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.

中文翻译：

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各向异性和p，q-非线性偏微分方程

各向异性偏微分方程由于在双相和多相变分能量以及积分形式的各向异性能量中的应用而引起了数学文献的极大兴趣。更具体地说，从数学的角度来看，我们需要考虑经典的拉普拉斯椭圆和抛物线偏微分方程以及非线性p -Laplacian方程的推广，后者自然会引起新的有趣的数学问题，如今仅能部分解决在各向异性的情况下p，  q-增长非线性椭圆和抛物线偏微分方程。在这种情况下，我们描述了一些“数学病理学”。更确切地说，是由某些各向异性能量产生的电势中的某些奇点。如果能量的各向异性在某些方向上太大，则会出现奇点，而不仅是能量在所有方向上都是各向同性的，而且即使我们允许具有轻度各向异性的能量积分，也不会出现这些奇点。