Geodesic
Necessary and sufficient conditions for a~generalized solution to the initial-boundary value problem for the wave equation to belong to $W^1_p$ with~$p\geq1$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 115-120

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We establish necessary and sufficient conditions on the boundary function under which a generalized solution to the initial–boundary value problem for the wave equation with boundary conditions of the first kind belongs to $W^1_p$. 
@article{TM_2013_283_a7,
     author = {V. A. Il'in and A. A. Kuleshov},
     title = {Necessary and sufficient conditions for a~generalized solution to the initial-boundary value problem for the wave equation to belong to $W^1_p$ with~$p\geq1$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {115--120},
     publisher = {mathdoc},
     volume = {283},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_283_a7/}
}
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V. A. Il'in; A. A. Kuleshov. Necessary and sufficient conditions for a~generalized solution to the initial-boundary value problem for the wave equation to belong to $W^1_p$ with~$p\geq1$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 115-120. http://geodesic.mathdoc.fr/item/TM_2013_283_a7/