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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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