The thin obstacle problem a survey
Article Sidebar
Main Article Content
Xavier Fernandez-Real
École Polytechnique Fédérale de Lausanne
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory with some proofs, from the optimal regularity of solutions and classification of free boundary points to more recent results on the non-regular part of the free boundary and generic regularity.
Paraules clau
thin obstacle problem, Signorini problem, survey, free boundary, fractional obstacle problem
Article Details
Com citar
Fernandez-Real, Xavier. «The thin obstacle problem: a survey». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 1, p. 3-55, https://raco.cat/index.php/PublicacionsMatematiques/article/view/396317.
Referències
F. J. Almgren, Jr., “Almgren’s Big Regularity Paper”, Q-Valued Functions
Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2, With a preface by Jean E. Taylor and
Vladimir Scheffer, World Scientific Monograph Series in Mathematics 1, World
Scientific Publishing Co., Inc., River Edge, NJ, 2000.
J. Andersson, Optimal regularity for the Signorini problem and its free boundary, Invent. Math. 204(1) (2016), 1–82. DOI: 10.1007/s00222-015-0608-6.
I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst.
Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor.
Funkts. 35 [34], 49–66, 226; reprinted in J. Math. Sci. (N.Y.) 132(3) (2006),
274–284. DOI: 10.1007/s10958-005-0496-1.
I. Athanasopoulos, L. Caffarelli, and E. Milakis, On the regularity of the
non-dynamic parabolic fractional obstacle problem, J. Differential Equations
265(6) (2018), 2614–2647. DOI: 10.1016/j.jde.2018.04.043.
I. Athanasopoulos, L. Caffarelli, and E. Milakis, Parabolic obstacle problems, quasi-convexity and regularity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
19(2) (2019), 781–825. DOI: 10.2422/2036-2145.201703_008.
I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The structure of the
free boundary for lower dimensional obstacle problems, Amer. J. Math. 130(2)
(2008), 485–498. DOI: 10.1353/ajm.2008.0016.
B. Barrios, A. Figalli, and X. Ros-Oton, Global regularity for the free
boundary in the obstacle problem for the fractional Laplacian, Amer. J. Math.
140(2) (2018), 415–447. DOI: 10.1353/ajm.2018.0010.
B. Barrios, A. Figalli, and X. Ros-Oton, Free boundary regularity in the
parabolic fractional obstacle problem, Comm. Pure Appl. Math. 71(10) (2018),
2129–2159. DOI: 10.1002/cpa.21745.
H. Beirao da Veiga ˜ , Sulla h¨olderianit`a delle soluzioni di alcune disequazioni
variazionali con condizioni unilatere al bordo, Ann. Mat. Pura Appl. (4) 83
(1969), 73–112. DOI: 10.1007/BF02411161.
H. Beirao da Veiga and F. Conti ˜ , Equazioni ellittiche non lineari con ostacoli
sottili. Applicazioni allo studio dei punti regolari, Ann. Scuola Norm. Sup. Pisa
Cl. Sci. (3) 26 (1972), 533–562.
J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media:
statistical mechanisms, models and physical applications, Phys. Rep. 195(4–5)
(1990), 127–293. DOI: 10.1016/0370-1573(90)90099-N.
L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta
Math. 139(3–4) (1977), 155–184. DOI: 10.1007/BF02392236.
L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial
Differential Equations 4(9) (1979), 1067–1075. DOI: 10.1080/036053079088201
19.
L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4(4–
5) (1998), 383–402. DOI: 10.1007/BF02498216.
L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math. 680 (2013), 191–233. DOI: 10.
1515/crelle.2012.036.
L. A. Caffarelli and N. M. Riviere ` , Asymptotic behaviour of free boundaries
at their singular points, Ann. of Math. (2) 106(2) (1977), 309–317. DOI: 10.
2307/1971098.
L. Caffarelli, X. Ros-Oton, and J. Serra, Obstacle problems for integrodifferential operators: regularity of solutions and free boundaries, Invent. Math.
208(3) (2017), 1155–1211. DOI: 10.1007/s00222-016-0703-3.
L. A. Caffarelli, S. Salsa, and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,
Invent. Math. 171(2) (2008), 425–461. DOI: 10.1007/s00222-007-0086-6.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional
Laplacian, Comm. Partial Differential Equations 32(7–9) (2007), 1245–1260.
DOI: 10.1080/03605300600987306.
L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional
diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171(3) (2010),
1903–1930. DOI: 10.4007/annals.2010.171.1903.
A. Carbotti, S. Dipierro, and E. Valdinoci, “Local Density of Solutions
to Fractional Equations”, De Gruyer Studies in Mathematics 74, De Gruyer,
Berlin, Boston, 2019. DOI: 10.1515/9783110664355.
J. A. Carrillo, M. G. Delgadino, and A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys.
343(3) (2016), 747–781. DOI: 10.1007/s00220-016-2598-7.
M. Colombo, L. Spolaor, and B. Velichkov, Direct epiperimetric inequalities
for the thin obstacle problem and applications, Comm. Pure Appl. Math. 73(2)
(2020), 384–420. DOI: 10.1002/cpa.21859.
R. Cont and P. Tankov, “Financial Modelling with Jump Processes”, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca
Raton, FL, 2004.
D. Danielli, N. Garofalo, A. Petrosyan, and T. To, Optimal regularity and
the free boundary in the parabolic Signorini problem, Mem. Amer. Math. Soc.
249(1181) (2017), 103 pp. DOI: 10.1090/memo/1181.
D. Danielli and S. Salsa, Obstacle problems involving the fractional Laplacian, in: “Recent Developments in Nonlocal Theory”, De Gruyter, Berlin, 2018,
pp. 81–164. DOI: 10.1515/9783110571561-005.
M. de Acutis, Regolarit`a di frontiere minimali con ostacoli sottili, Rend. Sem.
Mat. Univ. Padova 61 (1979), 133–144.
E. De Giorgi, Problemi di superfici minime con ostacoli: forma non cartesiana,
Boll. Un. Mat. Ital. (4) 8 (1973), suppl. no. 2, 80–88.
D. De Silva and O. Savin, Boundary Harnack estimates in slit domains and
applications to thin free boundary problems, Rev. Mat. Iberoam. 32(3) (2016),
891–912. DOI: 10.4171/RMI/902.
D. De Silva and O. Savin, A short proof of boundary Harnack principle, J. Differential Equations 269(3) (2020), 2419–2429. DOI: 10.1016/j.jde.2020.02.
004.
G. Duvaut and J. L. Lions, “Inequalities in Mechanics and Physics”, Translated from the French by C. W. John, Grundlehren der Mathematischen
Wissenschaften 219, Springer-Verlag, Berlin-New York, 1976. DOI: 10.1007/
978-3-642-66165-5.
L. C. Evans, “An Introduction to Stochastic Differential Equations”, American
Mathematical Society, Providence, RI, 2013. DOI: 10.1090/mbk/082.
X. Fernandez-Real ´ , C1,α estimates for the fully nonlinear Signorini problem,
Calc. Var. Partial Differential Equations 55(4) (2016), Art. 94, 20 pp. DOI: 10.
1007/s00526-016-1034-3.
X. Fernandez-Real and Y. Jhaveri ´ , On the singular set in the thin obstacle
problem: higher-order blow-ups and the very thin obstacle problem, Anal. PDE
14(5) (2021), 1599–1669. DOI: 10.2140/apde.2021.14.1599.
X. Fernandez-Real and X. Ros-Oton ´ , The obstacle problem for the fractional
Laplacian with critical drift, Math. Ann. 371(3–4) (2018), 1683–1735. DOI: 10.
1007/s00208-017-1600-9.
X. Fernandez-Real and X. Ros-Oton ´ , “Regularity Theory for Elliptic PDE”,
Submitted (2020) (available at the webpage of the authors).
X. Fernandez-Real and X. Ros-Oton ´ , Free boundary regularity for almost every solution to the Signorini problem, Arch. Ration. Mech. Anal. 240(1) (2021),
419–466. DOI: 10.1007/s00205-021-01617-8.
X. Fernandez-Real and J. Serra ´ , Regularity of minimal surfaces with lowerdimensional obstacles, J. Reine Angew. Math. 767 (2020), 37–75. DOI: 10.1515/
crelle-2019-0035.
G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl.
Sci. Fis. Mat. Natur. Sez. Ia (8) 7 (1963/64), 91–140.
A. Figalli and J. Serra, On the fine structure of the free boundary for the
classical obstacle problem, Invent. Math. 215(1) (2019), 311–366. DOI: 10.1007/
s00222-018-0827-8.
A. Figalli, X. Ros-Oton, and J. Serra, Generic regularity of free boundaries
for the obstacle problem, Publ. Math. Inst. Hautes Etudes Sci. ´ 132 (2020),
181–292. DOI: 10.1007/s10240-020-00119-9.
M. Focardi and E. Spadaro, An epiperimetric inequality for the thin obstacle
problem, Adv. Differential Equations 21(1–2) (2016), 153–200.
M. Focardi and E. Spadaro, On the measure and the structure of the free
boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal.
230(1) (2018), 125–184. DOI: 10.1007/s00205-018-1242-4.
M. Focardi and E. Spadaro, The local structure of the free boundary in the
fractional obstacle problem, Preprint (2019). arXiv:1903.05909.
M. Focardi and E. Spadaro, How a minimal surface leaves a thin obstacle,
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 37(4) (2020), 1017–1046. DOI: 10.
1016/j.anihpc.2020.02.005.
J. Frehse, On Signorini’s problem and variational problems with thin obstacles,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2) (1977), 343–362.
N. Garofalo and A. Petrosyan, Some new monotonicity formulas and the
singular set in the lower dimensional obstacle problem, Invent. Math. 177(2)
(2009), 415–461. DOI: 10.1007/s00222-009-0188-4.
N. Garofalo, A. Petrosyan, C. A. Pop, and M. Smit Vega Garcia, Regularity of the free boundary for the obstacle problem for the fractional Laplacian
with drift, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 34(3) (2017), 533–570.
DOI: 10.1016/j.anihpc.2016.03.001.
N. Garofalo, A. Petrosyan, and M. Smit Vega Garcia, An epiperimetric
inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients, J. Math. Pures Appl. (9) 105(6) (2016), 745–787.
DOI: 10.1016/j.matpur.2015.11.013.
M. Giaquinta and G. Modica, Regolarit`a Lipschitziana per la soluzione di
alcuni problemi di minimo con vincolo, Ann. Mat. Pura Appl. (4) 106 (1975),
95–117. DOI: 10.1007/BF02415024.
G. Grubb, Fractional Laplacians on domains, a development of H¨ormander’s
theory of µ-transmission pseudodifferential operators, Adv. Math. 268 (2015),
478–528. DOI: 10.1016/j.aim.2014.09.018.
N. Kikuchi and J. T. Oden, “Contact Problems in Elasticity: A Study of
Variational Inequalities and Finite Element Methods”, SIAM Studies in Applied Mathematics 8, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1988. DOI: 10.1137/1.9781611970845.
D. Kinderlehrer, Variational inequalities with lower dimensional obstacles,
Israel J. Math. 10 (1971), 339–348. DOI: 10.1007/BF02771651.
D. Kinderlehrer, Remarks about Signorini’s problem in linear elasticity, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(4) (1981), 605–645.
D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2) (1977), 373–391.
H. Koch, A. Petrosyan, and W. Shi, Higher regularity of the free boundary in
the elliptic Signorini problem, Nonlinear Anal. 126 (2015), 3–44. DOI: 10.1016/
j.na.2015.01.007.
B. Krummel and N. Wickramasekera, Fine properties of branch point singularities: Two-valued harmonic functions, Preprint (2013). arXiv:1311.0923.
P. Laurence and S. Salsa, Regularity of the free boundary of an American option on several assets, Comm. Pure Appl. Math. 62(7) (2009), 969–994.
DOI: 10.1002/cpa.20268.
H. Lewy, On a variational problem with inequalities on the boundary, J. Math.
Mech. 17(9) (1968), 861–884.
H. Lewy and G. Stampacchia, On the regularity of the solution of a variational
inequality, Comm. Pure Appl. Math. 22(2) (1969), 153–188. DOI: 10.1002/cpa.
3160220203.
J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl.
Math. 20(3) (1967), 493–519. DOI: 10.1002/cpa.3160200302.
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals
and rectifiability”, Cambridge Studies in Advanced Mathematics 44, Cambridge
University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813.
R. C. Merton, Option pricing when underlying stock returns are discontinuous,
J. Financ. Econ. 3(1–2) (1976), 125–144. DOI: 10.1016/0304-405X(76)90022-2.
R. Monneau, Pointwise estimates for Laplace equation. Applications to the free
boundary of the obstacle problem with Dini coefficients, J. Fourier Anal. Appl.
15(3) (2009), 279–335. DOI: 10.1007/s00041-009-9066-0.
A. Naber and D. Valtorta, Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. of Math. (2) 185(1) (2017),
131–227. DOI: 10.4007/annals.2017.185.1.3.
A. Naber and D. Valtorta, The singular structure and regularity of stationary
varifolds, J. Eur. Math. Soc. (JEMS) 22(10) (2020), 3305–3382. DOI: 10.4171/
jems/987.
A. Petrosyan and C. A. Pop, Optimal regularity of solutions to the obstacle
problem for the fractional Laplacian with drift, J. Funct. Anal. 268(2) (2015),
417–472. DOI: 10.1016/j.jfa.2014.10.009.
A. Petrosyan, H. Shahgholian, and N. Uraltseva, “Regularity of Free
Boundaries in Obstacle-Type Problems”, Graduate Studies in Mathematics 136,
American Mathematical Society, Providence, RI, 2012. DOI: 10.1090/gsm/136.
H. Pham, Optimal stopping, free boundary, and American option in a jumpdiffusion model, Appl. Math. Optim. 35(2) (1997), 145–164. DOI: 10.1007/
BF02683325.
D. J. A. Richardson, Variational problems with thin obstacles, Thesis (Ph.D.)-
The University of British Columbia, Canada (1978). Available on ProQuest.
X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ.
Mat. 60(1) (2016), 3–26. DOI: 10.5565/PUBLMAT_60116_01.
X. Ros-Oton, Obstacle problems and free boundaries: an overview, SeMA J.
75(3) (2018), 399–419. DOI: 10.1007/s40324-017-0140-2.
X. Ros-Oton and J. Serra, The structure of the free boundary in the fully
nonlinear thin obstacle problem, Adv. Math. 316 (2017), 710–747. DOI: 10.1016/
j.aim.2017.06.032.
A. Ruland and W. Shi ¨ , Optimal regularity for the thin obstacle problem with
C0,α coefficients, Calc. Var. Partial Differential Equations 56(5) (2017), Paper
No. 129, 41 pp. DOI: 10.1007/s00526-017-1230-9.
S. Salsa, The problems of the obstacle in lower dimension and for the fractional Laplacian, in: “Regularity Estimates for Nonlinear Elliptic and Parabolic
Problems”, Lecture Notes in Math. 2045, Fond. CIME/CIME Found. Subser.,
Springer, Heidelberg, 2012, pp. 153–244. DOI: 10.1007/978-3-642-27145-8_4.
W. Shi, An epiperimetric inequality approach to the parabolic Signorini problem, Discrete Contin. Dyn. Syst. 40(3) (2020), 1813–1846. DOI: 10.3934/dcds.
2020095.
A. Signorini, Sopra alcune questioni di elastostatica, Atti Soc. It. Progr. Sc.
21II (1933), 143–148.
A. Signorini, Questioni di elasticit`a non linearizzata e semilinearizzata, Rend.
Mat. e Appl. (5) 18 (1959), 95–139.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the
Laplace operator, Comm. Pure Appl. Math. 60(1) (2007), 67–112. DOI: 10.
1002/cpa.20153.
G. S. Weiss, A homogeneity improvement approach to the obstacle problem,
Invent. Math. 138(1) (1999), 23–50. DOI: 10.1007/s002220050340.
B. White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997), 1–35. DOI: 10.1515/crll.1997.
488.1
Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2, With a preface by Jean E. Taylor and
Vladimir Scheffer, World Scientific Monograph Series in Mathematics 1, World
Scientific Publishing Co., Inc., River Edge, NJ, 2000.
J. Andersson, Optimal regularity for the Signorini problem and its free boundary, Invent. Math. 204(1) (2016), 1–82. DOI: 10.1007/s00222-015-0608-6.
I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst.
Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor.
Funkts. 35 [34], 49–66, 226; reprinted in J. Math. Sci. (N.Y.) 132(3) (2006),
274–284. DOI: 10.1007/s10958-005-0496-1.
I. Athanasopoulos, L. Caffarelli, and E. Milakis, On the regularity of the
non-dynamic parabolic fractional obstacle problem, J. Differential Equations
265(6) (2018), 2614–2647. DOI: 10.1016/j.jde.2018.04.043.
I. Athanasopoulos, L. Caffarelli, and E. Milakis, Parabolic obstacle problems, quasi-convexity and regularity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
19(2) (2019), 781–825. DOI: 10.2422/2036-2145.201703_008.
I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The structure of the
free boundary for lower dimensional obstacle problems, Amer. J. Math. 130(2)
(2008), 485–498. DOI: 10.1353/ajm.2008.0016.
B. Barrios, A. Figalli, and X. Ros-Oton, Global regularity for the free
boundary in the obstacle problem for the fractional Laplacian, Amer. J. Math.
140(2) (2018), 415–447. DOI: 10.1353/ajm.2018.0010.
B. Barrios, A. Figalli, and X. Ros-Oton, Free boundary regularity in the
parabolic fractional obstacle problem, Comm. Pure Appl. Math. 71(10) (2018),
2129–2159. DOI: 10.1002/cpa.21745.
H. Beirao da Veiga ˜ , Sulla h¨olderianit`a delle soluzioni di alcune disequazioni
variazionali con condizioni unilatere al bordo, Ann. Mat. Pura Appl. (4) 83
(1969), 73–112. DOI: 10.1007/BF02411161.
H. Beirao da Veiga and F. Conti ˜ , Equazioni ellittiche non lineari con ostacoli
sottili. Applicazioni allo studio dei punti regolari, Ann. Scuola Norm. Sup. Pisa
Cl. Sci. (3) 26 (1972), 533–562.
J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media:
statistical mechanisms, models and physical applications, Phys. Rep. 195(4–5)
(1990), 127–293. DOI: 10.1016/0370-1573(90)90099-N.
L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta
Math. 139(3–4) (1977), 155–184. DOI: 10.1007/BF02392236.
L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial
Differential Equations 4(9) (1979), 1067–1075. DOI: 10.1080/036053079088201
19.
L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4(4–
5) (1998), 383–402. DOI: 10.1007/BF02498216.
L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math. 680 (2013), 191–233. DOI: 10.
1515/crelle.2012.036.
L. A. Caffarelli and N. M. Riviere ` , Asymptotic behaviour of free boundaries
at their singular points, Ann. of Math. (2) 106(2) (1977), 309–317. DOI: 10.
2307/1971098.
L. Caffarelli, X. Ros-Oton, and J. Serra, Obstacle problems for integrodifferential operators: regularity of solutions and free boundaries, Invent. Math.
208(3) (2017), 1155–1211. DOI: 10.1007/s00222-016-0703-3.
L. A. Caffarelli, S. Salsa, and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,
Invent. Math. 171(2) (2008), 425–461. DOI: 10.1007/s00222-007-0086-6.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional
Laplacian, Comm. Partial Differential Equations 32(7–9) (2007), 1245–1260.
DOI: 10.1080/03605300600987306.
L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional
diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171(3) (2010),
1903–1930. DOI: 10.4007/annals.2010.171.1903.
A. Carbotti, S. Dipierro, and E. Valdinoci, “Local Density of Solutions
to Fractional Equations”, De Gruyer Studies in Mathematics 74, De Gruyer,
Berlin, Boston, 2019. DOI: 10.1515/9783110664355.
J. A. Carrillo, M. G. Delgadino, and A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys.
343(3) (2016), 747–781. DOI: 10.1007/s00220-016-2598-7.
M. Colombo, L. Spolaor, and B. Velichkov, Direct epiperimetric inequalities
for the thin obstacle problem and applications, Comm. Pure Appl. Math. 73(2)
(2020), 384–420. DOI: 10.1002/cpa.21859.
R. Cont and P. Tankov, “Financial Modelling with Jump Processes”, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca
Raton, FL, 2004.
D. Danielli, N. Garofalo, A. Petrosyan, and T. To, Optimal regularity and
the free boundary in the parabolic Signorini problem, Mem. Amer. Math. Soc.
249(1181) (2017), 103 pp. DOI: 10.1090/memo/1181.
D. Danielli and S. Salsa, Obstacle problems involving the fractional Laplacian, in: “Recent Developments in Nonlocal Theory”, De Gruyter, Berlin, 2018,
pp. 81–164. DOI: 10.1515/9783110571561-005.
M. de Acutis, Regolarit`a di frontiere minimali con ostacoli sottili, Rend. Sem.
Mat. Univ. Padova 61 (1979), 133–144.
E. De Giorgi, Problemi di superfici minime con ostacoli: forma non cartesiana,
Boll. Un. Mat. Ital. (4) 8 (1973), suppl. no. 2, 80–88.
D. De Silva and O. Savin, Boundary Harnack estimates in slit domains and
applications to thin free boundary problems, Rev. Mat. Iberoam. 32(3) (2016),
891–912. DOI: 10.4171/RMI/902.
D. De Silva and O. Savin, A short proof of boundary Harnack principle, J. Differential Equations 269(3) (2020), 2419–2429. DOI: 10.1016/j.jde.2020.02.
004.
G. Duvaut and J. L. Lions, “Inequalities in Mechanics and Physics”, Translated from the French by C. W. John, Grundlehren der Mathematischen
Wissenschaften 219, Springer-Verlag, Berlin-New York, 1976. DOI: 10.1007/
978-3-642-66165-5.
L. C. Evans, “An Introduction to Stochastic Differential Equations”, American
Mathematical Society, Providence, RI, 2013. DOI: 10.1090/mbk/082.
X. Fernandez-Real ´ , C1,α estimates for the fully nonlinear Signorini problem,
Calc. Var. Partial Differential Equations 55(4) (2016), Art. 94, 20 pp. DOI: 10.
1007/s00526-016-1034-3.
X. Fernandez-Real and Y. Jhaveri ´ , On the singular set in the thin obstacle
problem: higher-order blow-ups and the very thin obstacle problem, Anal. PDE
14(5) (2021), 1599–1669. DOI: 10.2140/apde.2021.14.1599.
X. Fernandez-Real and X. Ros-Oton ´ , The obstacle problem for the fractional
Laplacian with critical drift, Math. Ann. 371(3–4) (2018), 1683–1735. DOI: 10.
1007/s00208-017-1600-9.
X. Fernandez-Real and X. Ros-Oton ´ , “Regularity Theory for Elliptic PDE”,
Submitted (2020) (available at the webpage of the authors).
X. Fernandez-Real and X. Ros-Oton ´ , Free boundary regularity for almost every solution to the Signorini problem, Arch. Ration. Mech. Anal. 240(1) (2021),
419–466. DOI: 10.1007/s00205-021-01617-8.
X. Fernandez-Real and J. Serra ´ , Regularity of minimal surfaces with lowerdimensional obstacles, J. Reine Angew. Math. 767 (2020), 37–75. DOI: 10.1515/
crelle-2019-0035.
G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl.
Sci. Fis. Mat. Natur. Sez. Ia (8) 7 (1963/64), 91–140.
A. Figalli and J. Serra, On the fine structure of the free boundary for the
classical obstacle problem, Invent. Math. 215(1) (2019), 311–366. DOI: 10.1007/
s00222-018-0827-8.
A. Figalli, X. Ros-Oton, and J. Serra, Generic regularity of free boundaries
for the obstacle problem, Publ. Math. Inst. Hautes Etudes Sci. ´ 132 (2020),
181–292. DOI: 10.1007/s10240-020-00119-9.
M. Focardi and E. Spadaro, An epiperimetric inequality for the thin obstacle
problem, Adv. Differential Equations 21(1–2) (2016), 153–200.
M. Focardi and E. Spadaro, On the measure and the structure of the free
boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal.
230(1) (2018), 125–184. DOI: 10.1007/s00205-018-1242-4.
M. Focardi and E. Spadaro, The local structure of the free boundary in the
fractional obstacle problem, Preprint (2019). arXiv:1903.05909.
M. Focardi and E. Spadaro, How a minimal surface leaves a thin obstacle,
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 37(4) (2020), 1017–1046. DOI: 10.
1016/j.anihpc.2020.02.005.
J. Frehse, On Signorini’s problem and variational problems with thin obstacles,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2) (1977), 343–362.
N. Garofalo and A. Petrosyan, Some new monotonicity formulas and the
singular set in the lower dimensional obstacle problem, Invent. Math. 177(2)
(2009), 415–461. DOI: 10.1007/s00222-009-0188-4.
N. Garofalo, A. Petrosyan, C. A. Pop, and M. Smit Vega Garcia, Regularity of the free boundary for the obstacle problem for the fractional Laplacian
with drift, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 34(3) (2017), 533–570.
DOI: 10.1016/j.anihpc.2016.03.001.
N. Garofalo, A. Petrosyan, and M. Smit Vega Garcia, An epiperimetric
inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients, J. Math. Pures Appl. (9) 105(6) (2016), 745–787.
DOI: 10.1016/j.matpur.2015.11.013.
M. Giaquinta and G. Modica, Regolarit`a Lipschitziana per la soluzione di
alcuni problemi di minimo con vincolo, Ann. Mat. Pura Appl. (4) 106 (1975),
95–117. DOI: 10.1007/BF02415024.
G. Grubb, Fractional Laplacians on domains, a development of H¨ormander’s
theory of µ-transmission pseudodifferential operators, Adv. Math. 268 (2015),
478–528. DOI: 10.1016/j.aim.2014.09.018.
N. Kikuchi and J. T. Oden, “Contact Problems in Elasticity: A Study of
Variational Inequalities and Finite Element Methods”, SIAM Studies in Applied Mathematics 8, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1988. DOI: 10.1137/1.9781611970845.
D. Kinderlehrer, Variational inequalities with lower dimensional obstacles,
Israel J. Math. 10 (1971), 339–348. DOI: 10.1007/BF02771651.
D. Kinderlehrer, Remarks about Signorini’s problem in linear elasticity, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(4) (1981), 605–645.
D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2) (1977), 373–391.
H. Koch, A. Petrosyan, and W. Shi, Higher regularity of the free boundary in
the elliptic Signorini problem, Nonlinear Anal. 126 (2015), 3–44. DOI: 10.1016/
j.na.2015.01.007.
B. Krummel and N. Wickramasekera, Fine properties of branch point singularities: Two-valued harmonic functions, Preprint (2013). arXiv:1311.0923.
P. Laurence and S. Salsa, Regularity of the free boundary of an American option on several assets, Comm. Pure Appl. Math. 62(7) (2009), 969–994.
DOI: 10.1002/cpa.20268.
H. Lewy, On a variational problem with inequalities on the boundary, J. Math.
Mech. 17(9) (1968), 861–884.
H. Lewy and G. Stampacchia, On the regularity of the solution of a variational
inequality, Comm. Pure Appl. Math. 22(2) (1969), 153–188. DOI: 10.1002/cpa.
3160220203.
J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl.
Math. 20(3) (1967), 493–519. DOI: 10.1002/cpa.3160200302.
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals
and rectifiability”, Cambridge Studies in Advanced Mathematics 44, Cambridge
University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813.
R. C. Merton, Option pricing when underlying stock returns are discontinuous,
J. Financ. Econ. 3(1–2) (1976), 125–144. DOI: 10.1016/0304-405X(76)90022-2.
R. Monneau, Pointwise estimates for Laplace equation. Applications to the free
boundary of the obstacle problem with Dini coefficients, J. Fourier Anal. Appl.
15(3) (2009), 279–335. DOI: 10.1007/s00041-009-9066-0.
A. Naber and D. Valtorta, Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. of Math. (2) 185(1) (2017),
131–227. DOI: 10.4007/annals.2017.185.1.3.
A. Naber and D. Valtorta, The singular structure and regularity of stationary
varifolds, J. Eur. Math. Soc. (JEMS) 22(10) (2020), 3305–3382. DOI: 10.4171/
jems/987.
A. Petrosyan and C. A. Pop, Optimal regularity of solutions to the obstacle
problem for the fractional Laplacian with drift, J. Funct. Anal. 268(2) (2015),
417–472. DOI: 10.1016/j.jfa.2014.10.009.
A. Petrosyan, H. Shahgholian, and N. Uraltseva, “Regularity of Free
Boundaries in Obstacle-Type Problems”, Graduate Studies in Mathematics 136,
American Mathematical Society, Providence, RI, 2012. DOI: 10.1090/gsm/136.
H. Pham, Optimal stopping, free boundary, and American option in a jumpdiffusion model, Appl. Math. Optim. 35(2) (1997), 145–164. DOI: 10.1007/
BF02683325.
D. J. A. Richardson, Variational problems with thin obstacles, Thesis (Ph.D.)-
The University of British Columbia, Canada (1978). Available on ProQuest.
X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ.
Mat. 60(1) (2016), 3–26. DOI: 10.5565/PUBLMAT_60116_01.
X. Ros-Oton, Obstacle problems and free boundaries: an overview, SeMA J.
75(3) (2018), 399–419. DOI: 10.1007/s40324-017-0140-2.
X. Ros-Oton and J. Serra, The structure of the free boundary in the fully
nonlinear thin obstacle problem, Adv. Math. 316 (2017), 710–747. DOI: 10.1016/
j.aim.2017.06.032.
A. Ruland and W. Shi ¨ , Optimal regularity for the thin obstacle problem with
C0,α coefficients, Calc. Var. Partial Differential Equations 56(5) (2017), Paper
No. 129, 41 pp. DOI: 10.1007/s00526-017-1230-9.
S. Salsa, The problems of the obstacle in lower dimension and for the fractional Laplacian, in: “Regularity Estimates for Nonlinear Elliptic and Parabolic
Problems”, Lecture Notes in Math. 2045, Fond. CIME/CIME Found. Subser.,
Springer, Heidelberg, 2012, pp. 153–244. DOI: 10.1007/978-3-642-27145-8_4.
W. Shi, An epiperimetric inequality approach to the parabolic Signorini problem, Discrete Contin. Dyn. Syst. 40(3) (2020), 1813–1846. DOI: 10.3934/dcds.
2020095.
A. Signorini, Sopra alcune questioni di elastostatica, Atti Soc. It. Progr. Sc.
21II (1933), 143–148.
A. Signorini, Questioni di elasticit`a non linearizzata e semilinearizzata, Rend.
Mat. e Appl. (5) 18 (1959), 95–139.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the
Laplace operator, Comm. Pure Appl. Math. 60(1) (2007), 67–112. DOI: 10.
1002/cpa.20153.
G. S. Weiss, A homogeneity improvement approach to the obstacle problem,
Invent. Math. 138(1) (1999), 23–50. DOI: 10.1007/s002220050340.
B. White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997), 1–35. DOI: 10.1515/crll.1997.
488.1