Sobolev-poincaré Implies


Ω |∇u| dx )1/p holds for 1 ≤ p < n whenever u is smooth, uΩ = |Ω|−1 ∫ Ω u dx, and Ω ⊂ R is bounded and satisfies the cone condition. By the density of smooth functions, (1.1) then holds for all functions in the Sobolev space W (Ω) consisting of all functions in L(Ω) whose distributional gradients belong to L(Ω). For 1 < p < n, inequality (1.1) was proved by… (More)


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