Tagged Questions
2
votes
0answers
131 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
0answers
56 views
Typical problem in functional analysis #2
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
150 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
35 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
58 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
82 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 ...
0
votes
0answers
55 views
Lagrange theorem
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
70 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
211 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
127 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
256 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
62 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
134 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 $\Omega=(0,1)$:
...
