Understanding singularitiesin free boundary problems

Xavier Ros-Oton; Joaquim Serra

Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana (2019)

  • Volume: 4, Issue: 2, page 107-118
  • ISSN: 2499-751X

Abstract

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Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfacesor boundaries. The most classical example is the melting of ice to water (the Stefan problem). In this case, the freeboundary is the liquid-solid interface between ice and water. A central mathematical challenge in this context is to understand the regularity and singularities of free boundaries. In this paper we provide a gentle introduction to this topic by presenting some classical results of Luis Caffarelli, as well as some important recent works due to Alessio Figalli and collaborators.

How to cite

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Ros-Oton, Xavier, and Serra, Joaquim. "Understanding singularitiesin free boundary problems." Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana 4.2 (2019): 107-118. <http://eudml.org/doc/295088>.

@article{Ros2019,
abstract = {Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfacesor boundaries. The most classical example is the melting of ice to water (the Stefan problem). In this case, the freeboundary is the liquid-solid interface between ice and water. A central mathematical challenge in this context is to understand the regularity and singularities of free boundaries. In this paper we provide a gentle introduction to this topic by presenting some classical results of Luis Caffarelli, as well as some important recent works due to Alessio Figalli and collaborators.},
author = {Ros-Oton, Xavier, Serra, Joaquim},
journal = {Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana},
language = {eng},
month = {8},
number = {2},
pages = {107-118},
publisher = {Unione Matematica Italiana},
title = {Understanding singularitiesin free boundary problems},
url = {http://eudml.org/doc/295088},
volume = {4},
year = {2019},
}

TY - JOUR
AU - Ros-Oton, Xavier
AU - Serra, Joaquim
TI - Understanding singularitiesin free boundary problems
JO - Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana
DA - 2019/8//
PB - Unione Matematica Italiana
VL - 4
IS - 2
SP - 107
EP - 118
AB - Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfacesor boundaries. The most classical example is the melting of ice to water (the Stefan problem). In this case, the freeboundary is the liquid-solid interface between ice and water. A central mathematical challenge in this context is to understand the regularity and singularities of free boundaries. In this paper we provide a gentle introduction to this topic by presenting some classical results of Luis Caffarelli, as well as some important recent works due to Alessio Figalli and collaborators.
LA - eng
UR - http://eudml.org/doc/295088
ER -

References

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  1. ALMGREN, F. J., Almgren's big regularity paper. Q-valued functions minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2, J. E. Taylor, V. Scheffer, eds. World Scientific Monograph Series in Mathematics, 2000. Zbl0985.49001MR1777737
  2. BAIOCCHI, C., Free boundary problems in the theory of fluid flow through porous media, in Proceedings of the ICM 1974. Zbl0347.76067MR421331
  3. BARRIOS, B., FIGALLI, A., ROS-OTON, X., Global regularity for the free boundary in the obstacle problem for the fractional Laplacian, Amer. J. Math.140 (2018), 415-447. Zbl1409.35044MR3783214DOI10.1353/ajm.2018.0010
  4. BARRIOS, B., FIGALLI, A., ROS-OTON, X., Free boundary regularity in the parabolic fractional obstacle problem, Comm. Pure Appl. Math.71 (2018), 2129-2159. MR3861075DOI10.1002/cpa.21745
  5. BLANCHET, A., On the singular set of the parabolic obstacle problem, J. Differential Equations231 (2006), 656-672. Zbl1121.35145MR2287901DOI10.1016/j.jde.2006.05.013
  6. CAFFARELLI, L., The regularity of free boundaries in higher dimensions, Acta Math.139 (1977), 155-184. Zbl0386.35046MR454350DOI10.1007/BF02392236
  7. CAFFARELLI, L., Compactness methods in free boundary problems, Comm. Partial Differential Equations5 (1980), 427-448. Zbl0437.35070MR567780DOI10.1080/0360530800882144
  8. CAFFARELLI, L., The obstacle problem revisited, J. Fourier Anal. Appl.4 (1998), 383-402. Zbl0928.49030MR1658612DOI10.1007/BF02498216
  9. CAFFARELLI, L., FIGALLI, A., Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math., 680 (2013), 191-233. Zbl1277.35088MR3100955DOI10.1515/crelle.2012.036
  10. CAFFARELLI, L., KOHN, R., NIRENBERG, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math.35 (1982), 771-831. Zbl0509.35067MR673830DOI10.1002/cpa.3160350604
  11. CAFFARELLI, L., RIVIÈRE, N. M., Asymptotic behavior of free boundaries at their singular points, Ann. of Math.106 (1977), 309-317. Zbl0364.35041MR463690DOI10.2307/1971098
  12. CAFFARELLI, L., SALSA, S., A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, vol. 68, AMS2005. Zbl1083.35001MR2145284DOI10.1090/gsm/068
  13. CAFFARELLI, L., SALSA, S., SILVESTRE, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math.171 (2008), 425-461. Zbl1148.35097MR2367025DOI10.1007/s00222-007-0086-6
  14. CARRILLO, J. A., DELGADINO, M. G., MELLET, A., Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys.343 (2016), 747-781. Zbl1337.49066MR3488544DOI10.1007/s00220-016-2598-7
  15. COLOMBO, M., SPOLAOR, L., VELICHKOV, B., A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal.28 (2018), 1029-1061. Zbl1428.49042MR3820438DOI10.1007/s00039-018-0451-1
  16. CONT, R., TANKOV, P., Financial Modelling With Jump Processes, Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. Zbl1052.91043MR2042661
  17. DE SILVA, D., SAVIN, O., Boundary Harnack estimates in slit domains and applications to thin free boundary problems, Rev. Mat. Iberoam.32 (2016), 891-912. Zbl1356.35020MR3556055DOI10.4171/RMI/902
  18. DUVAUT, G., Résolution d'un problème de Stefan (Fusion d'un bloc de glace a zero degrées), C. R. Acad. Sci. Paris276 (1973), 1461-1463. Zbl0258.35037MR328346
  19. DUVAUT, G., LIONS, J. L., Inequalities in Mechanics and Physics, Springer, 1976. Zbl0331.35002MR521262
  20. FERNÁNDEZ-REAL, X., JHAVERI, Y., On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem, preprint arXiv (2018). MR1463593DOI10.1088/0266-5611/13/4/011
  21. FERNÁNDEZ-REAL, X., ROS-OTON, X., The obstacle problem for the fractional Laplacian with critical drift, Math. Ann.371 (2018), 1683-1735. Zbl1395.35228MR3831283DOI10.1007/s00208-017-1600-9
  22. FIGALLI, A., Regularity of interfaces in phase transitions via obstacle problems, Prooceedings of the International Congress of Mathematicians (2018). MR3966720
  23. FIGALLI, A., SERRA, J., On the fine structure of the free boundary for the classical obstacle problem, Invent. Math.215 (2019), 311-366. Zbl1408.35228MR3904453DOI10.1007/s00222-018-0827-8
  24. FIGALLI, A., ROS-OTON, X., SERRA, J., On the singular set in the Stefan problem and a conjecture of Schaeffer, forthcoming (2019). 
  25. FOCARDI, M., SPADARO, E., On the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Rat. Mech. Anal.230 (2018), 125-184. Zbl1405.35257MR3840912DOI10.1007/s00205-018-1242-4
  26. FOCARDI, M., SPADARO, E., The local structure of the free boundary in the fractional obstacle problem, forthcoming. Zbl1405.35257
  27. FRIEDMAN, A., Variational Principles and Free Boundary Problems, Wiley, New York, 1982. Zbl0564.49002MR679313
  28. GAROFALO, N., PETROSYAN, A., Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math.177 (2009), no. 2, 414-461. Zbl1175.35154MR2511747DOI10.1007/s00222-009-0188-4
  29. GAROFALO, N., PETROSYAN, A., POP, C. A., SMIT VEGA GARCIA, M., Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift, Ann. Inst. H. Poincaré Anal. Non Linéaire34 (2017), 533-570. MR3633735DOI10.1016/j.anihpc.2016.03.001
  30. GAROFALO, N., ROS-OTON, X., Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam. (2019), in press. Zbl07144967MR4018099DOI10.4171/rmi/1087
  31. GIUSTI, E., Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics80, Birkhäuser, 1984. Zbl0545.49018MR775682DOI10.1007/978-1-4684-9486-0
  32. JHAVERI, Y., NEUMAYER, R., Higher regularity of the free boundary in the obstacle problem for the fractional Laplacian, Adv. Math.311 (2017), 748-795. Zbl1372.35061MR3628230DOI10.1016/j.aim.2017.03.006
  33. KINDERLEHRER, D., NIRENBERG, L., Regularity in free boundary problems, Ann. Sc. Norm. Sup. Pisa (1977), 373-391. Zbl0352.35023MR440187
  34. KINDERLEHRER, D., STAMPACCHIA, G., An Introduction to Variational Inequalities and Their Applications, SIAM, 1980. Zbl0457.35001MR567696
  35. KOCH, H., PETROSYAN, A., SHI, W., Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal.126 (2015), 3-44. Zbl1329.35362MR3388870DOI10.1016/j.na.2015.01.007
  36. KOCH, H., RÜLAND, A., SHI, W., Higher regularity for the fractional thin obstacle problem, preprint arXiv (2016). Zbl1423.35465MR3692912
  37. LAMÉ, G., CLAPEYRON, B. P., Mémoire sur la solidification par refroidissement d'un globe liquide, Ann. Chimie Physique47 (1831), 250-256. 
  38. LINDGREN, E., MONNEAU, R., Pointwise regularity of the free boundary for the parabolic obstacle problem, Calc. Var. Partial Differential Equations54 (2015), 299-347. Zbl1327.35455MR3385162DOI10.1007/s00526-014-0787-9
  39. MERTON, R., Option pricing when the underlying stock returns are discontinuous, J. Finan. Econ.5 (1976), 125-144. Zbl1131.91344
  40. MONNEAU, R., On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal.13 (2003), 359-389. Zbl1041.35093MR1967031DOI10.1007/BF02930701
  41. PETROSYAN, A., POP, C. A., Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift, J. Funct. Anal.268 (2015), 417-472. Zbl1366.35223MR3283160DOI10.1016/j.jfa.2014.10.009
  42. PETROSYAN, A., SHAHGHOLIAN, H., URALTSEVA, N., Regularity of Free Boundaries in Obstacle-type Problems, volume 136 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. Zbl1254.35001MR2962060DOI10.1090/gsm/136
  43. RODRIGUES, J. F., Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987. Zbl0606.73017MR880369
  44. SAKAI, M., Regularity of a boundary having a Schwarz function. Acta Math.166 (1991), 263-297. Zbl0728.30007MR1097025DOI10.1007/BF02398888
  45. SAKAI, M., Regularity of free boundaries in two dimensions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 3, 323-339. Zbl0851.35022MR1256071
  46. SCHAEFFER, D. G., An example of generic regularity for a nonlinear elliptic equation, Arch. Rat. Mech. Anal.57 (1974), 134-141. Zbl0319.35036MR387810DOI10.1007/BF00248415
  47. SCHAEFFER, D. G., Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa4 (1977), 133-144. Zbl0354.35033MR516201
  48. SERFATY, , Coulomb Gases and Ginzburg-Landau Vortices, Zurich Lectures in Advanced Mathematics, EMS books, 2015. MR3309890DOI10.4171/152
  49. SILVESTRE, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math.60 (2007), 67-112. Zbl1141.49035MR2270163DOI10.1002/cpa.20153
  50. SIMONS, J. H., Minimal varieties in riemannian manifolds, Ann. of Math.88 (1968) 62-105. Zbl0181.49702MR233295DOI10.2307/1970556
  51. SMALE, N., Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds, Comm. Anal. Geom.1 (1993), 217-228. Zbl0843.58027MR1243523DOI10.4310/CAG.1993.v1.n2.a2
  52. STEFAN, J., Ueber einige Probleme der Theorie der Wärmeleitung, Wien. Ber.98 (1889), 473-484. Zbl21.1197.01
  53. STEFAN, J., Ueber die Theorie der Eisbildung, insbesondere ueber die Eisbildung im Polarmeere, Ann. Physik Chemie42 (1891), 269-286. Zbl23.1188.04MR1546138DOI10.1007/BF01692459
  54. WEISS, G., A homogeneity improvement approach to the obstacle problem, Invent. Math.138 (1999) 23-50. Zbl0940.35102MR1714335DOI10.1007/s002220050340
  55. WEISS, G., Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems, SIAM J. Math. Anal.30 (1999) 623-644. Zbl0922.35193MR1677947DOI10.1137/S0036141097327409
  56. WHITE, B., The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc.13 (2000), 665-695. Zbl0961.53039MR1758759DOI10.1090/S0894-0347-00-00338-6

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