Metastable equilibrium of fluids with surface tension
Author:
Epifanio G. Virga
Journal:
Quart. Appl. Math. 46 (1988), 217-228
MSC:
Primary 58E30; Secondary 49F10, 49F22, 53A10, 58E12, 76E99, 76T05
DOI:
https://doi.org/10.1090/qam/950598
MathSciNet review:
950598
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper deals with the effects of the surface tension on the equilibrium of fluids. It shows that, when a fluid with an incompressible fluid inclusion is put in a fluid environment kept at constant pressure, equilibrium configurations may arise whose character of stability is affected by the sign of the perturbations of the environmental pressure. Such configurations we refer to as metastable.
- A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad. Univ. 13 (1958), no. 19, 5–8 (Russian, with English summary). MR 0102114
- A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315. MR 143162, DOI https://doi.org/10.1007/BF02413056
M. H. I. Baird, The stability of inverse bubbles, Trans. Faraday Soc., 56, 213–219 (1960)
- Ennio De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213 (Italian). MR 62214, DOI https://doi.org/10.1007/BF02412838
- Ennio De Giorgi, Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat. 4 (1955), 95–113 (Italian). MR 74499
E. De Giorgi, Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Mem. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. 5, 33–44 (1958)
J. T. Davies and E. K. Rideal, Interfacial phenomena, Academic Press, New York, 1961
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI https://doi.org/10.2307/1970227
- Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 114892, DOI https://doi.org/10.1007/BF01236935
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682
- Morton E. Gurtin and A. Ian Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57 (1975), 291–323. MR 371223, DOI https://doi.org/10.1007/BF00261375
- Morton E. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, vol. 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 636255
- Heinz Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4 (1951), 232–249 (German). MR 40042, DOI https://doi.org/10.1002/mana.3210040122
- G. H. Hardy, J. E. Littlewood, and G. PĂłlya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
H. Lamb, Hydrodynamics, Dover, New York, 1945
- Heinrich Liebmann, Ueber die Verbiegung der geschlossenen Flächen positiver Krümmung, Math. Ann. 53 (1900), no. 1-2, 81–112 (German). MR 1511083, DOI https://doi.org/10.1007/BF01456030
- Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI https://doi.org/10.1090/S0002-9904-1978-14553-4
- Robert C. Reilly, Applications of the integral of an invariant of the Hessian, Bull. Amer. Math. Soc. 82 (1976), no. 4, 579–580. MR 415540, DOI https://doi.org/10.1090/S0002-9904-1976-14110-9
- Robert C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), no. 3, 459–472. MR 474149, DOI https://doi.org/10.1512/iumj.1977.26.26036
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI https://doi.org/10.1007/BF02418013
A. D. Alexandrov, Uniqueness theorems for surfaces in the large, V, Vestnik Leningrad Univ. 13, 5–8 (1958), English translation, Amer. Math. Soc. Transl. 21, 412–413 (1962)
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. 58, 303–315 (1962)
M. H. I. Baird, The stability of inverse bubbles, Trans. Faraday Soc., 56, 213–219 (1960)
E. De Giorgi, Su una teoria generale della misura $\left ( {r - 1} \right )$-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36, 191–213 (1954)
E. De Giorgi, Nuovi teoremi relativi alle misure $\left ( {r - 1} \right )$-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4, 95–113 (1955)
E. De Giorgi, Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Mem. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. 5, 33–44 (1958)
J. T. Davies and E. K. Rideal, Interfacial phenomena, Academic Press, New York, 1961
H. Federer and W. Fleming, Normal and integral currents, Ann. Math. 72, 458–520 (1960)
W. Fleming and R. Rishel, An integral formula for total gradient variation, Arch. Math. 11, 218–222 (1960)
E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Boston, 1984
M. E. Gurtin and A. I. Murdoch, A continuum theory of elastic material surfaces, Arch. Rat. Mech. Anal. 57, 291–323 (1975)
M. E. Gurtin, An introduction to continuum mechanics, Academic Press, New York, 1981
H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4, 232–249 (1950–1951)
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952
H. Lamb, Hydrodynamics, Dover, New York, 1945
G. H. Liebmann, Ueber die Verbiegung der geschlossenen Flächen positiver Krümmung, Math. Ann. 53, 81–112 (1900)
R. Ossermann, The isoperimetric inequality, Bull. Amer. Math. Soc. 84, 1182–1237 (1978)
R. C. Reilly, Applications of the integral of an invariant of the Hessian, Bull. Amer. Math. Soc. 82, 579–580 (1976)
R. C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26, 459–472 (1977)
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110, 353–372 (1976)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
58E30,
49F10,
49F22,
53A10,
58E12,
76E99,
76T05
Retrieve articles in all journals
with MSC:
58E30,
49F10,
49F22,
53A10,
58E12,
76E99,
76T05
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society
