Elliptic boundary-value problems in some distribution spaces of generalized smoothness
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915523361505280 |
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| author | Anop, Anna Murach, Aleksandr |
| author_facet | Anop, Anna Murach, Aleksandr |
| contents | We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic boundary-value problems in some distribution spaces of generalized smoothness Anop, Anna Murach, Aleksandr Analysis of PDEs 35J40, 46E35, 46B70 We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems. |
| title | Elliptic boundary-value problems in some distribution spaces of generalized smoothness |
| topic | Analysis of PDEs 35J40, 46E35, 46B70 |
| url | https://arxiv.org/abs/2503.20747 |