Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Bekmaganbetov, Bekarys Dong, Hongjie |
| author_facet | Bekmaganbetov, Bekarys Dong, Hongjie |
| contents | We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08293 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights Bekmaganbetov, Bekarys Dong, Hongjie Analysis of PDEs 35K65, 35K67, 35K20, 35D30, 46E35 We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed. |
| title | Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights |
| topic | Analysis of PDEs 35K65, 35K67, 35K20, 35D30, 46E35 |
| url | https://arxiv.org/abs/2410.08293 |