Matching Tag: one-phase-free-boundary
Fernández-Real–Ros-Oton conjecture. Suppose that u : R n → ( 0 , ∞ ) u:\mathbb{R}^n\to(0,\infty) u : R n → ( 0 , ∞ ) I 1 I_1 I 1 n ≤ n e ∗ − 1 n\leq n_e^*-1 n ≤ n e ∗ − 1 u u u
Assume (A1): β \beta β C 1 C^1 C 1 supp β = [ 0 , 1 ] \textrm{supp}\,\beta=[0,1] supp β = [ 0 , 1 ] ∫ 0 1 β = 1 \int_0^1\beta=1 ∫ 0 1 β = 1 Φ ′ = β \Phi'=\beta Φ ′ = β
Assume (A1)–(A2): β \beta β C 1 C^1 C 1 supp β = [ 0 , 1 ] \textrm{supp}\,\beta=[0,1] supp β = [ 0 , 1 ] ∫ 0 1 β = 1 \int_0^1\beta=1 ∫ 0 1 β = 1 u ≥ 0 u\geq0 u ≥ 0 ∥ ∇ u ∥ L ∞ ( R n ) < ∞ \|\nabla u\|_{L^\infty(\mathbb{R}^n)}<\infty ∥∇ u ∥ L ∞ ( R n ) < ∞
