Abstract
In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L 1-data. We also extend the Lewy–Stampacchia inequalities to the general framework of L 1-data and show convergence and stability results. We then prove that the free boundary has finite (N − 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p > 2.
Article PDF
Similar content being viewed by others
References
Adams R.A.: Sobolev Spaces. Pure and Applied Mathematics. Vol. 65. Academic Press, New York-London (1975)
Aharouch L., Benkirane A., Rhoudaf M.: Existence results for some unilateral problems without sign condition with obstacle free in Orlicz spaces. Nonlinear Anal. 68(8), 2362–2380 (2008)
Benkirane A., Bennouna J.: Existence and uniqueness of solution of unilateral problems with L 1-data in Orlicz spaces. Ital. J. Pure Appl. Math. 16, 87–102 (2004)
Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vázquez J.L.: An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2), 241–273 (1995)
Boccardo L., Gallouët T., Orsina L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(5), 539–551 (1996)
Boccardo L., Cirmi G.R.: Existence and uniqueness of solution of unilateral problems with L 1 data. J. Convex Anal. 6(1), 195–206 (1999)
Caffarelli L.A.: A Remark on the Hausdorff Measure of a Free Boundar, and the Convergence of the Coincidence Sets. Bolletino UMI 18((5), 109–113 (1981)
Challal S., Lyaghfouri A.: Porosity of free boundaries in A-obstacle problems. Nonlinear Anal. Theory Methods Appl. 70(7), 2772–2778 (2009)
Challal S., Lyaghfouri A.: Hölder continuity of solutions to the A-Laplace equation involving measures. Commun. Pure Appl. Anal. 8(5), 1577–1583 (2009)
Challal S., Lyaghfouri A.: Second order regularity for the A-Laplace operator. Mediterr. J. Math. 7(3), 283–297 (2010)
Challal, S., Lyaghfouri, A.: Some properties of A-harmonic functions in the plane. In: Preparation
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
Karp L., Kilpeläinen T., Petrosyan A., Shahgholian H.: On the porosity of free boundaries in degenerate variational inequalities. J. Differ. Equ. 164, 110–117 (2000)
Lieberman G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations. Comm. Partial Differ. Equ. 16(2–3), 311–361 (1991)
Lieberman G.M.: Regularity of solutions to some degenerate double obstacle problems. Indiana Univ. Math. J. 40(3), 1009–1028 (1991)
Lee K., Shahgholian H.: Hausdorff measure and stability for the p-obstacle problem, 2 < p < ∞. J. Differ. Equ. 195(1), 14–24 (2003)
Martinez S., Wolanski N.: A minimum problem with free boundary in Orlicz-Sobolev spaces. Adv. Math. 218, 1914–1971 (2008)
Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI (1997)
Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, 134. Mathematical Notes, 114. North-Holland Publishing Co., Amsterdam (1987)
Rodrigues J.F.: Stability remarks to the obstacle problem for p-Laplacian type equations. Calc. Var. Partial Differ. Equ. 23(1), 51–65 (2005)
Rodrigues J.F., Sanchón M., Urbano J.M.: The obstacle problem for nonlinear elliptic equations with variable growth and L 1-data. Monatsh. Math. 154(4), 303–322 (2008)
Ziemer W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Challal, S., Lyaghfouri, A. & Rodrigues, J.F. On the A-obstacle problem and the Hausdorff measure of its free boundary. Annali di Matematica 191, 113–165 (2012). https://doi.org/10.1007/s10231-010-0177-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-010-0177-7
Keywords
- Obstacle problem
- Entropy solution
- A-Laplace operator
- Lewy–Stampacchia inequalities
- Stability
- Free boundary
- Hausdorff measure