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# Questions tagged [sobolev-spaces]

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.

564 questions
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### Question on relation between a parabolic sobolev space and a sobolev bochner space

For parabolic sobolev spaces I follow the following definition: According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$ ...
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### Construct a dense family in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ based on distance functions

Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function. Suppose $\mathcal{F}$ is a ...
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### Practical applications of Sobolev spaces

What are the examples of practical applications of Sobolev spaces? The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of ...
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### Open problems in Sobolev spaces

What are the open problems in the theory of Sobolev spaces? I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to ...
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### Sobolev extension operators

Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary and that $M\subset \tilde{M}$ with $(\tilde{M},g)$ also a compact Riemannian manifold with smooth boundary. Let us consider a one-...
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We define $L^p-$ Sobolev spaces as follows: $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ where $\langle \... 1answer 114 views ### Estimate of the difference quotients in terms of an$L^{1,\infty}$function Let$f \colon \mathbb R^d \to \mathbb R$be a measurable function. Consider the following property: (P) there exist a negligible set$N \subset \mathbb R^d$and function$T_f \in L^p(\mathbb R^d)$... 0answers 54 views ### Regularity of superposition operator generated by function between Banach spaces Let$E$,$F$be Banach spaces,$D$be open in$E$, and$K=[0,1]$. Given$\varphi\colon K\times D\to F$I call $$\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))$$ the ... 1answer 70 views ### Sobolev trace operator on hyperplanes For Sobolev spaces$H^s(R^d)$, with$s> \frac{d}{2}$every element of$H^s(R^d)$is an equivalence class$[f]$and in every such a class there exists a unique continuous function$f^{*}$. Can ... 0answers 104 views ### Sobolev Multiplication on non-compact manifold We know that for a compact Riemannian$n$-dim manifold$(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that$L^p_k\times L^q_l?L^r_m$, where$1/r?m/n>1/p?k/m+1/...

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