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Abstract The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.
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Mappings of generalized finite distortion and continuity
Abstract We study continuity properties of Sobolev mappings , , that satisfy the following generalized finite distortion inequalityfor almost every . Here and are measurable functions. Note that when , we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion , where a sharp condition for continuity is that is in the Zygmund space for some . We also show that one can slightly relax the boundedness assumption on to an exponential class with , and still obtain continuous solutions when with . On the other hand, for all with , we construct a discontinuous solution with and , including an example with and .
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- Award ID(s):
- 2154943
- PAR ID:
- 10482268
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 109
- Issue:
- 1
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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