The sobolev-spaces tag has no wiki summary.

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### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset ...

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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

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### Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

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### Reference request: Simple facts about vector-valued Sobolev space

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = ...

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### Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...

**3**

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100 views

### Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?

**4**

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263 views

### Negative real order Sobolev spaces: density and representation

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial ...

**2**

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**1**answer

465 views

### A comparison principle for parabolic equation

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...

**0**

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**1**answer

193 views

### A question about PDE argument involving monotone convergence theorem and Sobolev space

I'm reading this paper. In it there is the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...

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### Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...

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### Mixed (anisotropic) Sobolev spaces

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...

**3**

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127 views

### Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...

**3**

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### How do functions operate in a Sobolev space $H^{s}$?

Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...

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### Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...

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### Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that ...