Abstract / Description of output
In this paper we classify the nonnegative global minimizers of the functional
JF(u)=∫ΩF(|∇u|2)+λ2χ{u>0},
where F satisfies some structural conditions and is the characteristic function of a set. We compute the second variation of the energy and study the properties of the stability operator. The free boundary can be seen as a rectifiable varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.
JF(u)=∫ΩF(|∇u|2)+λ2χ{u>0},
where F satisfies some structural conditions and is the characteristic function of a set. We compute the second variation of the energy and study the properties of the stability operator. The free boundary can be seen as a rectifiable varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.
| Original language | English |
|---|---|
| Pages (from-to) | 981-999 |
| Journal | Annales de l'Institut Henri Poincaré C |
| Volume | 38 |
| Issue number | 4 |
| Early online date | 5 Oct 2020 |
| DOIs | |
| Publication status | Published - 31 Aug 2021 |