New results on some classical parabolic free-boundary problems
Authors:
A. Fasano and M. Primicerio
Journal:
Quart. Appl. Math. 38 (1981), 439-460
MSC:
Primary 35K85; Secondary 80A20
DOI:
https://doi.org/10.1090/qam/614552
MathSciNet review:
614552
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Additional Information
- A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), no. 3, 694–723. MR 487016, DOI https://doi.org/10.1016/0022-247X%2877%2990256-6
- Alfred Schatz, Free boundary problems of Stephan type with prescribed flux, J. Math. Anal. Appl. 28 (1969), 569–580. MR 267285, DOI https://doi.org/10.1016/0022-247X%2869%2990009-2
- B. Sherman, A general one-phase Stefan problem, Quart. Appl. Math. 28 (1970), 377–382. MR 282082, DOI https://doi.org/10.1090/S0033-569X-1970-0282082-8
- B. Sherman, General one-phase Stefan problems and free boundary problems for the heat equation with Cauchy data prescribed on the free boundary, SIAM J. Appl. Math. 20 (1971), 555–570. MR 287193, DOI https://doi.org/10.1137/0120058
- John Bather, Optimal stopping problems for Brownian motion, Advances in Appl. Probability 2 (1970), 259–286. MR 283920, DOI https://doi.org/10.2307/1426320
- A. Bensoussan and J.-L. Lions, Problèmes de temps d’arrêt optimal et inéquations variationnelles paraboliques, Applicable Anal. 3 (1973), 267–294 (French). MR 449843, DOI https://doi.org/10.1080/00036817308839070
- Daniel B. Kotlow, A free boundary problem connected with the optimal stopping problem for diffusion processes, Trans. Amer. Math. Soc. 184 (1973), 457–478 (1974). MR 365729, DOI https://doi.org/10.1090/S0002-9947-1973-0365729-0
- Avner Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Functional Analysis 18 (1975), 151–176. MR 0477461, DOI https://doi.org/10.1016/0022-1236%2875%2990022-1
- Avner Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal. 61 (1976), no. 2, 97–125. MR 407452, DOI https://doi.org/10.1007/BF00249700
- Avner Friedman and Robert Jensen, Convexity of the free boundary in the Stefan problem and in the dam problem, Arch. Rational Mech. Anal. 67 (1978), no. 1, 1–24. MR 473315, DOI https://doi.org/10.1007/BF00280824
- Robert Jensen, Smoothness of the free boundary in the Stephan problem with supercooled water, Illinois J. Math. 22 (1978), no. 4, 623–629. MR 503966
- Pierre van Moerbeke, An optimal stopping problem with linear reward, Acta Math. 132 (1974), 111–151. MR 376225, DOI https://doi.org/10.1007/BF02392110
P. van Moerbecke, About optimal stopping problems, Arch. Rat. Mech. Anal. 60, 101–148 (1976)
- E. Magenes, Topics in parabolic equations: some typical free boundary problems, Boundary value problems for linear evolution: partial differential equations (Proc. NATO Advanced Study Inst., Liège, 1976) Reidel, Dordrecht, 1977, pp. 239–312. NATO Advanced Study Inst. Ser., Ser. C: Math. and Phys. Sci., Vol. 29. MR 0470497
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford (1975)
- David George Wilson, Alan D. Solomon, and Paul T. Boggs (eds.), Moving boundary problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0466887
- John Crank and R. S. Gupta, A method for solving moving boundary problems in heat-flow using cubic splines or polynomials, J. Inst. Math. Appl. 10 (1972), 296–304. MR 347105
E. Hansen and P. Hougaard, On a moving boundary problem from biomechanics, J. Inst. Maths. Applics. 13, 385–398 (1974)
- C. Baiocchi and G. A. Pozzi, An evolution variational inequality related to a diffusion-absorption problem, Appl. Math. Optim. 2 (1975/76), no. 4, 304–314. MR 430538, DOI https://doi.org/10.1007/BF01448174
- C. Baiocchi and G. A. Pozzi, Error estimates and free-boundary convergence for a finite difference discretization of a parabolic variational inequality, RAIRO Anal. Numér. 11 (1977), no. 4, 315–340, iii (English, with French summary). MR 464607, DOI https://doi.org/10.1051/m2an/1977110403151
- J. C. W. Rogers, A free boundary problem as diffusion with non-linear absorption, J. Inst. Math. Appl. 20 (1977), no. 2, 261–268. MR 455492
- Alan E. Berger, Melvyn Ciment, and Joel C. W. Rogers, Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Numer. Anal. 12 (1975), no. 4, 646–672. MR 383779, DOI https://doi.org/10.1137/0712049
- Alan E. Berger, The truncation method for the solution of a class of variational inequalities, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 10 (1976), no. R-1, 29–42 (English, with Loose French summary). MR 0455489
H. Block, Sur les équations aux dérivées partielles de type parabolique, Ark. Matem. Astr. Fys. 6, n. 31 (1910)
- David G. Schaeffer, A new proof of the infinite differentiability of the free boundary in the Stefan problem, J. Differential Equations 20 (1976), no. 1, 266–269. MR 390499, DOI https://doi.org/10.1016/0022-0396%2876%2990106-6
- Antonio Fasano and Mario Primicerio, La diffusione del calore in uno strato di spessore variabile in presenza di scambi termici non lineari con l’ambiente. I. Deduzione di limitazioni a priori sulla temperatura e le sue derivate, Rend. Sem. Mat. Univ. Padova 50 (1973), 269–330 (1974) (Italian, with English summary). MR 366241
- Antonio Fasano and Mario Primicerio, Su un problema unidimensionale di diffusione in un mezzo a contorno mobile con condizioni ai limiti non lineari, Ann. Mat. Pura Appl. (4) 93 (1972), 333–357. MR 410084, DOI https://doi.org/10.1007/BF02412027
Carslaw and Jaeger, Conduction of heat in solids, Clarendon Press, Oxford (1959)
- S. N. Kruzhkov, On some problems with unknown boundaries for the heat conduction equation, J. Appl. Math. Mech. 31 (1967), 1014–1024 (1968). MR 0231063, DOI https://doi.org/10.1016/0021-8928%2867%2990205-5
A. Fasano and M. Primicerio, General free boundary problems for the heat equation I, J. Math. Anal. Appl. 57, 694–723 (1977); II, ibid. 58, 202–231 (1977); III, ibid. 59, 1–14 (1977)
A. Schatz, Free boundary problems of Stefan type with prescribed flux, J. Math. Anal. Appl. 28, 569–580 (1969)
B. Sherman, A general one-phase Stefan problem, Quart. Appl. Math. 28, 377–382 (1970)
B. Sherman, General one-phase Stefan problems and free boundary problems for the heat equation with Cauchy data prescribed on the free boundary, SIAM J. Appl. Math. 20, 555–570 (1971)
J. Bather, Optimal stopping problems for Brownian motion, Adv. Appl. Prob. 2, 259–286 (1970)
A. Bensoussan and J. L. Lions, Problèmes de temps d’arrêt optimal et inéquations variationelles paraboliques, Applic. Anal. 3, 267–295 (1973)
D. B. Kotlow, A free boundary problem connected with the optimal stopping problem for diffusion processes, Trans. Amer. Math. Soc. 194, 457–478 (1973)
A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Funct. Anal. 18, 151–176 (1975)
A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rat. Mech. Anal. 61, 97–125 (1976)
A. Friedman and R. Jensen, Convexity of the free boundary in the Stefan problem and in the dam problem, Arch. Rat. Mech. Anal. 67, 1–24 (1977)
R. Jensen, Smoothness of the free boundary in the Stefan problem with supercooled water, to appear
P. van Moerbecke, An optimal stopping problem for linear reward, Acta Math. 132, 1–41 (1974)
P. van Moerbecke, About optimal stopping problems, Arch. Rat. Mech. Anal. 60, 101–148 (1976)
E. Magenes, Topics in parabolic equations; some typical free boundary problems, in Boundary value problems for linear evolution partial equations (ed. Garnir), 1977, 239–312
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford (1975)
D. G. Wilson, A. D. Solomon and P. T. Boggs, Moving boundary problems, Academic Press (1978)
J. Crank and R. S. Gupta, A moving boundary problem arising from the diffusion of oxygen in absorbing tissues, J. Inst. Maths. Applics. 10, 19–33 (1972)
E. Hansen and P. Hougaard, On a moving boundary problem from biomechanics, J. Inst. Maths. Applics. 13, 385–398 (1974)
C. Baiocchi and G. A. Pozzi, An evolution variational inequality related to a diffusion-absorption problem, Appl. Math. Optim. 2, 304–314 (1976)
C. Baiocchi and G. A. Pozzi, Error estimates and free boundary convergence for a finite difference discretization of a parabolic variational inequality, to appear RAIRO.
J. C. W. Rogers, A free boundary problem as diffusion with nonlinear absorption, J. Inst. Maths. Applics. 20, 261–268 (1977)
A. E. Berger, M. Ciment and J. C. W. Rogers, Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Num. Anal. 12, 646–672 (1975)
A. E. Berger, The truncation method for the solution of a class of variational inequalities, RAIRO 10, 29–42 (1976)
H. Block, Sur les équations aux dérivées partielles de type parabolique, Ark. Matem. Astr. Fys. 6, n. 31 (1910)
D. G. Schaeffer, A new proof of the infinite differentiability of the free boundary in the Stefan problem, J. Diff. Eq. 20, 266–269 (1976)
A. Fasano and M. Primicerio, La diffusione del colore in uno strato di spessore variabile in presenza di scambi termici non lineari con l’ambiente, Rend. Sem. Mat. Univ. Padova 50, 269–330 (1973)
A. Fasano and M. Primicerio, Su un problema unidimensionale di diffusione in un mezzo a contorno mobile con condizioni ai limiti non lineari, Ann. Mat. Pura Appl. (IV) 93, 333–357 (1972)
Carslaw and Jaeger, Conduction of heat in solids, Clarendon Press, Oxford (1959)
S. N. Kruzkov, On some problems with unknown boundaries for the heat conduction equation, Prikl. Mat. Meh. 31, 1009–1020 (1967)
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Article copyright:
© Copyright 1981
American Mathematical Society