Galerkin approximations in problems with anisotropic p(·)-Laplacian

ABSTRACT

In this paper, we consider the Dirichlet problem in a bounded domain of ${\mathbb{R}}^d$ for an elliptic equation with an anisotropic $p(·)$-Laplace operator. Anisotropy is created by a measurable symmetric matrix A which stands under the divergence operator in the $p(·)$-Laplacian. A Cordes-type condition is imposed on the matrix A to ensure the monotonicity property of the operator. We study the so-called variational solutions to the Dirichlet problem and construct Galerkin approximations for them. We estimate the difference between the exact and approximate solutions and the difference between corresponding flows.

KEYWORDS:

• Galerkin approximations
• anisotropic Laplacian with variable exponent
• variable exponent spaces
• error estimate

AMS SUBJECT CLASSIFICATIONS:

• 35J92
• 35A35

Notes

No potential conflict of interest was reported by the authors.

In memory of Vasili Vasilievich Zhikov.

Additional information

Funding

The authors were supported by the Grant of the Ministry of Education and Science of the Russian Federation [grant number 1.3270.2017/4.6].