = u_{xy}\), \(D(A_ 0) = C^ \infty (\overline \Omega) \cap W_ 0^{1,2} (\Omega)\). Also, some regularity properties of the solutions are studied.