Boundary value problems for the Helmholtz equation in an octant
Let \(\Omega =\{ x\in \mathbb R^3\); \(x_j >0\), \(j=1,2,3 \}\), \(Q_j=\{ x\in \mathbb R^3\); \(x_j=0\), \(x_n>0\) for \(n\neq j\}\), \(k\) be a complex number with positive imaginary part. It is studied the following problem: Find a solution \(u\in H^s(\Omega)\), \(s\geq 1\), of the Helmholtz equation \((\Delta +k^2)u=0\) in \(\Omega \) which satisfies the boundary conditions \(B_j u=g_j\) on \(Q_j\). Here \(B_j u=\beta_{j0}u+\beta_{j1}\partial_1 u+\beta_{j2}\partial_2 u+\beta_{j3}\partial_3 u\) with given constant coefficients \(\beta_{jn}\). First it is studied a problem when \(B_ju=g_j\) is the Dirichlet condition or the Neumann condition on each \(Q_j\) (i.e. the Dirichlet problem, the Neumann problem and mixed Dirichlet-Neumann problems). The solution is given analytically. Using these results further boundary value problems are studied. New type of potentials are used for this goal.
scientific article; zbMATH DE number 5617575