TY - JOUR

T1 - The sub-elliptic obstacle problem

T2 - C1, α regularity of the free boundary in Carnot groups of step two

AU - Danielli, Donatella

AU - Garofalo, Nicola

AU - Petrosyan, Arshak

N1 - Funding Information:
* Corresponding author. E-mail addresses: danielli@math.purdue.edu (D. Danielli), garofalo@math.purdue.edu (N. Garofalo), arshak@math.purdue.edu (A. Petrosyan). 1 Supported in part by NSF grants DMS-0002801 and CAREER DMS-0239771. 2 Supported in part by NSF grant DMS-0300477. 3 Supported in part by NSF grant DMS-0401179.

PY - 2007/6/1

Y1 - 2007/6/1

N2 - The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361-398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland-Stein (optimal) Lipschitz class Γloc1, 1 (the analogue of the well-known Cloc1, 1 interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step r = 2, the free boundary is (Euclidean) C1, α near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3-4) (1977) 155-184; MR0454350 (56 #12601)].

AB - The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361-398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland-Stein (optimal) Lipschitz class Γloc1, 1 (the analogue of the well-known Cloc1, 1 interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step r = 2, the free boundary is (Euclidean) C1, α near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3-4) (1977) 155-184; MR0454350 (56 #12601)].

KW - Carnot groups

KW - NTA domains

KW - Regularity of free boundary

KW - Sub-elliptic obstacle problem

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U2 - 10.1016/j.aim.2006.08.008

DO - 10.1016/j.aim.2006.08.008

M3 - Article

AN - SCOPUS:33947604997

VL - 211

SP - 485

EP - 516

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -