A geometric evolution problem
Author:
Andréas Bergwall
Journal:
Quart. Appl. Math. 60 (2002), 37-73
MSC:
Primary 76D27; Secondary 35R35, 76A05
DOI:
https://doi.org/10.1090/qam/1878258
MathSciNet review:
MR1878258
Full-text PDF Free Access
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Abstract: A traditional approach to compression moulding of polymers involves the study of a generalized Hele-Shaw flow of a power-law fluid, and leads to the $p$-Poisson equation for the instantaneous pressure in the fluid. By studying the convex dual of an equivalent extremal problem, one may let the power-law index of the fluid tend to zero. The solution of the resulting extremal problem, referred to as the asymptotically dual problem, is known to have the property that the flow is always directed towards the closest point on the boundary.
G. Aronsson, Asymptotic solution of a compression molding problem, LiTH-MAT-R-95-1, Linköping, 1994
- Gunnar Aronsson and Ulf Janfalk, On Hele-Shaw flow of power-law fluids, European J. Appl. Math. 3 (1992), no. 4, 343–366. MR 1196816, DOI https://doi.org/10.1017/S0956792500000905
A. Bergwall, A geometric evolution problem arising in an asymptotic approach to compression moulding, LiU-TEK-LIC-1998:28, Linköping, 1998
- Luis A. Caffarelli and Avner Friedman, The free boundary for elastic-plastic torsion problems, Trans. Amer. Math. Soc. 252 (1979), 65–97. MR 534111, DOI https://doi.org/10.1090/S0002-9947-1979-0534111-0
L. C. Evans, Personal communication
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- D. H. Fremlin, Skeletons and central sets, Proc. London Math. Soc. (3) 74 (1997), no. 3, 701–720. MR 1434446, DOI https://doi.org/10.1112/S0024611597000233
- 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
C. Hyltén-Cavallius and L. Sandgren, Matematisk analys I, Lund, 1962
- Ulf Janfalk, Behaviour in the limit, as $p\to \infty $, of minimizers of functionals involving $p$-Dirichlet integrals, SIAM J. Math. Anal. 27 (1996), no. 2, 341–360. MR 1377478, DOI https://doi.org/10.1137/S0036141093252619
- Ulf Janfalk, On a minimization problem for vector fields in $L^1$, Bull. London Math. Soc. 28 (1996), no. 2, 165–176. MR 1367165, DOI https://doi.org/10.1112/blms/28.2.165
U. Janfalk, Personal communication
C. Lee, F. Folgar, and C. L. Tucker, Simulation of compression molding for fiber-reinforced thermosetting polymers, Trans. of the ASME 106, 114–125 (1984)
- David Milman and Zeev Waksman, On topological properties of the central set of a bounded domain in ${\bf R}^{m}$, J. Geom. 15 (1981), no. 1, 1–7. MR 605061, DOI https://doi.org/10.1007/BF01919351
- B. N. Pshenichnyi, Necessary conditions for an extremum, Pure and Applied Mathematics, vol. 4, Marcel Dekker, Inc., New York, 1971. Translated from the Russian by Karol Makowski; Translation edited by Lucien W. Neustadt. MR 0276845
G. Aronsson, Asymptotic solution of a compression molding problem, LiTH-MAT-R-95-1, Linköping, 1994
G. Aronsson and U. Janfalk, On Hele-Shaw flow of power-law fluids, European J. Appl. Math. 3, 343–336 (1992)
A. Bergwall, A geometric evolution problem arising in an asymptotic approach to compression moulding, LiU-TEK-LIC-1998:28, Linköping, 1998
L. A. Caffarelli and A. Friedman, The free boundary for elastic-plastic torsion problems, Trans. Amer. Math. Soc. 252, 65–97 (1979)
L. C. Evans, Personal communication
W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc. 54, 141–175 (1987)
D. H. Fremlin, Skeletons and central sets, Proc. London Math. Soc. 74, 701–720 (1997)
M. E. Gurtin, An introduction to continuum mechanics, Academic Press, San Diego, 1981
C. Hyltén-Cavallius and L. Sandgren, Matematisk analys I, Lund, 1962
U. Janfalk, Behaviour in the limit, as $p \to \infty$, of minimizers of functionals involving $p$-Dirichlet integrals, SIAM J. Math. Anal. 27, 341–360 (1996)
U. Janfalk, On a minimization problem for vector fields in $L^{1}$, Bull. London Math. Soc. 28, 165–176 (1996)
U. Janfalk, Personal communication
C. Lee, F. Folgar, and C. L. Tucker, Simulation of compression molding for fiber-reinforced thermosetting polymers, Trans. of the ASME 106, 114–125 (1984)
D. Milman and Z. Waksman, On topological properties of the central set of a bounded domain in ${\mathbb {R}^n}$, J. of Geometry 15, 1–7 (1981)
B. N. Pshenichnyi, Necessary conditions for an extremum, Marcel Dekker, New York, 1971
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© Copyright 2002
American Mathematical Society