ABSTRACT

This work deals with the mth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space $L_{q)}\left(\mathrm{\Omega }\right),1<q<+\mathrm{\infty }$. These spaces are non-separable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this aim, we consider some subspace, where the infinitely differentiable functions are dense. Then we prove that this subspace is invariant with respect to the singular integral operator and with respect to the multiplication operator by a function from $L_{\mathrm{\infty }}$. Finally, using classical method of parametrics, we prove the existence in the small of the solution to the considered equation in $W_{q)}^m\left(\mathrm{\Omega }\right)$.

摘要

这项工作处理由grand-Lebesgue空间的范数生成的grand-Sobolev空间中具有非光滑系数的m阶椭圆方程${升}_{q)}\left(\mathrm{\Omega }\right),1<q<+\mathrm{\infty }$. 这些空间是不可分的，因此，要使用经典方法来处理这些空间中的可解性问题，您需要修改这些方法。为此，我们考虑一些子空间，其中无限可微函数是密集的。然后我们通过一个函数证明这个子空间对于奇异积分算子和乘法算子是不变的${升}_{\mathrm{\infty }}$. 最后，使用经典的参数化方法，我们证明了所考虑方程的解的小处存在性${宽}_{q)}^{米}\left(\mathrm{\Omega }\right)$.