The two-sided Stefan problem with a spatially dependent latent heat

McConnell, Terry R.

doi:10.1090/S0002-9947-1991-1008699-5

The two-sided Stefan problem with a spatially dependent latent heat
HTML articles powered by AMS MathViewer

by Terry R. McConnell PDF

Trans. Amer. Math. Soc. 326 (1991), 669-699 Request permission

Abstract:

We prove existence and uniqueness of solutions to a problem which generalizes the two-sided Stefan problem. The initial temperature distribution and variable latent heat may be given by positive measures rather than point functions, and the free boundaries which result are essentially arbitrary increasing functions which need not exhibit any degree of smoothness in general. Nevertheless, the solutions are "classical" in the sense that all derivatives and boundary values have the classical interpretation. We also study connections with the Skorohod embedding problem of probability theory and with a general class of optimal stopping problems.

References

Jacques Azéma and Marc Yor, Une solution simple au problème de Skorokhod, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 90–115 (French). MR 544782
Richard F. Bass, Skorokhod imbedding via stochastic integrals, Seminar on probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 221–224. MR 770414, DOI 10.1007/BFb0068318
J. R. Baxter and R. V. Chacon, Potentials of stopped distributions, Illinois J. Math. 18 (1974), 649–656. MR 358960
Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267

The one-dimensional heat equation

The filling scheme and barrier stopping times

R. V. Chacon and J. B. Walsh, One-dimensional potential embedding, Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976, pp. 19–23. MR 0445598
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Lester E. Dubins, On a theorem of Skorohod, Ann. Math. Statist. 39 (1968), 2094–2097. MR 234520, DOI 10.1214/aoms/1177698036
Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
Neil Falkner, On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin-New York, 1980, pp. 357–391. MR 580142
David Heath, Skorokhod stopping via potential theory, Séminaire de Probabilités, VIII (Univ. Strasbourg, année universitaire 1972-1973), Lecture Notes in Math., Vol. 381, Springer, Berlin, 1974, pp. 150–154. MR 0368185
Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
R. M. Loynes, Stopping times on Brownian motion: Some properties of Root’s construction, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 211–218. MR 292170, DOI 10.1007/BF00534597
Isaac Meilijson, On the Azéma-Yor stopping time, Seminar on probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 225–226. MR 770415, DOI 10.1007/BFb0068319
P. A. Meyer, Le schéma de remplissage en temps continu, d’après H. Rost, Séminaire de Probabilités, VI (Univ. Strasbourg, année universitaire 1970–1971), Lecture Notes in Math., Vol. 258, Springer, Berlin, 1972, pp. 130–150. MR 0402946
Edwin Perkins, The Cereteli-Davis solution to the $H^1$-embedding problem and an optimal embedding in Brownian motion, Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985) Progr. Probab. Statist., vol. 12, Birkhäuser Boston, Boston, MA, 1986, pp. 172–223. MR 896743
D. H. Root, The existence of certain stopping times on Brownian motion, Ann. Math. Statist. 40 (1969), 715–718. MR 238394, DOI 10.1214/aoms/1177697749
Hermann Rost, The stopping distributions of a Markov Process, Invent. Math. 14 (1971), 1–16. MR 346920, DOI 10.1007/BF01418740
H. Rost, Skorokhod stopping times of minimal variance, Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976, pp. 194–208. MR 0445600
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 10.1016/0022-0396(76)90106-6
Alfred Schatz, Free boundary problems of Stephan type with prescribed flux, J. Math. Anal. Appl. 28 (1969), 569–580. MR 267285, DOI 10.1016/0022-247X(69)90009-2
A. V. Skorohod, A limit theorem for independent random variables, Soviet Math. Dokl. 1 (1960), 810–811. MR 0125605
A. V. Skorohod, Issledovaniya po teorii sluchaĭ nykh protsessov (Stokhasticheskie differentsial′nye uravneniya i predel′nye teoremy dlya protsessov Markova), Izdat. Kiev. Univ., Kiev, 1961 (Russian). MR 0185619
Pierre Vallois, Le problème de Skorokhod sur $\textbf {R}$: une approche avec le temps local, Seminar on probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 227–239 (French). MR 770416, DOI 10.1007/BFb0068320
Pierre van Moerbeke, On optimal stopping and free boundary problems, Arch. Rational Mech. Anal. 60 (1975/76), no. 2, 101–148. MR 413250, DOI 10.1007/BF00250676
Pierre van Moerbeke, An optimal stopping problem with linear reward, Acta Math. 132 (1974), 111–151. MR 376225, DOI 10.1007/BF02392110
D. V. Widder, The heat equation, Pure and Applied Mathematics, Vol. 67, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0466967