zbMATH — the first resource for mathematics

Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. (English) Zbl 1275.49016
The paper under review establishes Lewy-Stampacchia type estimates for several classes of nonlinear problems, starting with classes driven by nonlocal operators. The approach developed in this paper is of high interest and it covers the settings corresponding to the standard Laplace or \(p\)-Laplace operators, as well as the Laplacian on the Heisenberg group. The case of integral operators with even kernel is also covered by the Lewy-Stampacchia estimates established in this paper. The proofs combine modern techniques in the theory of nonlocal operators with arguments in the theory of nonlinear partial differential equations.

49J40 Variational inequalities
35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35R11 Fractional partial differential equations
35R35 Free boundary problems for PDEs
Full Text: DOI
[1] Boccardo, L. and Gallouet, T: Probl‘ emes unilatéraux avec données dans L1 (Unilateral problems with L1 data). C. R. Acad. Sci. Paris, Sér. I 311 (1990), no. 10, 617-619. · Zbl 0715.49013
[2] Brézis, H.: Analyse fonctionelle. Théorie et applications. Masson, Paris, 1983. · Zbl 0511.46001
[3] Caffarelli, L. A., Salsa, S. and Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171 (2008), no. 2, 425-461. · Zbl 1148.35097 · doi:10.1007/s00222-007-0086-6 · arxiv:math/0702392
[4] Challal, S., Lyaghfouri, A. and Rodrigues, J. F.: On the A-obstacle problem and the Hausdorff measure of its free boundary. Ann. Mat. Pura Appl. (4) 191 (2012), no. 1, 113-165. · Zbl 1235.35285 · doi:10.1007/s10231-010-0177-7
[5] DiBenedetto, E.: C1+\alpha -local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983), no. 8, 827-850. · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5
[6] Di Nezza, E., Palatucci, G. and Valdinoci, E.: Hitchhiker’s guide to the frac- tional Sobolev spaces. Bull. Sci. Math. 136 (2012), no. 5, 521-573. · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 · arxiv:1104.4345
[7] Donati, F.: A penalty method approach to strong solutions of some nonlinear parabolic unilateral problems. Nonlinear Anal. 6 (1982), no. 6, 585-597. · Zbl 0489.49006 · doi:10.1016/0362-546X(82)90050-5
[8] Frehse, J. and Mosco, U.: Irregular obstacles and quasivariational inequalities of stochastic impulse control. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), no. 1, 109-157. · Zbl 0503.49008 · numdam:ASNSP_1982_4_9_1_105_0 · eudml:83873
[9] Gilbarg, D. and Trudinger, N. S.: Elliptic partial differential equations of second order, 2nd ed. Springer Verlag, Berlin, 1983. · Zbl 0562.35001
[10] Hanouzet, B. and Joly, J. L.: Méthodes d’ordre dans l’interprétation de certaines inéquations variationelles et applications. J. Funct. Anal. 34 (1979), no. 2, 217-249. · Zbl 0425.49009 · doi:10.1016/0022-1236(79)90032-6
[11] Landkof, N. S.: Foundations of modern potential theory. Springer Verlag, New York-Heidelberg, 1972. · Zbl 0253.31001
[12] Lewy, H. and Stampacchia, G.: On the smoothness of superharmonics which solve a minimum problem. J. Analyse Math. 23 (1970), 227-236. · Zbl 0206.40702 · doi:10.1007/BF02795502
[13] Lieb, E. H. and Loss, M.: Analysis. Amer. Math. Soc., Providence, RI, 1997.
[14] Lieberman, G. M.: Boundary regularity for solutions of degenerate elliptic equa- tions. Nonlinear Anal. 12 (1988), no. 11, 1203-1219. · Zbl 0675.35042 · doi:10.1016/0362-546X(88)90053-3
[15] Mastroeni, L. and Matzeu, M.: An integro-differential parabolic variational in- equality connected with the problem of the American option pricing. Z. Anal. An- wendungen 14 (1995), no. 4, 869-880. · Zbl 0878.45005 · doi:10.4171/ZAA/654
[16] Mastroeni, L. and Matzeu, M.: Nonlinear variational inequalities for jump- diffusion processes and irregular obstacles with a financial application. Nonlinear Anal. 34 (1998), no. 6, 889-905. · Zbl 1096.49506 · doi:10.1016/S0362-546X(97)00553-1
[17] Milakis, E. and Silvestre, L.: Regularity for the nonlinear Signorini problem. Adv. Math. 217 (2008), no. 3, 1301-1312. · Zbl 1132.35025 · doi:10.1016/j.aim.2007.08.009
[18] Mokrane, A. and Murat, F.: A proof of the Lewy-Stampacchia’s inequality by a penalization method. Potential Anal. 9 (1998), no. 2, 105-142. · Zbl 0918.35058 · doi:10.1023/A:1008649609888
[19] Mokrane, A. and Murat, F.: The Lewy-Stampacchia inequality for bilateral problems. Ricerche Mat. 53 (2004), no. 1, 139-182. · Zbl 1121.35066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.