2019春-张永兵-微分方程II

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次数
内容
上交日期
6
[第七章] 7 8 10
5
[第六章] 12 13 15
[第七章] 1 2 3 4 6
2019.6.10
4
[第六章] 5 6 7 8 9 10 11
[附] 1. 设\(H\)是Hilbert空间,\(K:H\to H\)紧,则\(\operatorname{dim}N(I-K)=\operatorname{dim}N(I-K^*)\).
  2. 证明Hopf引理:设\(u\in C^2(U)\cap C^1(\overline{U})\),且在\(U\)上\(c\equiv0\),\(Lu\leqslant0\). 除此之外,还存在\(x^0\in\partial U\),使得\(\forall x\in U\),\(u(x^0)>u(x)\),且存在开球\(B\subset U\),使得\(x^0\in\partial U\),则
    (1)\(\displaystyle\frac{\partial u}{\partial\nu}(x^0)>0\),其中\(\nu\)为\(B\)在\(x^0\)处的外法向导数;
    (2)若在\(U\)上\(c\geqslant0\),\(u(x^0)\geqslant0\),则(1)依旧成立.
2019.5.27
3
[第五章] 12 13 15 20 21
[第六章] 1 2 3 4
[附] 1. 设\(1\leqslant p<\infty\),\(u\in W_0^{1,p}(V_0)\),\(V_0\Subset\mathbb{R}^n\),\(V=\{x:\operatorname{dist}(x,V_0)<1\}\). 求证:对任意\(0<\varepsilon<1\),\(\displaystyle\int_V\left|u^\varepsilon(x)-u(x)\right|\mathrm{d}x\leqslant\varepsilon\int_V|\mathrm{D}u|\mathrm{d}x.\)
  2. 设\(U=B_1(0)\cap\{x_n>0\}\),\(V=B_{\frac 1 2}(0)\cap\{x_n>0\}\),\(u\in W^{1,p}(U)\). 求证:\(\forall 0<|h|<<1\),\(\displaystyle\int_V|\mathrm{D}_i^hu|^p\mathrm{d}x\leqslant\int_U|\mathrm{D}_iu|^p\mathrm{d}x\ (i=1,\cdots,n-1).\)
2019.4.29
2
[第五章] 4 7 8 14 17 18 19
[5.8.3] 定理5的证明
2019.4.8
1
[第五章] 1 2 3 5 6 9 10 11
[附] 1. 证明:\(C_0^\infty(\mathbb{R}^n)\)在\(W^{k,p}(\mathbb{R}^n)\)中稠密.
2019.3.25

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