On Helmholtz’s decomposition theorem and Poisson’s equation with an infinite domain
Author:
Ton Tran Cong
Journal:
Quart. Appl. Math. 51 (1993), 23-35
MSC:
Primary 31A25; Secondary 53A45
DOI:
https://doi.org/10.1090/qam/1205933
MathSciNet review:
MR1205933
Full-text PDF Free Access
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H. Helmholtz, Ueber Integral der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelle 55, 25–55 (1858)
G. G. Stokes, On the dynamical theory of diffraction, Cambridge Philos. Trans. 9, 1–62 (1849)
- Arnold Sommerfeld, Mechanics of Deformable Bodies. Lectures on Theoretical Physics, Vol. II, Academic Press, Inc., New York, N. Y., 1950. Translated from the second German edition by G. Kuerti. MR 0035165
H. Lass, Vector and Tensor Analysis, McGraw-Hill, Kogakusha, Tokyo, 1950
R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1962
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O. Nikodym, Sur un théorème de M. S. Zaremba concernant les fonctions harmoniques, J. de Math. 12, 95–108 (1933)
- Kurt Friedrichs, On Differential Operators in Hilbert Spaces, Amer. J. Math. 61 (1939), no. 2, 523–544. MR 1507392, DOI https://doi.org/10.2307/2371518
- Hermann Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444. MR 3331
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G. J. Hirasaki, Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics, Quart. Appl. Math. 28, 293–296 (1970)
Y. A. S. Aregrebesola and D. M. Burley, The vector and scalar potential method for the numerical solution of two- and three-dimensional Navier-Stokes equations, J. Comp. Phys. 24, 398–415 (1977)
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E. Schmidt, Bemerkung zur Potentialtheorie. Mathematische Abhandlugen Hermann Amandus Schwarz zu seinem funfzigjahrigen Doktorjubilaum, Springer, Berlin, 1914
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- Mitchell H. Taibleson, The preservation of Lipschitz spaces under singular integral operators, Studia Math. 24 (1964), 107–111. MR 162133, DOI https://doi.org/10.4064/sm-24-1-107-111
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- Ton Tran Cong, On the completeness of the Papkovich-Neuber solution, Quart. Appl. Math. 47 (1989), no. 4, 645–659. MR 1031682, DOI https://doi.org/10.1090/qam/1031682
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H. Helmholtz, Ueber Integral der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelle 55, 25–55 (1858)
G. G. Stokes, On the dynamical theory of diffraction, Cambridge Philos. Trans. 9, 1–62 (1849)
A. Sommerfeld, Mechanics of Deformable Bodies, Academic Press, New York, 1950
H. Lass, Vector and Tensor Analysis, McGraw-Hill, Kogakusha, Tokyo, 1950
R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1962
R. M. Bowen and C. C. Wang, Introduction to vectors and tensors, Vectors and Tensor Analysis, Vol. 2, Plenum Press, New York, 1976
H. B. Phillips, Vector Analysis, John Wiley & Sons, New York, 1933
C. E. Weatherburn, Advanced Vector Analysis, G. Bell & Sons, London, 1924
O. Blumenthal, Uber die Zerlegung unendlicher Vectorfelder, Math. Ann. 61, 235–250 (1905)
M. E. Gurtin, On Helmholtz’s Theorem and the completeness of the Papkovich-Neuber stress functions for infinite domains, Arch. Rational Mech. Anal. 9, 225–233 (1962)
O. Nikodym, Sur un théorème de M. S. Zaremba concernant les fonctions harmoniques, J. de Math. 12, 95–108 (1933)
K. Friedrichs, On differential operators in Hilbert spaces, Amer. J. Math. 61, 523–544 (1939)
H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7, 411–444 (1940)
E. B. Bykhovsky and N. V. Smirnov, On orthogonal expression of the space of vector functions which are square-summable over a given domain, Trudy Mat. Inst. Steklova 59, 5–36 (1960)
D(aisuke) Fujiwara and H(iroko) Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. I Math. 24, 685–700 (1977)
M. E. Gurtin, The linear theory of elasticity, Mechanics of Solids, vol. 2, edited by C. Truesdell, Springer-Verlag, Berlin, 1973
R. F. Millar, On the completeness of the Papkovich potentials, Quart. Appl. Math. 41, 385–393 (1984)
G. J. Hirasaki, Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics, Quart. Appl. Math. 28, 293–296 (1970)
Y. A. S. Aregrebesola and D. M. Burley, The vector and scalar potential method for the numerical solution of two- and three-dimensional Navier-Stokes equations, J. Comp. Phys. 24, 398–415 (1977)
S. M. Richardson and A. R. H. Cornish, Solution of three-dimensional incompressible flow problems, J. Fluid Mech. 82, 309–319 (1977)
L. Morino, Helmholtz decomposition revisited: Vorticity generation and trailing edge condition, Comp. Mech. 1, 65–90 (1986)
O. D. Kellogg, Foundations of Potential Theory, Springer, New York, 1929; Dover, New York, 1953
R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, 1962
M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl. 77, 482–492 (1980)
W. O. Amrein, A. M. Berthier, and V. Georcescu, $L^{p}$-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble) 31, 153–168 (1981)
D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121, 463–494 (1985)
L. Hormander, The Analysis of Linear Partial Differential Operators II, Springer-Verlag, Berlin, 1983
R. L. Fosdick, On the vector potential and the representation of a polyharmonic function in n-dimensions, J. Math. Mech. 14, 573–587 (1965)
T(on) Tran-Cong, Notes on some relationship between the Galerkin, Papkovich-Neuber and Naghdi-Hsu solutions in linear elasticity, Mech. Res. Comm. 8, 207–211 (1981)
T(on) Tran-Cong and J. R. Blake, General solutions of the Stokes’s flow equations, J. Math. Anal. Appl. 90, 72–84 (1982)
E. Schmidt, Bemerkung zur Potentialtheorie. Mathematische Abhandlugen Hermann Amandus Schwarz zu seinem funfzigjahrigen Doktorjubilaum, Springer, Berlin, 1914
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer-Verlag, Berlin, 1983
S. G. Mihlin, Singular integral equations, Translation No. 24, Amer. Math. Soc., Providence, RI, 1950
A. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88, 85–139 (1952)
M. H. Taibleson, The preservation of Lipschitz spaces under singular integral operators, Studia Math. 24, 107–111 (1963)
T. Apostol, Mathematical Analysis, 2nd. ed., Addison-Wesley, Reading, MA, 1974
H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, 1963
R. D. Mindlin, Note on the Galerkin and Papkovich stress functions, Bull. Amer. Math. Soc. 42, 373–376 (1936)
T(on) Tran-Cong, On the completeness of the Papkovich-Neuber solution, Quart. Appl. Math. 47, 645–659 (1989)
A. F. Stevenson, Note on the existence and determination of a vector potential, Quart. Appl. Math. 12, 194–198 (1954)
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