RECENT PAPERS

 

 ● D. Danielli, N. Garofalo, A. Petrosyan, and T. To, Optimal regularity and the Free Boundary in the Parabolic Signorini Problem. To appear in Memoirs AMS.

Abstract: 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.

● D. Danielli  and S. Salsa, Obstacle problems involving the fractional Laplacian. To appear in Recent developments in the Nonlocal Theory (T. Kuusi and G. Palatucci, Eds.), Book Series on Measure Theory, De Gruyter, Berlin, 2017.

● D. Danielli, A singular perturbation approach to a two phase parabolic free boundary problem arising in combustion theory, preprint. 

Abstract: We study the uniform properties of solutions to a singular perturbation problem associated to a general second order parabolic operator.  In particular, our main results show that, under suitable assumptions, the limit function is a pointwise solution to a free boundary problem that naturally arises in combustion theory.

 ● D. Danielli, A. Petrosyan, and C. Pop, Obstacle problems for nonlocal operators. To appear in Contemporary Mathematics.

Abstract: 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.

● D. Danielli, A. Petrosyan, and C. Pop, Obstacle problems for nonlocal operators: A brief overview, To appear in ISNPS 2018 Proceedings.

Abstract: In this note, we give a brief overview of obstacle problems for nonlocal operators, focusing on the applications to financial mathematics. The class of nonlocal operators that we consider can be viewed as infinitesimal generators of non-Gaussian asset price models, such as Variance Gamma Processes and Regular Levy Processes of Exponential type. In this context, we analyze the existence, uniqueness and regularity of viscosity solutions to obstacle problems which correspond to prices of perpetual and finite expiry American options.

● A. Banerjie, D. Danielli, N. Garofalo, and A. Petrosyan, The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, preprint.

Abstract: 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. Our main result establishes the complete structure and regularity of the singular free boundary. To achieve it, we prove new Weiss and Monneau type monotonicity formulas.

● T. Backing, D. Danielli, and R. Jain, Regularity results for a penalized boundary obstacle problem, in preparation.

Abstract: We study the optimal regularity of solutions and the structure of the free boundary in a two-penalty boundary obstacle problem modeling fluid flow through a permeable membrane.