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.


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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