On a Weiss-type Almost Monotonicity Formula
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917438367465472 |
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| author | Sobral, Aelson |
| author_facet | Sobral, Aelson |
| contents | We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt--Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed in Araújo et al. [Calc. Var. Partial Differential Equations, 65, no.~1, Paper No.~24 (2026)]. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result. We conclude with a discussion of further extensions, including two-phase analogues. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12712 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a Weiss-type Almost Monotonicity Formula Sobral, Aelson Analysis of PDEs Primary 35R35. Secondary 35A21 We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt--Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed in Araújo et al. [Calc. Var. Partial Differential Equations, 65, no.~1, Paper No.~24 (2026)]. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result. We conclude with a discussion of further extensions, including two-phase analogues. |
| title | On a Weiss-type Almost Monotonicity Formula |
| topic | Analysis of PDEs Primary 35R35. Secondary 35A21 |
| url | https://arxiv.org/abs/2604.12712 |