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We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren’s monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.

This volume contains the proceedings of the AMS Special Session on New Developments in the Analysis of Nonlocal Operators, held from October 28-30, 2016, at the University of St. Thomas, Minneapolis, Minnesota. Over the last decade there has been a resurgence of interest in problems involving nonlocal operators, motivated by applications in many areas such as analysis, geometry, and stochastic processes. Problems represented in this volume include uniqueness for weak solutions to abstract parabolic equations with fractional time derivatives, the behavior of the one-phase Bernoulli-type free boundary near a fixed boundary and its relation to a Signorini-type problem, connections between fractional powers of the spherical Laplacian and zeta functions from the analytic number theory and differential geometry, and obstacle problems for a class of not stable-like nonlocal operators for asset price models widely used in mathematical finance. The volume also features a comprehensive introduction to various aspects of the fractional Laplacian, with many historical remarks and an extensive list of references, suitable for beginners and more seasoned researchers alike.

This volume contains the proceedings of the AMS Special Session on New Developments in the Analysis of Nonlocal Operators, held from October 28-30, 2016, at the University of St. Thomas, Minneapolis, Minnesota. Problems represented in this volume include uniqueness for weak solutions to abstract parabolic equations with fractional time derivatives and the behavior of the one-phase Bernoulli-type free boundary near a fixed boundary and its relation to a Signorini-type problem.

This volume contains the proceedings of the AMS Special Session on Harmonic Analysis and Partial Differential Equations, held from April 21-22, 2018, at Northeastern University, Boston, Massachusetts. The book features a series of recent developments at the interface between harmonic analysis and partial differential equations and is aimed toward the theoretical and applied communities of researchers working in real, complex, and harmonic analysis, partial differential equations, and their applications. The topics covered belong to the general areas of the theory of function spaces, partial differential equations of elliptic, parabolic, and dissipative types, geometric optics, free boundary problems, and ergodic theory, and the emphasis is on a host of new concepts, methods, and results.

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 .

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient conditions on the coefficients of the operator to obtain Hölder and Lipschitz continuous solutions. The class of nonlocal operators that we consider include non-Gaussian asset price models widely used in mathematical finance, such as Variance Gamma Processes and Regular Lévy Processes of Exponential type. In this context, the viscosity solutions that we analyze coincide with the prices of perpetual and finite expiry American options.