Abstract
In my childhood, I read oftentimes in various Russian fairy tales the expression from the epigraph. It was a standard task for a fairy tale hero, to be accomplished by any means. Nowadays, a similar task is faced by mathematicians studying free boundary problems. You may ask, why is this so? I will try to explain.
Go I know not whitherand fetch I know not what
Russian Fairy Tales
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2), 431–461 (1984)
J. Andersson, N. Matevosyan, H. Mikayelyan, On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem. Ark. Mat. 44(1), 1–15 (2006)
D.E. Apushkinskaya, N. Matevosyan, N.N. Uraltseva, The behavior of the free boundary close to a fixed boundary in a parabolic problem. Indiana Univ. Math. J. 58(2), 583–604 (2009)
I. Athanasopoulos, S. Salsa, An application of a parabolic comparison principle to free boundary problems. Indiana Univ. Math. J. 40(1), 29–32 (1991)
D.E. Apushkinskaya, H. Shahgholian, N.N. Uraltseva, Boundary estimates for solutions of a parabolic free boundary problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271, 39–55 (2000)
D.E. Apushkinskaya, N.N. Uraltseva, On the behavior of the free boundary near the boundary of the domain. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221(Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 26), 5–19, 253 (1995). English transl. in J. Math. Sci. (N.Y.) 87(2), 3267–3276 (1997)
D.E. Apushkinskaya, N.N. Uraltseva, Boundary estimates for solutions of two-phase obstacle problems. Probl. Math. Anal. (34), 3–11 (2006). English transl. in J. Math. Sci. (N.Y.) 142(1), 1723–1732 (2007)
D.E. Apushkinskaya, N.N. Uraltseva, Boundary estimates for solutions to the two-phase parabolic obstacle problem. Probl. Math. Anal. (38), 3–10 (2008). English transl. in J. Math. Sci. (N.Y.) 156(4), 569–576 (2009)
D. E. Apushkinskaya and N. N. Uraltseva. Uniform estimates near the initial state for solutions of the two-phase parabolic problem. Algebra i Analiz 25(2), 63–74 (2013)
D.E. Apushkinskaya, N.N. Uraltseva, Free boundaries in problems with hysteresis. Phil. Trans. Roy. Soc. A 373(2050), 20140271 (2015)
D.E. Apushkinskaya, N.N. Ural’tseva, Kh. Shakhgolyan, On global solutions of a parabolic problem with an obstacle. Algebra i Analiz 14(1), 3–25 (2002)
D.E. Apushkinskaya, N.N. Ural’tseva, Kh. Shakhgolyan, On the Lipschitz property of the free boundary in a parabolic problem with an obstacle. Algebra i Analiz 15(3), 78–103 (2003)
C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. (4) 92, 107–127 (1972)
H. Berestycki, L.A. Caffarelli, L. Nirenberg, Uniform estimates for regularization of free boundary problems, in Analysis and Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 122 (Dekker, New York, 1990), pp. 567–619
A. Blanchet, J. Dolbeault, R. Monneau, On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients. J. Math. Pures Appl. (9) 85(3), 371–414 (2006)
H. Brézis, D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23(9), 831–844 (1974)
H. Berestycki, B. Larrouturou, A semi-linear elliptic equation in a strip arising in a two-dimensional flame propagation model. J. Reine Angew. Math. 396, 14–40 (1989)
A. Blanchet, On the singular set of the parabolic obstacle problem. J. Diff. Equ. 231(2), 656–672 (2006)
H. Berestycki, B. Larrouturou, L. Nirenberg, A nonlinear elliptic problem describing the propagation of a curved premixed flame, in Mathematical Modeling in Combustion and Related Topics (Lyon, 1987). NATO Advanced Science Institutes Series E: Applied Sciences, vol. 140 (Nijhoff, Dordrecht, 1988), pp. 11–28
L.A. Caffarelli, The smoothness of the free surface in a filtration problem. Arch. Rational Mech. Anal. 63(1), 77–86 (1976)
L.A. Caffarelli, The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977)
L.A. Caffarelli, Compactness methods in free boundary problems. Commun. Partial Diff. Equ. 5(4), 427–448 (1980)
L.A. Caffarelli, A monotonicity formula for heat functions in disjoint domains, in Boundary Value Problems for Partial Differential Equations and Applications. RMA Research Notes in Applied Mathematics, vol. 29 (Masson, Paris, 1993), pp. 53–60
L.A. Caffarelli, The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998)
L.A. Caffarelli, C.E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems. Am. J. Math. 120(2), 391–439 (1998)
L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151(1), 269–292 (2000)
L. Caffarelli, A. Petrosyan, H. Shahgholian, Regularity of a free boundary in parabolic potential theory. J. Am. Math. Soc. 17(4), 827–869 (electronic) (2004)
J. Crank, Free and Moving Boundary Problems (The Clarendon Press Oxford University Press, New York, 1987)
L.A. Caffarelli, H. Shahgholian, The structure of the singular set of a free boundary in potential theory. Izv. Nats. Akad. Nauk Armenii Mat. 39(2), 43–58 (2004)
L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, vol. 68 (American Mathematical Society, Providence, RI, 2005)
G. Duvaut, J.-L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976). Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219
G. Duvaut, Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré). C. R. Acad. Sci. Paris Sér. A-B 276, A1461–A1463 (1973)
J. Frehse, On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital. (4) 6, 312–315 (1972)
A. Friedman, Variational Principles and Free-Boundary Problems. Pure and Applied Mathematics (Wiley, New York, 1982). A Wiley-Interscience Publication
A. Friedman, M. Sakai, A characterization of null quadrature domains in R N. Indiana Univ. Math. J. 35(3), 607–610 (1986)
A. Gurevich, Boundary regularity for free boundary problems. Commun. Pure Appl. Math. 52(3), 363–403 (1999)
R. Jensen, Boundary regularity for variational inequalities. Indiana Univ. Math. J. 29(4), 495–504 (1980)
L. Karp, On the Newtonian potential of ellipsoids. Complex Variables Theory Appl. 25(4), 367–371 (1994)
L. Karp, A.S. Margulis, Newtonian potential theory for unbounded sources and applications to free boundary problems. J. Anal. Math. 70, 1–63 (1996)
L. Kapr, A.S. Margulis, Null quadrature domains and a free boundary problem for the Laplacian. Indiana Univ. Math. J. 61(2), 859–882 (2012)
N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. Math. USSR Izv. 22(1), 67–98 (1984)
D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics, vol. 88 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980)
G. Lamé, B.P. Clapeyron, Mémoire sur la solidification par refroidissement d’um globe solide. Ann. Chem. Phys. 47, 250–256 (1831)
E. Lindgren, R. Monneau, Pointwise regularity of the free boundary for the parabolic obstacle problem. Calc. Var. 54(1), 299–347 (2015)
K. Nyström, A. Pascucci, S. Polidoro, Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators. J. Diff. Equ. 249(8), 2044–2060 (2010)
K. Nyström, On the behaviour near expiry for multi-dimensional American options. J. Math. Anal. Appl. 339(1), 644–654 (2008)
V.I. Oliker, N.N. Uraltseva, Evolution of nonparametric surfaces with speed depending on curvature. II. The mean curvature case. Commun. Pure Appl. Math. 46(1), 97–135 (1993)
V.I. Oliker, N.N. Uraltseva, Evolution of nonparametric surfaces with speed depending on curvature. III. Some remarks on mean curvature and anisotropic flows, in Degenerate Diffusions (Minneapolis, MN, 1991). The IMA Volumes in Mathematics and Its Applications, vol. 47, pp. 141–156 (Springer, New York, 1993)
V.I. Oliker, N.N. Ural’tseva, Long time behavior of flows moving by mean curvature, in Nonlinear Evolution Equations. American Mathematical Society Translations: Series 2, vol. 164, pp. 163–170 (American Mathematical Society, Providence, RI, 1995)
V.I. Oliker, N.N. Ural’tseva, Long time behavior of flows moving by mean curvature. II. Topol. Methods Nonlinear Anal. 9(1), 17–28 (1997)
A. Petrosyan, H. Shahgholian, Parabolic obstacle problems applied to finance, in Recent Developments in Nonlinear Partial Differential Equations. Contemporary Mathematics, vol. 439, pp. 117–133 (American Mathematical Society, Providence, RI, 2007)
A. Petrosyan, H. Shahgholian, N. Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics, vol. 136 (American Mathematical Society, Providence, RI, 2012)
J.-F. Rodrigues, Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, vol. 134 (North-Holland Publishing Co., Amsterdam, 1987)
M.V. Safonov, Smoothness near the boundary of solutions of elliptic Bellman equations. Boundary value problems of mathematical physics and related problems in the theory of functions, no. 17 [Russian. English summary]. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 147, 150–154 (1985)
M. Sakai, Null quadrature domains. J. Anal. Math. 40, 144–154 (1982), 1981
M. Sakai, Regularity of a boundary having a Schwarz function. Acta Math. 166(3–4), 263–297 (1991)
D.G. Schaeffer, Some examples of singularities in a free boundary. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 4(1), 133–144 (1977)
H. Shahgholian, On quadrature domains and the Schwarz potential. J. Math. Anal. Appl. 171(1), 61–78 (1992)
H. Shahgholian, C 1, 1 regularity in semilinear elliptic problems. Commun. Pure Appl. Math. 56(2), 278–281 (2003)
H. Shahgholian, Free boundary regularity close to initial state for parabolic obstacle problem. Trans. Am. Math. Soc. 360(4), 2077–2087 (2008)
J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Sitzungsber. Österreich. Akad. Wiss. Math. Naturwiss. Kl. Abt. 2, Math. Astron. Phys. Meteorol. Tech. 98, 965–983 (1889)
H. Shahgholian, N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary. Duke Math. J. 116(1), 1–34 (2003)
H. Shahgholian, N. Uraltseva, G.S. Weiss, The two-phase membrane problem—regularity of the free boundaries in higher dimensions. Int. Math. Res. Not. IRMN (8), 16 (2007)
H. Shahgholian, N. Uraltseva, G.S. Weiss, A parabolic two-phase obstacle-like equation. Adv. Math. 221(3), 861–881 (2009)
H. Shahgholian, G.S. Weiss, The two-phase membrane problem—an intersection-comparison approach to the regularity at branch points. Adv. Math. 205(2), 487–503 (2006)
N.N. Uraltseva, C 1 regularity of the boundary of a noncoincident set in a problem with an obstacle. Algebra i Analiz 8(2), 205–221 (1996)
N.N. Uraltseva, Two-phase obstacle problem. J. Math. Sci. (New York) 106(3), 3073–3077 (2001). Function theory and phase transitions
N.N. Uraltseva, Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities, in Nonlinear Equations and Spectral Theory. American Mathematical Society Translations Series 2, vol. 220, pp. 235–246 (American Mathematical Society, Providence, RI, 2007)
G.S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem. Commun. Partial Diff. Equ. 23(3–4), 439–455 (1998)
G.S. Weiss, Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems. SIAM J. Math. Anal. 30(3), 623–644 (electronic) (1999)
G.S. Weiss, An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound. 3(2), 121–128 (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Apushkinskaya, D. (2018). Introduction. In: Free Boundary Problems. Lecture Notes in Mathematics, vol 2218. Springer, Cham. https://doi.org/10.1007/978-3-319-97079-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-97079-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-97078-3
Online ISBN: 978-3-319-97079-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)