# Tagged Questions

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### Small question about strong convergence

I have a small question I have that $\lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
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### Operator compact on $H^1 (0,\pi)$

Consider the operator $K\colon H^1(0,\pi)\to H^1(0,\pi)$ defined by duality (Riesz. Theorem) as $$\langle K\phi,\psi\rangle = \int_{0}^{\pi}{\phi(x)\psi(x)\,dx}$$ for all $\psi \in H^1(0,\pi)$, ...
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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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### 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 $m \ge 1$. ...
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### Folland PDE chapter 6.C problem 1

Problem 6.C.1: Suppose $0 \neq \phi \in C^\infty_c(\mathbb{R}^n)$ and $\{ a_j \}$ is a sequence in $\mathbb{R}^n$ with $|a_j| \to \infty$, and let $\phi_j(x) = \phi(x - a_j)$. Show that $\{\phi_j\}$ ...
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### Show properties of elements of $\mathcal{H}^2$
I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
This is a homework problem and I don't know how to solve it: Consider the following minimization problem on the real-valued sobolev space $H^{1,2}(\Omega)$ with dimension $n=1$ and $\Omega=(0,1)$: ...