Separable anisotropic elliptic operators and applications (Q653842)

The author focuses on boundary value problems (BVPs) for anisotropic differential-operator equations (DOE) of higher-order with variable coefficients. In Section 3, the author discusses BVPs for anisotropic type DOEs with constant coefficients of the form \[ (O+\lambda)u:=\sum_{k=1}^na_kD_k^{l_k}u(x)+(A+\lambda)u=f(x),\;x\in{\mathbb{R}}^n. \] An isomorphism between \(L_p({\mathbb{R}};E)\) and Sobolev-Lions spaces is established. In Section 4, separability and positivity of the operator \(O\) are established. In Section 5, maximal regularity properties are derived for the parabolic problem \[ \begin{multlined}\frac{\partial{u(t, x)}}{{\partial{t}}}+\sum_{k=1}^na_k(t, x)D^{l_k}_ku(t, x)+\\ + A(x) u(t,x)+\sum_{| \alpha:l| <1}A_{\alpha}(x)D^{\alpha}u(t,x)=f(t, x),\;t\in (0, 1),\;x\in{\mathbb{R}}^n_+,\end{multlined} \] under the condition \[ \sum_{i=0}^{m_k,j} \alpha_{ji}u^{[i]}(t, x^1, 0) = 0, \] where \( j= 1, 2,\dots, d\), \(0<d< l_n,\) \(u(0, x) = 0\), \(x^1 = (x_1, x_2,\dots, x_{n-1}).\) Finally, in section 6, the results are applied to infinite systems of anisotropic partial differential equations with variable coefficients.