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Anisotropic and p, q-nonlinear partial differential equations

  • Physicochemical Properties of Matter
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Abstract

Anisotropic partial differential equations recently received a large interest in the mathematical literature, due to their applications to double and multiphase variational energies, as well as to anisotropic energies in integral form. More specifically, from a mathematical point of view, we need to consider a generalization of the classical Laplacian elliptic and parabolic partial differential equations, as well as the nonlinear p-Laplacian equations, which naturally arises new and interesting mathematical questions, nowadays only partially solved in the context of anisotropic p, q-growth nonlinear elliptic and parabolic partial differential equations. In this context, we describe some “mathematical pathologies”; more precisely, some singularities in the potential generated by some anisotropic energies. The singularities appear if the anisotropy of the energy is too large in some directions, while these singularities do not appear, not only if the energy is isotropic with respect to all directions, but also even if we allow an energy integral with a mild anisotropy.

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Authors and Affiliations

  1. University of Firenze, Firenze, Italy

    Paolo Marcellini

  2. Dipartimento di Matematica e Informatica, Viale Morgagni 67/A, 50134, Firenze, Italy

    Paolo Marcellini

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This paper is the peer-reviewed version of a contribution presented at the Conference on “Anisotropic Properties of Matter”, organized by Giovanni Ferraris and held at Accademia Nazionale dei Lincei in Rome, October 16–17, 2019.

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Marcellini, P. Anisotropic and p, q-nonlinear partial differential equations. Rend. Fis. Acc. Lincei 31, 295–301 (2020). https://doi.org/10.1007/s12210-020-00885-y

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