OAI.Bernoulli.nonflat_in_seven

The theorem states that there exists a function u: ℝ⁷ → ℝ that is nonnegative almost everywhere, belongs locally to H¹, and is a global minimizer of the one-phase Bernoulli energy E_B(u) = ∫B (|∇u|² + 1{u>0}) dx. Here ∇u is a distributional gradient, and global minimality means that, for every open ball B of positive radius and every nonnegative almost-everywhere Sobolev competitor v ∈ H¹(B) with v − u ∈ H¹₀(B), one has E_B(u) ≤ E_B(v). Membership in H¹₀(B) requires approximation by smooth functions compactly supported in B, with both the functions and their gradients converging in L². The function is nonzero in the sense that it is not equal to zero almost everywhere, and it is homogeneous of degree one: for each real r > 0, u(rx) = r u(x) for almost every x. Nevertheless, it is not flat: there is no unit vector e ∈ ℝ⁷ for which u(x) = max(⟨x,e⟩, 0) almost everywhere. All almost-everywhere statements and integrals use Lebesgue measure.