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The wave equation in a wedge with general boundary conditions. (English) Zbl 0790.35055
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.

MSC:
35L05 Wave equation
35A20 Analyticity in context of PDEs
35C15 Integral representations of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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