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Sobolev Spaces
Mathematics — Calculus · sobolev-spaces-membership
Problem Denote by U U U { x ∈ R 2 ∣ ∣ x 1 ∣ < 1 , ∣ x 2 ∣ < 1 } \{x \in \mathbb{R}^2 \mid |x_1| < 1, |x_2| < 1\} { x ∈ R 2 ∣ ∣ x 1 ∣ < 1 , ∣ x 2 ∣ < 1 } u ( x ) = { 1 − x 1 if x 1 > 0 , ∣ x 2 ∣ < x 1 1 + x 1 if x 1 < 0 , ∣ x 2 ∣ < − x 1 1 − x 2 if x 2 > 0 , ∣ x 1 ∣ < x 2 1 + x 2 if x 2 < 0 , ∣ x 1 ∣ < − x 2 . u(x) = \begin{cases} 1 - x_1 & \text{if } x_1 > 0, \ |x_2| < x_1 \\ 1 + x_1 & \text{if } x_1 < 0, \ |x_2| < -x_1 \\ 1 - x_2 & \text{if } x_2 > 0, \ |x_1| < x_2 \\ 1 + x_2 & \text{if } x_2 < 0, \ |x_1| < -x_2. \end{cases} u ( x ) = ⎩ ⎨ ⎧ 1 − x 1 1 + x 1 1 − x 2 1 + x 2 if x 1 > 0 , ∣ x 2 ∣ < x 1 if x 1 < 0 , ∣ x 2 ∣ < − x 1 if x 2 > 0 , ∣ x 1 ∣ < x 2 if x 2 < 0 , ∣ x 1 ∣ < − x 2 . 1 ≤ p ≤ ∞ 1 \leq p \leq \infty 1 ≤ p ≤ ∞ u u u W 1 , p ( U ) W^{1,p}(U) W 1 , p ( U ) Browse the problem library