A new monotonicity formula for quasilinear elliptic free boundary problems
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916948465418240 |
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| author | Karakhanyan, Aram |
| author_facet | Karakhanyan, Aram |
| contents | We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_ΩF(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap Ω), \] where $F$ is a density function satisfying some structural conditions.
The onus of proof lies with the careful analysis of the ghost function, the gradient part in the Helmholtz-Wéyl decomposition of a nonlinear flux that appears in the domain variation formula for $J(u)$.
As an application we prove full regularity for a class of quasilinear Bernoulli type free boundary problems in $\R^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new monotonicity formula for quasilinear elliptic free boundary problems Karakhanyan, Aram Analysis of PDEs We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_ΩF(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap Ω), \] where $F$ is a density function satisfying some structural conditions. The onus of proof lies with the careful analysis of the ghost function, the gradient part in the Helmholtz-Wéyl decomposition of a nonlinear flux that appears in the domain variation formula for $J(u)$. As an application we prove full regularity for a class of quasilinear Bernoulli type free boundary problems in $\R^3$. |
| title | A new monotonicity formula for quasilinear elliptic free boundary problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.01175 |