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In this paper, we consider the obstacle problem for the fractional Laplace operator (- Δ) s in the Euclidian space R n in the case where 1 < s < 2 . As first observed in Yang (On higher order extensions for the fractional Laplacian arXiv:1302.4413, 2013), the problem can be extended to the upper half-space R + n + 1 to obtain a thin obstacle problem for the weighted b-biharmonic operator Δ b 2 , where Δ b U = y - b ∇ · (y b ∇ U) . Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and C loc 1 , 1 (R n) ∩ H 1 + s (R n) -regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in Schild (Ann Scuola Norm Sup Pisa Cl Sci (4) 11(1):87–122, 1984) and Schild (Ann Scuola Norm Sup Pisa Cl Sci (4) 13(4):559–616, 1986) from the case b = 0 to the general case - 1 < b < 1 . [ABSTRACT FROM AUTHOR]

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight | y | a for a ∈ (- 1 , 1) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator (∂ t - Δ x) s for s ∈ (0 , 1) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ( a = 0 ). [ABSTRACT FROM AUTHOR]

We provide a solution to the isoperimetric problem in the Heisenberg group n when the competing sets belong to a restricted class of C2 graphs. Within this restricted class we characterize the isoperimetric profiles as the bubble sets (1.5) (modulo nonisotropic dilations and left-translations). We also compute the isoperimetric constant. 2000 Mathematics Subject Classification: 53C17, 49K20. [ABSTRACT FROM AUTHOR]

Abstract: 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 (the analogue of the well-known 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 , the free boundary is (Euclidean) 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)]. [Copyright &y& Elsevier]

In this paper we proveregularity (near flat points) of the free boundaryin the Alt-Caffarelli type minimum problem for thep-Laplace operator: [ABSTRACT FROM AUTHOR]

In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian (− Δ) 3 / 2 , as first observed in Yang (2013). We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type. [ABSTRACT FROM AUTHOR]

Establishes sharp capacitary estimates for Carnot-Caratheodory rings associated to a system of vector fields of Hormander type. Generalization of fundamental estimates obtained independently by Sanchez-Calle and Nagel, Stein and Wainger; Fundamental solutions in groups of Heisenberg type; Maximum principle; Removable singularities; Behavior of singular solutions.

In this paper we establish the space-like strong unique continuation for nonlocal equations of the type (∂ t − Δ) s u = V u , for 0 < s < 1. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis. [ABSTRACT FROM AUTHOR]