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4 votes
2 answers
467 views

When is a Nemytskii map between Sobolev spaces compact?

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with bounded derivative. Define the Nemytskii map $F:H^1(\Omega) \to H^1(\Omega)$ by $F(u)(x) := f(u(x))$. Here $\Omega$ is a bounded smooth ...
GuestUser's user avatar
9 votes
1 answer
1k views

Noncompactness of the Sobolev embedding in the critical exponent case

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
anonymous's user avatar
  • 446
5 votes
1 answer
242 views

Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells

A theorem of Corson and Lindenstrauss in: Corson, H. H. and Lindenstrauss, J. “On weakly compact subsets of Banach spaces”. In: Proceedings of the American Mathematical Society 17.2 (1966), pp. 407–...
Amin's user avatar
  • 128
1 vote
1 answer
66 views

Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference. Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
Holi's user avatar
  • 111
6 votes
1 answer
916 views

Rellich-Kondrachov compacteness Theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
Lorenzo Cavallina's user avatar
3 votes
0 answers
392 views

Compact embedding of ${\rm L}^1_{loc}$ space

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...
Semmel's user avatar
  • 165
1 vote
1 answer
618 views

Condition to obtain a not compact embedding

I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
Vrouvrou's user avatar
  • 277
7 votes
2 answers
459 views

Compact operators on Lebesgue spaces

Let $K:{\rm L}^p({\bf R}^d)\to {\rm L}^p({\bf R}^d)$ be a bounded linear operator for every $p\in(1,\infty)$. Assume that for some $r\in(2, \infty)$ it holds that $K$ is compact on ${\rm L}^q({\bf R}^...
Semmel's user avatar
  • 165
2 votes
1 answer
274 views

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$

Let $f_n \to f$ on compact subsets of the real line (these are functions defined on the real line) satisfying some conditions: $f$ has linear growth (but is nonlinear function) and is continuous and ...
LapLace's user avatar
  • 173
4 votes
0 answers
404 views

Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
user61038's user avatar
  • 289
1 vote
0 answers
380 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
Jozsi's user avatar
  • 11
7 votes
1 answer
775 views

Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds

Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
Barrett's user avatar
  • 143
2 votes
0 answers
428 views

Weak relative compactness in $L^1_{loc}$.

In my work I stumbled upon a proposition (without proof, alas), which I can't really prove. Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
TZakrevskiy's user avatar
4 votes
0 answers
291 views

trace-class embeddings

There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
Delio Mugnolo's user avatar

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