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On the solvability of a spatial problem of Darboux type for the wave equation. (English) Zbl 0842.35052
The solvability of the Darboux problem for the wave equation (1) \(\square u = F\) with the boundary condition (2) \(u|_{S_1} = f_1\), (3) \({\partial u \over \partial n} |_{S_0} = 0\) is proved, where \(S_0\), \(S_1\) are some given surfaces.
The main result is: For any \(f_1 \in W^1_2 (S_1)\), \(F \in L_2 (D_+)\) there exists a unique strong solution \(u\) of the problem (1)–(3) in the class \(W^1_2\). The estimate \(|u |_{W^1_2 (D_+)}\leq C (|f_1 |_{W^1_2 (S_1)} + |F |_{L_2 (D_+)})\) is also proved.

35L20 Initial-boundary value problems for second-order hyperbolic equations
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