Constructing convex solutions via Perron’s method
Mikhail Feldman,John McCuan +1 more
TLDR
In this paper, the authors construct convex solutions for certain elliptic boundary value problems via Perron's method via weak solutions in the viscosity sense, and their construction follows work of Ishii (Duke Math. Differential Equations, 182 (2) 298-343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves.Abstract:
In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J.,
55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05read more
Citations
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State-Constraint Static Hamilton--Jacobi Equations in Nested Domains
TL;DR: In this paper, state-constraint static Hamilton-Jacobi equations were studied in a sequence of domains such that a subset of the Hamilton--Jacobi equation can be computed for all domains in the sequence.
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Viscosity solutions of HJB equations arising from the valuation of European passport options
TL;DR: The mathematical foundation for pricing the European passport option is established and the pricing equation is derived using the dynamic programming principle, and the comparison principle, uniqueness and convexity preserving of the viscosity solutions of related HJB equation are proved.
Journal ArticleDOI
The Valuation of American Passport Options: A Viscosity Solution Approach
TL;DR: The mathematical foundation for pricing the American passport option is rigorously established, using the dynamic programming principle, and it is proved that the option value is a viscosity solution of variational inequality, which is a fully nonlinear equation.
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Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model
TL;DR: It is proved the comparison principle, uniqueness and convexity preserving for the viscosity solutions of related pricing equations of degenerate parabolic type are proved.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Elliptic Partial Differential Equations of Second Order
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TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
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User’s guide to viscosity solutions of second order partial differential equations
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Perron’s method for Hamilton-Jacobi equations
TL;DR: On considere l'existence des solutions d'equations aux derivees partielles non lineaires scalaires d'ordre 1: F(x, u, Du) = 0 dans Ω, ou Ω est un sous-ensemble ouvert de R N, F: Ω×R×R N →R →R est continue, u:Ω→R est l'inconnue as mentioned in this paper.