partial-differential-equations-by-evans · Problem 17

Sobolev Spaces

Mathematics — Calculus · sobolev-spaces-absolute-value

Problem

Assume 1<p<∞1 < p < \infty, and UU is bounded. (i) Prove that if u∈W1,p(U)u \in W^{1,p}(U), then ∣u∣∈W1,p(U)|u| \in W^{1,p}(U). (ii) Prove u∈W1,p(U)u \in W^{1,p}(U) implies u+,u−∈W1,p(U)u^+, u^- \in W^{1,p}(U), and Du+={Dua.e. on {u>0}0a.e. on {u≤0},Du^+ = \begin{cases} Du & \text{a.e. on } \{u > 0\} \\ 0 & \text{a.e. on } \{u \leq 0\}, \end{cases} Du−={0a.e. on {u≥0}−Dua.e. on {u<0}.Du^- = \begin{cases} 0 & \text{a.e. on } \{u \geq 0\} \\ -Du & \text{a.e. on } \{u < 0\}. \end{cases} (Hint: u+=lim⁡ε→0Fε(u)u^+ = \lim_{\varepsilon \to 0} F_\varepsilon(u), for Fε(z):={(z2+ε2)1/2−εif z≥00if z<0.F_\varepsilon(z) := \begin{cases} (z^2 + \varepsilon^2)^{1/2} - \varepsilon & \text{if } z \geq 0 \\ 0 & \text{if } z < 0. \end{cases}) (iii) Prove that if u∈W1,p(U)u \in W^{1,p}(U), then Du=0 a.e. on the set {u=0}.Du = 0 \text{ a.e. on the set } \{u = 0\}.

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