×

About mathematical model or some diffuse processes in the Mediterranean and exterior Poincare problem for the Helmholtz equation. (English) Zbl 0758.35026

Let \(S\subset E^ 2\) be a closed Lyapunov’s curve of \({\mathcal C}^{1,h}\) class and denote by \(D^ -\) the external region \((\infty\in D^ -)\). The authors investigate the following problem: finding the solution \(u\) of the Helmholtz equation \(\Delta u+c_ 0u=0\), \(c_ 0\) const., in \(D^ -\), possessing Hölder-continuous first derivatives in \(D^ -\cup S\) and satisfying the boundary condition \(l\cdot\lim_{z\to t_ 0,z\in D^ -}u(z)=f(t_ 0)\), \(t_ 0\in S\), where \(l(t_ 0)=(l_ 1(t_ 0),l_ 2(t_ 0))\) is a unit vector, \(f\) is a Hölder-continuous function on \(S\). Metaharmonic potentials are used for the solution of this problem.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
31B35 Connections of harmonic functions with differential equations in higher dimensions
PDFBibTeX XMLCite
Full Text: EuDML