1
vote
1answer
41 views

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 ...
1
vote
1answer
34 views

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 $\psi \in H^1(0,\pi)$, ...
1
vote
1answer
42 views

minimizes the functional to solve a pde

I am trying to do this exercise: Let $\Omega$ a open bounded domain in $R^n$. Consider the Dirichlet problem $$ \left\{ \begin{array}{ccccccc} -\Delta u = \lambda \sin (u) + f , \ \text{in ...
0
votes
1answer
76 views

Bounded subsequence in Sobolev Space

The following is an exercise. Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$, First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p ...
2
votes
1answer
79 views

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 ...
1
vote
2answers
70 views

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 $m \ge 1$. ...
5
votes
1answer
219 views

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 $\{\phi_j\}$ ...
1
vote
1answer
170 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
4
votes
1answer
120 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
0
votes
0answers
29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
2
votes
1answer
40 views

Show that $v \in H^1(\Omega)$ if $v\in C^0(\Omega)$ and $v|_{\Omega_j} \in H^1(\Omega_j)$

Let $\Omega$ be an open set in $\Bbb R^d$. Let $\{\Omega_j\}_{j=1}^{N}$ be a fi nite collection of open disjoint subsets of $\Omega$ such that $\overline\Omega=\cup_{j=1}^{N}\overline\Omega_j$. ...
2
votes
2answers
94 views

Is $H_0^1([a,b]) \subset C([a,b],\mathbb{R})$?

i have a small question : how to see that $H_0^1([a,b])\subset C([a,b],\mathbb{R})$? Please Thank you
1
vote
1answer
207 views

weak and strong convergence

I don't know that why $||v_m||_{W^{1,p}_0}=1$, then we can assume that $v_m$ converges to $v_0$ weakly in ${W^{1,p}_0}(\Omega)$, and strongly in $L^p(\Omega)$ I can't understand clearly about weak ...
2
votes
0answers
157 views

A bounded sequence

I have a question please : Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq ...
1
vote
1answer
122 views

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
2
votes
1answer
599 views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
1
vote
1answer
78 views

Is the gradient of a function in $H^2_0$ in $H^1_0$?

Suppose we have $f\in H^2_0(U)$, so $f$ is the limit of some sequence $(g_n)$ of smooth compactly supported functions on $U\in\mathbb{R}^n$ (assume bounded & smooth boundary) and $f$ is in the ...
2
votes
1answer
129 views

Proving that a weak solution of a BVP satisfies the boundary condition

I am given the smooth function $u$ which satisfies $\int_U (\nabla u \cdot \nabla v +uv)\,dx = \int_U fv\,dx$ for all functions $v$ in the Sobolev space $H^1(U)$, where $f\in ...
3
votes
1answer
136 views

$W^{1,p}$ compact in $L^\infty$?

Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
1
vote
1answer
84 views

Critical exponent in O.D.E and Gâteaux derivative

I have this example and I don't understand its resolution: Let $\Omega \subset \mathbb{R}^n , n\geq3$ be a bounded open set (with smooth boundary), let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be ...
1
vote
1answer
619 views

weak subsolution

Assume $u\in H^1(U)$ is a bounded weak solution of $$-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}=0 ~~~in~ U$$ Let $\phi:R\rightarrow R$ be convex and smooth,and set $w=\phi(u)$ Show $w$ is a weak ...
3
votes
0answers
220 views

Why is this functional coercive?

Let $h\in L^2(0,1)$, such that $|\int h \mathrm{d}x|<1$. On the space $H^1(0,1)$, consider $$J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$$ ...
3
votes
1answer
439 views

“Poincaré” inequality for $H^1$

I have to show the following: Let $U\subset \mathbb{R}^n$ be nice (i.e. bounded, open and boundary of class $C^1$). Further there's a function $$f:H^1(U) \to \mathbb{R}^n$$ which is continuous and ...
1
vote
0answers
64 views

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 ...
3
votes
2answers
155 views

Minimization problem in Sobolev spaces

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 $\Omega=(0,1)$: ...