CPAA
Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients
Shigeaki Koike Andrzej Świech
We establish local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.
keywords: viscosity solutions Fully nonlinear elliptic PDE local maximum principle.
DCDS
Optimal transport and large number of particles
Wilfrid Gangbo Andrzej Świech
We present an approach for proving uniqueness of ODEs in the Wasserstein space. We give an overview of basic tools needed to deal with Hamiltonian ODE in the Wasserstein space and show various continuity results for value functions. We discuss a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space.
keywords: Hamilton-Jacobi equations. conservative systems infinite dimensional Hamiltonian systems Optimal transport
CPAA
Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE
Robert Jensen Andrzej Świech
Two results are proved in the paper. The first is a uniqueness theorem for viscosity solutions of Dirichlet boundary value problems for Bellman-Isaacs equations with just measurable lower order terms. The second is a proof that there always exist maximal and minimal viscosity solutions of Dirichlet boundary value problems for fully nonlinear, uniformly elliptic PDE that are measurable in the $x$-variable.
keywords: viscosity solutions. Fully nonlinear elliptic PDE

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