an:00034116
Zbl 0790.35055
Eskin, Gregory
The wave equation in a wedge with general boundary conditions
EN
Commun. Partial Differ. Equations 17, No. 1-2, 99-160 (1992).
00221394
1992
j
35L05 35A20 35C15 35D05
Fourier transform; pseudodifferential operator; uniqueness; wave equation; uniform Lopatinsky condition; integral equations on the boundary; two Riemann-Hilbert problems; existence
The paper is devoted to the wave equation in \(\mathbb{R}\times\Omega\) with the zero initial conditions and boundary conditions \(B_ i u= h_ i\) on \(\Gamma_ i\), \(i=1,2\), where \(\Omega\subset \mathbb{R}^ n\) is a wedge bounded by \(\Gamma_ 1\), \(\Gamma_ 2\), \(\Gamma_ 1= \{x\in\mathbb{R}^ n\); \(x_ 1\geq 0\), \(x_ 2=0\}\), \(\Gamma_ 2= \{x\in\mathbb{R}^ n\); \(x_ 1 \sin\alpha-x_ 2 \cos\alpha=0\), \(x_ 1 \cos\alpha+ x_ 2\sin \alpha\geq 0\}\) and homogeneous (in the derivatives) polynomials \(B_ 1\), \(B_ 2\) satisfying a uniform Lopatinsky condition. The problem is equivalent to the solution of integral equations on the boundary, which is reduced to two Riemann-Hilbert problems with a shift and these are solved explicitly. Uniqueness and existence of the solution in the appropriate spaces of distributions is proved.
M.Kop????kov?? (Praha)