5
votes
0answers
103 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
0
votes
1answer
30 views

If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
3
votes
0answers
43 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
2
votes
2answers
48 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
2
votes
1answer
58 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
1
vote
1answer
44 views

Implications of Weak convergence in Sobolev Spaces

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in ...
1
vote
1answer
41 views

If $(f_n)$ is Cauchy in the $L^2$-norm, then is $(f'_n)$ Cauchy in the $L^2$-norm?

Let $(f_n)$ be a sequence in $H^1(a,b)=\{f\in L^2(a,b);\;f'\in L^2(a,b)\}$, where $-\infty<a<b<+\infty$. If $(f_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$, is it possible to ...
3
votes
1answer
135 views

If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
2
votes
1answer
44 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
1
vote
3answers
482 views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that ...
0
votes
1answer
34 views

Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
-3
votes
2answers
163 views

convergence exercice

I have a question please, thanks to help me. Let $\Omega$ an open bounded, connexe and regular Let $(v_n)$ an sequence in $H^1(\Omega)$ and let $v \in H^1(\Omega)$ such that $v_n$ converge weakly in ...
2
votes
0answers
83 views

Limit in norm of a Sobolev space

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
3
votes
2answers
93 views

Weakly convergence in $W^{1,q}, 1<q<\infty$

Let $(x_n)$ be weakly convergent against $x$ in the Sobolev space $W^{1,q},1<q<\infty$. Now I have to show, that $(\dot{x_n})$ converges weakly against $\dot{x}$ in $L^q$. (With the point I ...
3
votes
1answer
260 views

Space Sobolev $W^{m,p}$ complete

Show that Sobolev space is complete. I am trying Than $L^p(\Omega)$ is complete then If $f_n \in L^p(\Omega)$ then $f_n \to f \in L^p(\Omega)$. But rest show that $D^{\alpha}f \in L^p(\Omega)$. How I ...
2
votes
0answers
259 views

Weak convergence and limit of this sequence

Let $f_n$ be bounded uniformly in the $H^1$ norm, so we have (weak convergence) $$f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$$ Then by compact embedding, we have strong ...
6
votes
1answer
186 views

Natural question about weak convergence.

Let $u_k, u \in H^{1}(\Omega)$ such that $u_k \rightharpoonup u$ (weak convergence) in $H^{1}(\Omega)$. Is true that $u_{k}^{+}\rightharpoonup u^{+}$ in $\{u\geqslant 0\}$? You can do hypothesis on ...