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A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian

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Abstract

In this manuscript we study an inhomogeneous obstacle type problem involving a fractional p-Laplacian type operator. First, we focus our attention in establishing existence and uniform estimates for any family of solutions \(\{u_p\}_{p \ge 2}\) which depend on the data of the problem and universal parameters. Next, we analyze the asymptotic behavior of such a family as \(p \rightarrow \infty \). At this point, we prove that \(\displaystyle \lim \nolimits _{p\rightarrow \infty } u_p(x) = u_{\infty }(x)\) there exists (up to a subsequence), verifies a limiting obstacle type problem in the viscosity sense, and it is an s-Hölder continuous function. We also present several explicit examples, as well as further features of the limit solutions and their free boundaries. In order to establish our results we overcome several technical difficulties and develop new strategies, which were not present in the literature for this type of problems. Finally, we remark that our results are new even for problems governed by fractional p-Laplacian operator, as well as they extend the previous ones by dealing with more general non-local operators, source terms and boundary data.

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Acknowledgements

We would like to thank Prof. Julio D. Rossi for sharing his intuition about the s-Hölder extension with obstacle constraint in the Sect. 2.3, as well as his insightful clarifications about the correct definition of limit operator. This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina) under grant PIP GI No. 11220150100036CO and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (PNPD/CAPES-UnB-Brazil). J.V. da Silva and A. Salort are members of CONICET.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Brasília, DF, 70910-900, Brazil

    João Vitor da Silva

  2. Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n, Buenos Aires, Argentina

    João Vitor da Silva & Ariel M. Salort

  3. Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n., Buenos Aires, Argentina

    Ariel M. Salort

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  1. João Vitor da Silva
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Correspondence to João Vitor da Silva.

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Communicated by L. Ambrosio.

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da Silva, J.V., Salort, A.M. A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian. Calc. Var. 58, 127 (2019). https://doi.org/10.1007/s00526-019-1573-5

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