Abstract
Let Ω be a domain in R N, N ≥ 2, whose boundary ∂Ω is of class C 1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Pierre Simon, Marquis de Laplace, 1749–1827; author of Traité de Mécanique céleste, (17991825). Also known for the frequent use of the phrase “il est aisé de voir,” which has unfortunately become all too popular in modern mathematical writings.
The same equation had been introduced, in the context of potential fluids, by Joseph Louis, Compte de Lagrange, 1736–1813, author of Traité de Mécanique Analytique, (1788).
See Section 2 of the Preliminaries.
Gustav Peter Lejeune Dirichlet, 1805–1859.
Carl Gottfried Neumann, 1832–1925.
See Section 5 of Chapter I.
Jacques Hadamard, 1865–1963. A first discussion on ill-posed problems is in J. Hadamard, Sur les problèmes aux derivées partielles et leur signification physique, Bull. Univ. Princeton, #13 (1902), pp. 49–52. The example is taken from J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1952.
The orientation of the vectors n(y) is arbitrary but fixed so that the function y-+ n(y) is continuous foryE E.
George Green, 1793–1841, George Gabriel Stokes, 1819–1903.
Introduced by Green in G.Green, An Essay on the Applications of Mathematical Analysis to the Theories of Electricity and Magnetism Nottingham, 1828.
Simeon Denis Poisson, 1781–1840.
See 4.1 of the Complements.
See 4.1 of the Complements.
A convex function defined in some interval [a, b], is continuous in (a, b). See Lemma 6.2c of the Complements of Chapter IV.
Johan Ludwig William Valdemar Jensen, 1859–1925. See 4.13 of the Complements.
Alex Harnack, 1851–1888; A. Harnack, Grundlagen der Theorie des logarithmischen Potentials, Leipzig, 1887.
Joseph Liouville, 1809–1882.
See Section 9.1 of Chapter I.
O. Perron, Eine neun Behandlung der Randewertaufgabe für Au = 0. Math. Z. #18 (1923) pp. 42–54.
See 5.5 of the Complements.
See 7.1 of the Complements.
Henri Léon Lebesgue, 1875–1941; H.L. Lebesgue, Sur des cas d’impossibilité du problème de Dirichlet, Comptes Rendus Soc. Math. de France, (1913), pp. 17. H.L. Lebesgue, Conditions de régularité, conditions d’irrégularité, conditions d’im possibilité dans le le problème de Dirichlet, Comptes Rendus Acad. Sci. Paris, Vol. 178 (1924,I), pp. 349–354. A spectrum of contributions of H. Lebesgue to the classical solvability of the Dirichlet problem is in OEuvres Scientifiques de H. L. Lebesgue, Inst. Math. Univ. de Gèneve, Gèneve 1975, Vol. I, Chap. V. A clear discussion on the role played by the local behavior of d S2 in the classical solvability of the Dirichlet problem is given by N. Wiener, Une condition nécessaire et suffisante de possibilité pour le problème de Dirichlet, Comptes Rendus Acad. Sci. Paris, Vol. 178 (1924), pp. 1050–1054 (présentée par M.H. Lebesgue). This note is a summary of a more complete investigation, i.e., N. Wiener, The Dirichlet Problem, J. Math. Phys. #3, (1924), pp. 127–147. Also in, Collected papers of Norbert Wiener, with contributions of Y.W. Lee, N. Levinson and W.T. Martin, SIAM, Philadelphia and the M.I.T. Press, Cambridge MA, 1965, pp. 361–371.
See 7.2 of the Complements.
See also 5.7 of the Complements.
See 8.1 of the Complements.
See 9.1 and 9.2 of the Complements.
Guido Fubini, 1879–1943. G. Fubini, Sugli integrali multipli, Rend. Accad. Lincei Roma, Vol. 16 (1907), pp. 608–614. See also Opere Scelte di Guido Fubini, a cura dell’U.M.I., Ed. Cremonese, Roma 1957, Vol.3 page 53.
See 10.1 of the Complements.
See 11.1 of the Complements.
See 4.12 of the Complements. For a proof see Lemma 9.2 of Chapter III.
See 12.1 of the Complements.
See Kellogg, [19], p. 22 and p. 193.
The sufficient part of the theorem is due to Newton. The necessary part in the space dimension N = 2 is in P. Dive, Attraction d’ellipsoides homogènes et réciproque d’un théorème de Newton, Bull. Soc. Math. France, #59 (1931), pp. 128–140. The necessary condition for N> 3 is in E. DiBenedetto and A. Friedman, Bubble growth in porous media, Indiana Univ. Math. J. Vol. 35 #3 (1986), pp. 573–606. The assumption of uniform distribution cannot be removed as shown in H. Shangolian, On the Newtonian potential of a heterogeneous ellipsoid, SIAM J. Math. Anal., Vol. 22 #5 (1991), pp. 1246–1255.
See N.S. Krylov, Controlled Diffusion Processes, Springer-Verlag, Series Applications of Mathematics #14, New York, 1980.
For information on harmonic polynomials and spherical harmonics see E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Pub. Co., New York, 1965.
This follows from the characteristic condition of holomorphy, i.e., the Cauchy—Riemann equations. See Cartan, [2] pp. 124–125.
J. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. #30 (1906), pp. 175–193. The inequality still holds if dy/ISZI is replaced by any Radon measure dµ such that fa dµ = 1. In the case of a discrete measure, the inequality had been established by Hölder in O. Hölder, Über einen Mittelwertsatz, Göttingen Nachr. (1889), pp. 38–47.
This fact is intuitive and it can be proved using Lemma 6.1c of the Complements of Chapter IV.
F. Riesz, Sur les fonctions subharmoniques at leur rapport à la théorie du potentiel, Part I and II, Acta Math. Vol. 48, (1926), pp. 329–343., and ibid., Vol. 54, (1930), pp. 321–360.
See F. Riesz cited above and also T. Rado, Subharmonic Functions, Chelsea Pub. Co., New York.
See Hörmander, An Introduction to Complex Analysis in Several Variables D.Van Nostrand Co. Inc., Princeton, New Jersey, 1966, page 16.
Brook Taylor, 1685–1731.
J.B. Serrin, On the Phragmen-Lindelöf principle for elliptic differential equations. J. Rat. Mech. Anal. #3 (1954) pp. 395–413.
See footnote #6.
See Problem 4.4.
D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. d’Analyse Math. #4 (1955), pp. 309–340.
Norbert Wiener, 1894–19M; N.Wiener, Certain notions in potential theory, J. Math. Phys. Mass. Inst. Tech. Vol. III, pp. 24–51, (1924).
William Thomson Kelvin, 1824–1907; W. Thomson Kelvin, Extraits de deux lettres adressées à M. Liouville, J. de Mathématiques Pures et Appliquées Vol. 12 (1847), pp. 256.
This is a particular case of general estimates proved by Julius Pawel Schauder, 1899–1943, in J.P. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. #38, (1934), pp. 257–282; J.P. Schauder, Numerische Abschatzungen in elliptischen linearen Differentialgleichungen, Studia Math. #5, (1935), pp. 34–42. A version of these estimates is in [12] and [21]. Their parabolic counterpart is in [8] and [22].
Godfrey Harold Hardy, 1877–1947.
COMPLEMENT: IO. POTENTIAL ESTIMATES IN LP(Q) & HARDY’S INEQUALITY 113
See Section 4.3 of the Complements of the Preliminaries.
A.P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. #88 (1952), pp. 85–139. See also E. Stein [34]; Anthony Zygmund, 1900–1992.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this chapter
Cite this chapter
DiBenedetto, E. (1995). The Laplace Equation. In: Partial Differential Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2840-5_3
Download citation
DOI: https://doi.org/10.1007/978-1-4899-2840-5_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-2842-9
Online ISBN: 978-1-4899-2840-5
eBook Packages: Springer Book Archive