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Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds

  • Alessandro Goffi and Francesco Pediconi ORCID logo EMAIL logo
From the journal Forum Mathematicum


In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner identity, and leads to “semilinear Calderón–Zygmund” type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed.

Communicated by Karin Melnick

Award Identifier / Grant number: 2017JZ2SW5

Funding statement: This work has been written while the second-named author was a postdoctoral fellow at the Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze. He is grateful to the department for the hospitality. The first-named author is member of GNAMPA of INdAM and has been partially supported by the GNAMPA project “Mean Field Games: modelli e sviluppi”. The second-named author is member of GNSAGA of INdAM and has been supported by the project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5).


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Received: 2022-04-15
Published Online: 2023-01-30
Published in Print: 2023-03-01

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