A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913617110106112 |
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| author | Wang, Xing Zhang, Cheng |
| author_facet | Wang, Xing Zhang, Cheng |
| contents | On smooth compact manifolds with smooth boundary, we first establish the sharp lower bounds for the restrictions of harmonic functions in terms of their frequency functions, by using a combination of microlocal analysis and frequency function techniques by Almgren and Garofalo-Lin. The lower bounds can be saturated by Steklov eigenfunctions on Euclidean balls and a family of symmetric warped product manifolds. Moreover, as in Sogge and Taylor, we analyze the interior behavior of harmonic functions by constructing a parametrix for the Poisson integral operator and calculate its composition with the spectral cluster. By using microlocal analysis, we obtain several sharp estimates for the harmonic functions whose traces are quasimodes on the boundary. As applications, we establish the almost-orthogonality, bilinear estimates and transversal restriction estimates for Steklov eigenfunctions, and discuss the numerical approximation of harmonic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions Wang, Xing Zhang, Cheng Analysis of PDEs Numerical Analysis Classical Analysis and ODEs Spectral Theory 58C40, 58J50, 58J32, 65N15 On smooth compact manifolds with smooth boundary, we first establish the sharp lower bounds for the restrictions of harmonic functions in terms of their frequency functions, by using a combination of microlocal analysis and frequency function techniques by Almgren and Garofalo-Lin. The lower bounds can be saturated by Steklov eigenfunctions on Euclidean balls and a family of symmetric warped product manifolds. Moreover, as in Sogge and Taylor, we analyze the interior behavior of harmonic functions by constructing a parametrix for the Poisson integral operator and calculate its composition with the spectral cluster. By using microlocal analysis, we obtain several sharp estimates for the harmonic functions whose traces are quasimodes on the boundary. As applications, we establish the almost-orthogonality, bilinear estimates and transversal restriction estimates for Steklov eigenfunctions, and discuss the numerical approximation of harmonic functions. |
| title | A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions |
| topic | Analysis of PDEs Numerical Analysis Classical Analysis and ODEs Spectral Theory 58C40, 58J50, 58J32, 65N15 |
| url | https://arxiv.org/abs/2412.13955 |