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  1. Zeitschrift für angewandte Mathematik und Physik
  2. Zeitschrift für angewandte Mathematik und Physik : Volume 62
  3. Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 5, October 2011
  4. Vekua theory for the Helmholtz operator
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Zeitschrift für angewandte Mathematik und Physik : Volume 67
Zeitschrift für angewandte Mathematik und Physik : Volume 66
Zeitschrift für angewandte Mathematik und Physik : Volume 65
Zeitschrift für angewandte Mathematik und Physik : Volume 64
Zeitschrift für angewandte Mathematik und Physik : Volume 63
Zeitschrift für angewandte Mathematik und Physik : Volume 62
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 6, December 2011
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 5, October 2011
Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems
Vekua theory for the Helmholtz operator
Plane wave approximation of homogeneous Helmholtz solutions
Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms
Mathematical study of the small oscillations of a pendulum containing an almost homogeneous, incompressible, inviscid liquid and a barotropic gas (Oscillations of a pendulum with almost homogeneous liquid and gas)
Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations
Vortices in the three-dimensional thin-film Ginzburg–Landau model of superconductivity
Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity
Propagation of a plane wave to a materially uniform but inhomogeneous body
Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures
Radiation effects on the MHD flow near the stagnation point of a stretching sheet: revisited
Author’s response to Dr. Wadee and Yiatros’ discussion of my paper: “On interactive buckling in a sandwich structure” (ZAMP, vol. 61, pp. 565–577, 2010)
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 4, August 2011
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 3, June 2011
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 2, April 2011
Zeitschrift für angewandte Mathematik und Physik : Volume 62, Issue 1, February 2011
Zeitschrift für angewandte Mathematik und Physik : Volume 61
Zeitschrift für angewandte Mathematik und Physik : Volume 60
Zeitschrift für angewandte Mathematik und Physik : Volume 59
Zeitschrift für angewandte Mathematik und Physik : Volume 58
Zeitschrift für angewandte Mathematik und Physik : Volume 57
Zeitschrift für angewandte Mathematik und Physik : Volume 56
Zeitschrift für angewandte Mathematik und Physik : Volume 55
Zeitschrift für angewandte Mathematik und Physik : Volume 54
Zeitschrift für angewandte Mathematik und Physik : Volume 53
Zeitschrift für angewandte Mathematik und Physik : Volume 52
Zeitschrift für angewandte Mathematik und Physik : Volume 51
Zeitschrift für angewandte Mathematik und Physik : Volume 50
Zeitschrift für angewandte Mathematik und Physik : Volume 49
Zeitschrift für angewandte Mathematik und Physik : Volume 48

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Vekua theory for the Helmholtz operator

Content Provider SpringerLink
Author Moiola, A. Hiptmair, R. Perugia, I.
Copyright Year 2011
Abstract Vekua operators map harmonic functions defined on domain in $${\mathbb R^{2}}$$ to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.
Ending Page 807
Page Count 29
Starting Page 779
File Format PDF
ISSN 00442275
e-ISSN 14209039
Journal Zeitschrift für angewandte Mathematik und Physik
Issue Number 5
Volume Number 62
Language English
Publisher SP Birkhäuser Verlag Basel
Publisher Date 2011-07-16
Publisher Place Basel
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Mathematical Methods in Physics Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation Helmholtz equation Vekua transform Integral representations of solutions Theoretical and Applied Mechanics Generalized harmonic polynomials Sobolev continuity
Content Type Text
Resource Type Article
Subject Applied Mathematics Physics and Astronomy
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