New answers tagged sobolev-spaces
0
votes
References for well-posedness of weak solutions to Stefan problem
The existence and uniqueness theorems (and thus the well posedness) for weak solutions (in some suitable sense) to the Stefan problem were proved by Shoshana Kamin (neé Shoshana L'vovna ...
- 5,632
2
votes
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This is a somewhat delicate topic and I think what has been proposed so far did not capture fully what is going on. (Probably I am missing something, too, and I also did not fully answer OP's question,...
- 2,195
2
votes
Accepted
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This may be an overkill : you can use the closed graph theorem. If $(u_n)_n$ converges to $u$ in $W^{s,p}_0(\Omega)$ and $(E(u_n))_n$ converges to $v$ in $W^{s,p}(\mathbf{R}^d)$, then both ...
- 2,710
0
votes
Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains
In many cases people only consider bounded domain because of some convenience on using compactness and finiteness of partition of unity. This question is very important in the field of analysis on ...
- 896
1
vote
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This is "trivial" in the sense that $W^{s,p}_0(\Omega)$ can be regarded as a subspace of $W^{s,p}(\mathbb R^n)$ where the functions are vanishing on $\Omega^c$. And you don't need to take ...
- 896
3
votes
Accepted
Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality
For $p>\max(2,d)$ you have by Sobolev embedding
$$\|f-\overline{f}\|_\infty \lesssim_{\Omega,p} \|f-\overline{f}\|_p + \|\nabla f\|_p.$$
The interpolation $L^p(\Omega) = [L^2(\Omega),L^\infty(\...
- 2,710
3
votes
Accepted
On the weak derivative of $|u|^{(p-2)/2}u$
Only a partial answer : (2) seems strange. In dimension $1$, if $u$ does not change sign, your setting includes the one of $u^\alpha \in H^1(0,1)$ (boundary values are irrelevant here) for some $\...
- 2,710
3
votes
Does the union of fractional Sobolev spaces fills $L^p$?
Let us assume that $p=2$, and let us consider
$$
\cup_{s>0} H^s(\mathbb R^d)\subset H^0(\mathbb R^d)=L^2(\mathbb R^d).
$$
The above inclusion is strict. Let us consider $u\in L^2(\mathbb R^d)$ ...
- 15.2k
2
votes
$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Too long for comment. take for instance $f=0, g=0$. Then the mapping $h\mapsto u$ is a pseudo-differential operator which will have some Sobolev continuity properties for spaces $W^{s,p}$ with $p\in (...
- 15.2k
Top 50 recent answers are included