no. 5
Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1259-1277

We consider fully nonlinear obstacle-type problems of the form

{F(D2u,x)=f(x)a.e. in B1∩Ω,|D2u|≤Ka.e. in B1\Ω,
where Ω is an open set and K>0. In particular, structural conditions on F are presented which ensure that W2,n(B1) solutions achieve the optimal C1,1(B1/2) regularity when f is Hölder continuous. Moreover, if f is positive on B‾1, Lipschitz continuous, and {u≠0}⊂Ω, we obtain interior C1 regularity of the free boundary under a uniform thickness assumption on {u=0}. Lastly, we extend these results to the parabolic setting.

DOI : 10.1016/j.anihpc.2015.03.009
Classification : 35J60, 35K55, 35R35
Keywords: Nonlinear elliptic equations, Nonlinear parabolic equations, Free boundaries, Regularity theory, Obstacle problems 
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     title = {Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems},
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Indrei, Emanuel; Minne, Andreas. Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1259-1277. doi: 10.1016/j.anihpc.2015.03.009

[1] Caffarelli, L.A. The obstacle problem revisited, J. Fourier Anal. Appl., Volume 4 (1998) no. 4–5, pp. 383–402 | MR | Zbl | DOI

[2] Caffarelli, L.A. The regularity of free boundaries in higher dimensions, Acta Math., Volume 139 (1977) no. 3–4, pp. 155–184 | MR | Zbl

[3] Caffarelli, L.; Salazar, J. Solutions of fully nonlinear elliptic equations with patches of zero gradient: existence, regularity and convexity of level curves, Trans. Am. Math. Soc., Volume 354 (2002) no. 8, pp. 3095–3115 | MR | Zbl | DOI

[4] Rodrigues, J. Obstacle Problems in Mathematical Physics, North-Holl. Math. Stud., Elsevier Science, 1987 http://books.google.se/books?id=B_8ep0ZHMo4C | MR | Zbl

[5] Kinderlehrer, D.; Nirenberg, L. Regularity in free boundary problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 4 (1977) no. 2, pp. 373–391 | MR | Zbl | Numdam

[6] Andersson, J.; Lindgren, E.; Shahgholian, H. Optimal regularity for the no-sign obstacle problem, Commun. Pure Appl. Math., Volume 66 (2013) no. 2, pp. 245–262 | MR | Zbl | DOI

[7] Lee, K. Obstacle problems for the fully nonlinear elliptic operators, 1998 (Ph.D. thesis) | MR

[8] Lee, K.-A.; Shahgholian, H. Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., Volume 54 (2001) no. 1, pp. 43–56 | MR | Zbl | DOI

[9] Figalli, A.; Shahgholian, H. A general class of free boundary problems for fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., Volume 213 (2014) no. 1, pp. 269–286 | MR | Zbl | DOI

[10] Caffarelli, L.A.; Huang, Q. Estimates in the generalized Campanato–John–Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., Volume 118 (2003) no. 1, pp. 1–17 | MR | Zbl | DOI

[11] Figalli, A.; Shahgholian, H. An overview of unconstrained free boundary problems, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. (2015) (in press) | MR | Zbl | DOI

[12] Caffarelli, L.A. Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math. (2), Volume 130 (1989) no. 1, pp. 189–213 | MR | Zbl | DOI

[13] Figalli, A.; Shahgholian, H. A general class of free boundary problems for fully nonlinear parabolic equations, Ann. Mat. Pura Appl. (2014), pp. 1–12 (in press) | MR | Zbl | DOI

[14] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer, 2001 | MR | Zbl | DOI

[15] Caffarelli, L.A.; Cabré, X. Fully Nonlinear Elliptic Equations, Colloq. Publ. – Am. Math. Soc., vol. 43, American Mathematical Society, Providence, RI, 1995 | MR | Zbl

[16] Schaeffer, D.G. Some examples of singularities in a free boundary, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 4 (1977) no. 1, pp. 133–144 | MR | Zbl | Numdam

[17] Petrosyan, A.; Shahgholian, H.; Uraltseva, N. Regularity of Free Boundaries in Obstacle-Type Problems, Grad. Stud. Math., vol. 136, American Mathematical Society, Providence, RI, 2012 | MR | Zbl | DOI

[18] Caffarelli, L.; Crandall, M.G.; Kocan, M.; Swiech, A. On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., Volume 49 (1996) no. 4, pp. 365–397 | MR | Zbl | DOI

[19] Wang, L. On the regularity theory of fully nonlinear parabolic equations. I, Commun. Pure Appl. Math., Volume 45 (1992) no. 1, pp. 27–76 | MR | Zbl | DOI

[20] Krylov, N.V. Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982) no. 3, pp. 487–523 (670) | MR | Zbl

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