<ruby id="d9npn"></ruby>

      <sub id="d9npn"><progress id="d9npn"></progress></sub>

      <nobr id="d9npn"></nobr>

      <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

      <th id="d9npn"><meter id="d9npn"></meter></th>

      Stack Exchange Network

      Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

      Visit Stack Exchange

      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.

      3
      votes
      1answer
      45 views

      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 $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$ ...
      4
      votes
      1answer
      83 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 ...
      2
      votes
      1answer
      128 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 ...
      2
      votes
      0answers
      47 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 $...
      4
      votes
      0answers
      165 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 $\...
      20
      votes
      1answer
      310 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 ...
      1
      vote
      1answer
      75 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^{\...
      1
      vote
      0answers
      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 ...
      2
      votes
      0answers
      106 views

      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&...
      1
      vote
      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-...
      2
      votes
      0answers
      46 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 \...
      5
      votes
      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 $T_f \in L^p(\mathbb R^d)$ ...
      1
      vote
      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 ...
      0
      votes
      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 ...
      2
      votes
      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/...

      15 30 50 per page
      特码生肖图
      <ruby id="d9npn"></ruby>

          <sub id="d9npn"><progress id="d9npn"></progress></sub>

          <nobr id="d9npn"></nobr>

          <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

          <th id="d9npn"><meter id="d9npn"></meter></th>

          <ruby id="d9npn"></ruby>

              <sub id="d9npn"><progress id="d9npn"></progress></sub>

              <nobr id="d9npn"></nobr>

              <rp id="d9npn"><big id="d9npn"><th id="d9npn"></th></big></rp>

              <th id="d9npn"><meter id="d9npn"></meter></th>