What the study found
The study found that, in outward cuspidal domains with a weighted measure, Sobolev spaces can be embedded into weighted Lebesgue spaces. The authors use this to establish solvability of a nonlinear Neumann spectral problem and to derive estimates for the corresponding weighted Neumann eigenvalues.
Why the authors say this matters
The authors indicate that these embeddings provide solvability for the Neumann spectral problem in the setting they study. They also conclude that the embeddings allow estimates of the weighted Neumann eigenvalues.
What the researchers tested
The researchers studied a nonlinear Neumann eigenvalue problem in outward cuspidal domains, which are domains with outward-pointing cusp-shaped boundaries, under a weighted measure. They used composition operators on Sobolev spaces to establish the needed embeddings.
What worked and what didn't
The embeddings from Sobolev spaces into weighted Lebesgue spaces were established in this setting. According to the abstract, these embeddings led to solvability of the Neumann spectral problem and provided estimates for the weighted Neumann eigenvalues; no negative or failed results are described.
What to keep in mind
The abstract does not give numerical eigenvalue bounds or detailed assumptions beyond outward cuspidal domains and a weighted measure. It also does not describe limitations, comparisons with other settings, or examples.
Key points
- The article studies a nonlinear Neumann eigenvalue problem in outward cuspidal domains.
- Sobolev spaces are shown to embed into weighted Lebesgue spaces in the setting considered.
- The embeddings are used to establish solvability of the Neumann spectral problem.
- The study provides estimates for the corresponding weighted Neumann eigenvalues.
- The abstract does not report detailed bounds or limitations.
Disclosure
- Research title:
- Weighted Neumann eigenvalues are estimated in outward cuspidal domains
- Authors:
- Alexander Menovschikov, Alexander Ukhlov
- Institutions:
- Ben-Gurion University of the Negev, National Research University Higher School of Economics
- Publication date:
- 2026-04-21
- OpenAlex record:
- View
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