Sunto
Nel presente lavoro si studia il problema al contorno per l'equazione di Helmholtz all'esterno di più tagli nel piano. Le due condizioni al contorno sono assegnate sui tagli. Una di queste prescrive il salto della funzione incognita, l'altra contiene il salto della derivata normale di una funzione incognita ed un valore limite di questa funzione sui tagli. La soluzione univoca di questo problema è ricondotta all'equazione di Fredholm di seconda specie ed indice zero, univocamente risolubiles, per mezzo dei potenziali di singolo strato ed angolare. Si studiano, inoltre, le singolarità agli estremi dei tagli.
Abstract
The boundary value problem for the Helmholtz equation outside several cuts in a plane is studied. The 2 boundary conditions are given on the cuts. One of them specifies the jump of the unkown function. Another one contain the jump of the normal derivative of an unknown function and a limit value of this function on the cuts. The unique solution of this problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.
This is a preview of subscription content, log in via an institution to check access.
References
P. Wolfe,An existence theorem for the reduced wave equation, Proc. Amer. Math. Soc.,21 (1969), pp. 663–666.
M. Durand,Layer potentials and boundary value problems for the Helmoholtz equation in the compliment of a thin obstacle, Math. Meth. Appl. Sci.11 (1969), pp. 185–213.
I. K. Lifanov,Singular integral equations and discrete vortices, VSP, Zeist, 1996.
P. A. Krutitskii,Dirichlet problem for the Helmholtz equation outside cuts in a plane, Comp. Math. Math. Phys.,34 (1994), pp. 1073–1090.
P. A. Krutitskii,Neumann problem for the Helmholtz equation outside cuts in a plane, Comp. Math. Math. Phys.,34 (1994), pp. 1421–1431.
P. A. Krutitskii,The Dirichlet problem for the 2-D Helmholtz equation in a multiply connected domain with cuts. ZAMM,77, No. 12 (1997) 883–890.
P. A. Krutitskii,The Neumann problem for the 2-D Helmholtz equation in a multiply connected domain with cuts, Zeitschrift für Analysis und ihre Anwendungen,16 (1997), pp. 349–361.
P. A. Krutitskii,Wave propagation in a 2-D external domains with cuts, Applicable Analysis,62 (1996), pp. 297–309.
P. A. Krutitskii,The Neumann problem on wave propagation in a 2-D external domain with cuts. J. Math. Kyoto Univ.,38, No. 3 (1998), pp. 439–452.
S. A. Gabov,An angular potential and its applications, Math. U.S.S.R. Sbornik,32 (1977), pp. 423–436.
R. H. Torres—G. V. Welland,The Helmholtz equation and transmission problems with Lipschitz interfaces, Indiana Univ. Math. J.,42 (1993), No. 4, pp. 1457–1485.
T. V. Petersdoff,Boundary integral equations for mixed Dirichlet, Neumann and transmission problems, Math. Meth. Appl. Sci.,11 (1989), pp. 185–213.
P. Wilde,Transmission problems for the vector Helmholtz equation, Proc. Roy. Soc. Edinburgh,105 A (1987), pp. 61–76.
D. Colton—R. Kress,Integral equation methods in scattering theory, Wiley, N.Y., 1983.
C. J. S. Alves—T. Ha-Duong,On inverse scattering by screens, Inverse Problems,13 (1995), pp. 1161–1176.
C. J. S. Alves—T. Ha-Duong,Inverse scattering for elastic plane cracks, Inverse Problems,15 (1999), pp. 91–97.
A. Friedman—M. Vogelius,Determining cracks from boundary measurements, Indiana Univ. Math. J.,38 (1989), pp. 497–525.
V. S. Vladimirov,Equations of Mathematical Physics, Marcel Dekkers, N.Y., 1971.
A. F. Nikiforov—V. B. Uvarov,Special functions of mathematical physics, Birkhäuser, Basel, 1988.
P. A. Krutitskii,The jump problem for the Helmholtz equation and singularities at the edges, Appl. Math. Letters,13 (2000), pp. 71–76.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krutitskii, P.A. The modified jump problem for the Helmholtz equation. Ann. Univ. Ferrara 47, 285–296 (2001). https://doi.org/10.1007/BF02838188
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02838188