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Pólya–Szegö type inequality and imbedding theorems for weighted Sobolev spaces

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Abstract

In this paper we will establish a new Pólya–Szegö type inequality for a weighted gradient of a function on \({\mathbb {R}}^2\) with respect to a weighted area. In order to do that we need to study an isoperimetric problem for the weighted area. We then apply the inequality to prove embedding theorems for weighted Sobolev spaces and to calculate the best constant in the Sobolev imbedding theorems. In our upcoming manuscript the obtained results in this note will be used to study boundary value problems for semilinear degenerate elliptic equations, see Luyen et al. (arXiv:2303.14661).

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Acknowledgements

This research is funded by the International Center for Research and Post-graduate Training in Mathematics-Institute of Mathematics-Vietnam Academy of Science and Technology under the Grant ICRTM01 2021.03. We thank the referees for helpful comments.

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Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    N. Q. Nga & N. M. Tri

  2. University of Sciences, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    D. A. Tuan

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  1. N. Q. Nga
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  2. N. M. Tri
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  3. D. A. Tuan
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All authors reviewed the manuscript. All authors did the work toghether.

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Correspondence to D. A. Tuan.

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Nga, N.Q., Tri, N.M. & Tuan, D.A. Pólya–Szegö type inequality and imbedding theorems for weighted Sobolev spaces. Anal.Math.Phys. 14, 20 (2024). https://doi.org/10.1007/s13324-024-00877-3

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  • DOI: https://doi.org/10.1007/s13324-024-00877-3

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