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Results tagged with sobolev-spaces
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user 33135
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
2
votes
1
answer
234
views
A bound in Sobolev spaces of negative order
Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$.
I wonder if the following bound is true:
$$
\|f g_{x_1}\|_{H^{-0.5}(U)}\leq C( …
- 843
2
votes
If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lip...
We have the bound
$$
|fg|_{H^{0.5}}\leq C |f|_{H^{0.5+\delta}}|g|_{H^{0.5}}.
$$
So, if you have a Lipschitz function, due to the compactness of the domain, the function is $H^1$.
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2
votes
References for well-posedness of weak solutions to Stefan problem
There is a huge literature in free boundaries. Here I collected some papers (in no particular order) addressing different physical systems (coming mainly from fluid dynamics) and questions (so, not on …
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2
votes
1
answer
774
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $T$ sa …
- 843