Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

1,267 questions
806 views

212 views

Question about boundedness of a sequence in $W^{3,q}$ for any $1\leq q < \frac{N}{N-1}$

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
244 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt$$ where $P$ is a strictly hyperbolic linear operator. For ...
131 views

Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
397 views

136 views

Sanity check: self-adjoint operator on Sobolev space

I just wanted to check if the conclusion below is true, and whether the following reasoning works: Let $H^i$ be the Sobolev spaces on a compact manifold $M$ and $D$ a self-adjoint (in the $H^0$-inner ...
369 views

Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
130 views

216 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ (...
347 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
208 views

66 views

How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
41 views

Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact. The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
96 views

Is being in the Sobolev space of power $\frac{d}{2}$ necessary for having well defined point evaluations?

From the Sobolev embedding theorem we know that for $\alpha = \frac{d}{2}$, $W^{\alpha, 2}(\mathbb{R}^d)$ is continuously embedded in $C^0(\mathbb{R}^d)$. Especially the point evaluations are in the ...
29 views

When Sobolev maps are localizable?

Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. A Sobolev map $f \in W^{1,p}(M,N)$ is called localizable if for every $x_0 \in M$ there exists a neighbourhood $U$ of $x_0$ in $M$ and a ...
114 views

Compact embedding in weighted Sobolev spaces

I have a question concerning Sobolev's embedding. Let (for simplicity) $\Omega=\left( 0,1\right)$. Then it is well known by Rellich's theorem that $H^{1}\left( \Omega\right)$ is compactly ...
85 views

Sobolev spaces for conformal metrics on a Riemannian manifold

Suppose we have a compact Riemmanian manifold $(M^n,g)$ without boundary, and some class $\{g_u\}$ of conformal metrics $g_u:=e^{2u}g$, where $\{u\}\subset C^\infty(M^n)$. For some $k,p$ suppose there ...
91 views

177 views

What is a distribution in $H^{-1}(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E$, where $\mathcal E$ denotes the ...