Free boundary regularity in obstacle problems with a degenerate forcing term
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908827587182592 |
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| author | Liu, Yong |
| author_facet | Liu, Yong |
| contents | In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional unit ball, where $f(x)=|x|$ is a degenerate forcing term. The key challenge stems from the degeneracy of $f(x)$, which leads to a more complex structure of the free boundary compared to the classical setting. To analyze it, we introduce the epiperimetric inequality developed by Weiss (Invent Math 138:23-50, 1999). Although this powerful tool was firstly introduced for the classical obstacle problem characterized by $f(x)>0$ in B_1, it also proves effective in our degenerate setting. This allows us to first obtain the decay rate of the Weiss energy for all blow-ups at the origin, which in turn implies the uniqueness of the blow-up profiles. With this uniqueness established, we then prove a very weak directional monotonicity properties satisfied by the solutions. This finally yields the regularity of the free boundary at the origin if the origin is a regular point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Free boundary regularity in obstacle problems with a degenerate forcing term Liu, Yong Analysis of PDEs In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional unit ball, where $f(x)=|x|$ is a degenerate forcing term. The key challenge stems from the degeneracy of $f(x)$, which leads to a more complex structure of the free boundary compared to the classical setting. To analyze it, we introduce the epiperimetric inequality developed by Weiss (Invent Math 138:23-50, 1999). Although this powerful tool was firstly introduced for the classical obstacle problem characterized by $f(x)>0$ in B_1, it also proves effective in our degenerate setting. This allows us to first obtain the decay rate of the Weiss energy for all blow-ups at the origin, which in turn implies the uniqueness of the blow-up profiles. With this uniqueness established, we then prove a very weak directional monotonicity properties satisfied by the solutions. This finally yields the regularity of the free boundary at the origin if the origin is a regular point. |
| title | Free boundary regularity in obstacle problems with a degenerate forcing term |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.10617 |