Letters in Mathematical Physics

, Volume 108, Issue 4, pp 1109–1135 | Cite as

The Riemann–Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann, Robin and Dirichlet boundary conditions

  • Alexander Its
  • Elizabeth Its


We revisit the Helmholtz equation in a quarter-plane in the framework of the Riemann–Hilbert approach to linear boundary value problems suggested in late 1990s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas’ scheme.


Helmholtz equation Riemann–Hilbert problem Lax pair 

Mathematics Subject Classification

35J25 35Q15 31B20 



This work was partially supported by the National Science Foundation (NSF) under Grant No. DMS-0203104, by the Russian Science Foundation Grant No.17-11-01126, and by a Grant of the London Mathematical Society. The authors are grateful to M. Lyalinov for very useful discussions.


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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University – Purdue University IndianapolisIndianapolisUSA
  2. 2.St. Petersburg State UniversitySt. PetersburgRussia

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