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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.

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        77 views

        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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        1answer
        120 views

        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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        45 views

        Traceless sobolev forms on compact manifolds with boundary

        Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
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        150 views

        Green's formula and traces in weighted Sobolev spaces

        Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let \begin{equation} \rho(x) = 1-|x| \quad \text{ for } x \in B_1, \end{equation} and let $\sigma >0$ be given. As per the comments, notice that $\...
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        1answer
        286 views

        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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        1answer
        72 views

        Equivalence of Sobolev spaces for different metrics

        Consider $M$ a manifold and $g_1, g_2$ two different Riemannian metrics. I want to know how the condition $|\nabla^{g_1,k}(g_1-g_2)|_{g_1}\leq C$ implies that the norms of $|\nabla^{g_1,i}u|_{T^{\...
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        62 views

        Sobolev embedding in complete manifold

        Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry. Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we ...
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        Hardy-Littlewood in Sobolev Spaces

        For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
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        1answer
        83 views

        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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        45 views

        example in $L^p_{s}-$Sobolev spaces

        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 \...
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        1answer
        107 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 $T_f \in L^p(\mathbb R^d)$ ...
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        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 ...
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        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 ...
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        103 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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        votes
        1answer
        197 views

        Density in fractional Sobolev space

        Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1?\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right), $$ $$ H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}. $$ Q: Is $C^\...

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