Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant
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| Language: | Vietnamese |
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2023
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| Online Access: | https://dlib.phenikaa-uni.edu.vn/handle/PNK/7629 |
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oai:localhost:PNK-76292023-04-06T04:07:35Z Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant Josep M., Gallegos Let Ω⊂Rd be a C1 domain Assume u is harmonic in Ω CC BY Let Ω⊂Rd be a C1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d×d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div(A(x)∇u)=0 in Ω, and u vanishes on Σ=∂Ω∩B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i such that u|Bi∩Ω does not change sign and K∖⋃iBi has Minkowski dimension smaller than d−1−ϵ for any compact K⊂Σ. We also find upper bounds for the (d−1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Σ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Σ is bounded except for a set of Hausdorff dimension at most d−1−ϵ. 2023-04-06T04:07:35Z 2023-04-06T04:07:35Z 2023 Book https://dlib.phenikaa-uni.edu.vn/handle/PNK/7629 vi application/pdf Springer |
| institution |
Digital Phenikaa |
| collection |
Digital Phenikaa |
| language |
Vietnamese |
| topic |
Let Ω⊂Rd be a C1 domain Assume u is harmonic in Ω |
| spellingShingle |
Let Ω⊂Rd be a C1 domain Assume u is harmonic in Ω Josep M., Gallegos Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| description |
CC BY |
| format |
Book |
| author |
Josep M., Gallegos |
| author_facet |
Josep M., Gallegos |
| author_sort |
Josep M., Gallegos |
| title |
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| title_short |
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| title_full |
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| title_fullStr |
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| title_full_unstemmed |
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant |
| title_sort |
size of the zero set of solutions of elliptic pdes near the boundary of lipschitz domains with small lipschitz constant |
| publisher |
Springer |
| publishDate |
2023 |
| url |
https://dlib.phenikaa-uni.edu.vn/handle/PNK/7629 |
| _version_ |
1762456093184753664 |
| score |
8.891787 |