All Questions
Tagged with lp-spaces sobolev-spaces
395
questions
3
votes
1
answer
67
views
Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
- 105
0
votes
0
answers
27
views
Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
- 133
0
votes
0
answers
44
views
Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$
I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
- 133
0
votes
1
answer
99
views
Integral function of bounded variation function derivative
Let $f: [a,b] \to \mathbb{R}$ be bounded variation. So $f’$ exists almost everywhere, and let
$g(x):=\int_a^x f’(y)dy$.
(Due to the fact that it is possible that $f\notin AC([a,b])$ it is not ...
- 133
0
votes
1
answer
55
views
Are Sobolev–Hölder functions continuous up to the boundary?
Let $U$ be an open subset of $\mathbb{R}^{n}$, let $k$ be a nonnegative integer, and let $W^{k,p}(U)$ ($1 \leq p < \infty$) be the Sobolev space consisting of all real-valued functions on $U$ whose ...
2
votes
1
answer
46
views
A property of solution operator of a elliptic PDE involving positive part of a function
For a given $u \in L^2(\mathbb{R}^N)$, there is a unique $S(u) \in H^1(\mathbb{R}^N)$ such that
$$
\int_{\mathbb{R}^N} \nabla S(u) \nabla \varphi + S(u) \varphi = \int_{\mathbb{R}^N} u \varphi, \quad \...
- 1,219
1
vote
1
answer
43
views
$\log \log (\frac{2}{||x||})$ in $W^{1,2} \setminus L^\infty$ of a ball in $\mathbb{R}^2$
Let $\Omega = B(0,1/2) \subset \mathbb{R}^2$, and let
$$
u(x) = \log \log \left(\frac{2}{||x||}\right) \quad for \quad x \in \Omega.
$$
I have to show that $u$ is in $W^{1,2}(\Omega) \setminus L^\...
- 47
1
vote
0
answers
25
views
Integration by Parts for not so regular Sobolev functions
I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$
and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta ...
- 41
2
votes
0
answers
170
views
$W^{1,p}(\Omega)$ estimates of solutions of elliptic equation
Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$:
$$
-\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega
$$
with Neumann boundary conditions in $\...
- 41
1
vote
1
answer
49
views
Showing that Sobolev Space $H^m$ is in $L^\infty$
I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too ...
2
votes
1
answer
44
views
Interchanging limits and integrals for Cauchy sequences in $L^p$
NOTE: $U\subseteq \mathbb R^n$ is an open, simply connected set.
I am reading L.C Evans's PDE book. I am on page 263. In order to show that Sobolev spaces are a kind of Banach space, we need to show ...
- 12.5k
1
vote
1
answer
49
views
Pointwise convergence up to a subsequence and Sobolev spaces
Let $u_m\to u$ in $W^{k,p}(\Omega)$ for some domain $\Omega\subset\mathbb{R}^n$. I feel like, up to a subsequence, i can say that $D^\alpha u_m\to D^\alpha u$ a.e. for every $|\alpha|\leq k$.
The ...
- 91
1
vote
0
answers
25
views
$H^1(\mathbb{R})$ function has a finite moment?
I have a function $u_0(x) \in H^1(\mathbb{R})$. It is nonnegative, compactly supported, and has a mass $M>0$.
I have to prove that its first moment is finite.
My attempt: $\int_{\mathbb{R}} xu_0(x) ...
- 194
1
vote
1
answer
68
views
If $u(.,t) \in H^2(\mathbb{R})$, then $u (.,t) \in C^1(\mathbb{R})$?
The statement is as such:
$u \in L^2([0,T]; H^2(\mathbb{R}))$ for any $T>0$, thus $u (.,t) \in
> C^1(\mathbb{R})$ a.e. in $t>0$.
My guess was that this must be a result of Sobolev embedding....
- 194
1
vote
1
answer
95
views
Brezis' exercise 8.28.11: how to prove $\sum_{k=0}^{\infty} \left | \frac{1}{2} \alpha_k (f) - a \right |^2 < + \infty$?
Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by
$$
(Tf) (x) = \int_0^x t ...
- 17.6k