Anisotropic and p,q-nonlinear partial differential equations

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.