# Tagged Questions

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### two-point boundary value problem for elliptic equations (ODE)

we consider two-point boundary value problem $$Au=-au''+bu'+cu=f~~~~~~~~~~~~~~~~ in ~~\Omega=(0,1)$$ $$u(0)=u_0,u(1)=u_1$$ where $a=a(x)>0$, $b=b(x)\ne 0$ and $c=c(x) \ge 0$ We must prove ...
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### Problem from Evans' PDE book, chapter 5, problem 5

I'm taking my first theoretical math course in a year and am bashing my head against a rock with this problem. "The sets $U,V$ are open, with $V \subset \subset U$ (compactly contained). Show that ...
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### Evans PDE chapter 2 problem 4

Problem is Give a direct proof that if $u \in C^{2}(U) \cap C(\overline{U})$ is harmonic within a bounded open set $U$, then $\max_{\overline{U}} u =\max_{\partial U} u$. What I think is ...
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### Find Solution in Similarity Form

Given equation: $$2r\cos\theta{\partial u\over\partial r}-2\sin\theta{\partial u\over\partial\theta}={\partial^2 u\over\partial r^2}$$ with $u=1$ at $r=0$ and $u\rightarrow0$ as $r\rightarrow\infty$ ...
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### For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...
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### Are these PDE's necessarily compatible?

In question 9, we're asked to show that the equations are compatible if and only if $u$ satisfies a certain equation. I can show that compatibility implies the constraint on $u$ and that the ...
### Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$
I have a problem: For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$. ...