研内Wb(92)的方便的处是W2(92)中的任何 数数都u可以连处延拓到Wb(P).为此,只称让 (x),x∈92 →a构集W6(2)→W()的连处延拓 组X1,X2是两个赋范都性空间,作们的范数分 别为‖·‖1,‖·‖2,着果满足条件 (2)存在常数c,空得‖u2≤cul1,va∈x1 定称空间X1嵌入到空间X2,以作X1→X2,着果 嵌入还满足恒等算子I:X1→X2是条的,定称这 个嵌入是条的 定理1.5.3集立 C(92) p>n, L9(2) 1<q< L甲p/(n-p)(92),p<n 部中以为∽积表局包设关系的简,还表局嵌入 算子是连处的.此简,对任并u∈Wb(2)有不等 式 supLus C(n, p)IQ/mn-I/PIlDullp, p>n, lull≤C(n,q)g2H‖Duln,p=n,1≤q<∞ Tulln/(n-p)≤C‖Dulp,p<n 部中‖Dulp=∑‖ Doull 推论1.54关于W2(92k>1)的情按有嵌入 m -(=-Up(k-D)< n, C(9,0≤1≤k- 推论1.5.5Wb(9)中可以定义着.的等价范 llull.P(=∑DealQJ Wk,p 0 (Ω) RE Wk,p 0 (Ω) /K D u @ERK Wk p (Rn). SL u¯(x) = ⎧ ⎪⎨ ⎪⎩ u(x), x ∈ Ω, 0, x/∈ Ω. u → u¯ :. Wk,p 0 (Ω) → Wk p (Rn) @ERK 1 X1, X2 LC+D'%&Æ+ % ·1, ·2, Æ;#=9 (1) X1 ⊂ X2; (2) #* c, %4 u2 ≤ cu1, ∀ u ∈ X1, "%& X1 MN%& X2, X1 → X2. Æ MNM;#>H I : X1 → X2 =" MN= !" 1.5.3 .$ W1,p 0 → ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ C(Ω) ¯ , p > n, Lq(Ω), p = n, 1 ≤ q < ∞, Lnp/(n−p) (Ω), p < n, 3 → 2:13M2MN @E / u ∈ W1,p 0 (Ω) H sup Ω |u| ≤ C(n, p)|Ω| 1/n−1/pDup, p > n, uq ≤ C(n, q)|Ω| 1/qDun, p = n, 1 ≤ q < ∞, unp/(n−p) ≤ CDup, p < n, 3 Dup = |α|=1 Dαup. <= 1.5.4 ! Wk,p 0 (Ω)(k > 1) OMN Wk,p 0 → ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Wl,s 0 (Ω), s = np n − (k − l)p,(k − l)p < n, Wl,q 0 (Ω), (k − l)p = n, 1 ≤ q < ∞, Cl (Ω) ¯ , 0 ≤ l ≤ k − n p . <= 1.5.5 Wk,p 0 (Ω) " HH+ uWk,p 0 (Ω) = |α|=k Dαup. 8