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.

3,525 questions
222 views

87 views

Functional Analysis, Why this statement true?

Why this statement true? If $f \in C^0([0,1], W^{2,2}(K))$ then $f \in C^0 ([0,1] \times K)$. $K \subset R^n$ , W : Sobolev Space.
55 views

What is the meaning of “approximation” in Sobolev spaces?

For example, I want to know that the statements as below. (1) there is a sequence $\{ f_k \} \subset W^{3,2}$ approximating $f$ in $W^{2,2}$. (2) we can approximate $f$ in $K \subset R^n$ by a ...
689 views

Why do we talk about Trace Operator?

What is the importance of a Trace operator in PDE . Although I have read the Wiki page on this but I am not able to connect it to the aspect of solving PDE's. Particularly why do we define Trace ...
2k views

Does weak convergence in Sobolev spaces imply pointwise convergence?

I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that $\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$ ...
2k views

Reference of Sobolev space

currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level). Also, Adams 1975 version has been widely cited. So besides these ones, which book in your ...
872 views

Is the Sobolev space $W^{k ,\infty}$ a Banach algebra?

Some Sobolev spaces are closed under multiplication, making them Banach algebras. My question is whether $W^{k ,\infty}$ is a Banach algebra? Since $L^\infty$ is closed under multiplication, I ...
188 views

Sufficient conditions to hold the following inequality.

If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the ...
209 views

111 views

What is “approximation in a Sobolev Space”? For example,

I want to know the meaning of the statement as below. $$\text{There is a sequence} \;\{f_k\} \subset W^{3,2}\; \text{approximating}\;\; f \;\;\text{in}\;\; W^{2,2}.$$ Here $W^{n,m}$ means a ...
572 views

156 views

The reflexivity of the product $L^p(I)\times L^p(I)$

I am starting to read about the Sobolev spaces $W^{1,p}(I),$ where $I$ is an open interval in $\mathbb{R}.$ In order to establish the reflexivity of $W^{1,p}(I)$ for $p\in ]1,\infty[,$ I need the ...
2k views

Poincaré inequality in unbounded domain

Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
125 views

119 views

198 views

201 views

557 views

Show properties of elements of $\mathcal{H}^2$
I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...