Existencia y unicidad de la solución del Problema de Diricfhlet para la ecuacion de Helmholtz en ángulos exteriores

Ponente(s): Anatoli Merzon Merzon, Dr: Jose EligionDe La Paz Mendez

We consider the following model boundary value problem in a plane angle Q of magnitude Φ > π with a complex wave number ω ∈ C^+. (−∆ − ω^2) u(x) = 0, x ∈ Q, u(x)|_ Γ1 = f_1(x), u(x)|_Γ2 = f_2(x). Here Γ |_l are the sides of the angle Q, , f_l = e^ {ik_l x|} , k_ l > 0, l = 1, 2. Problems of this type arise in many areas of mathematical physics, for example ih diffusion of desintegration gas . Besides, some problems of the diffraction theory are reduced to it as a model problem. The problem differs from numerous similar problems in which the boundary data are summable functions. Using the method of complex characteristics [1], we reduce the problem to the Riemann–Hilbert problem for a Neumann data on the Riemann surface of zeros of the symbol of the Helmholtz operator. We solve this problem in quadratures and we give the solution u in the Sommerfeld type form. We find the asymptotics of the solution at the vertex and we describe its uniqueness class. This paper was supported by UMICH and CONACYT (M ́exico). REFERENCES [1] Komech A., Merzon A. Stationary Diffraction by Wedges. Spring, 2019. [2] A. Merzon, P. Zhevandrov, J.E. De la Paz M ́endez, and M.I. Romero Rodr ́ıguez, Exact solution of a Dirichlet problem in nonconvex angle. Contemporary Mathematics. Fundamental Directions, 2022, Vol. 68, No. 4, 653-670