Let $\Omega $ be a bounded open set in $\mathbb R^n$, $n \geq 2$. In a well-known paper {\it Indiana Univ. Math. J.}, 20, 1077--1092 (1971) Moser found the smallest value of $K$ such that $$ \sup \bigg \{\int _{\Omega } \exp \Big (\Big (\frac {\left |f(x)\right |}K\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\|\nabla f\|_{L^n}\leq 1\bigg \}<\infty . $$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n\log ^{n-1}L \log ^{\alpha }\log L$ $(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $\exp (\exp (t^{n/(n-1-\alpha )}))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.
We show that for every $\varepsilon >0$ there is a set $A\subset \mathbb{R}^3$ such that ${\Cal H}^1\llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ${\Cal H}^1\llcorner A$ has the $1$-dimensional density between $1$ and $2+\varepsilon $ everywhere in the support.
Let Ω \subset \mathbb{R}^{n} be a domain and let α < n − 1. We prove the Concentration-Compactness Principle for the embedding of the space W_{0}^{1}L^{n} log^{\alpha } L(Ω) into an Orlicz space corresponding to a Young function which behaves like (t^{n/n-1-\alpha }) for large t. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem 1.6 where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula P: = \left( {1 - \left\| {\Phi (|\nabla u|)} \right\|_{L^1 (\mathbb{R}^n )} } \right)^{ - 1/(n - 1)} ., Robert Černý., and Obsahuje seznam literatury
Germ-layers, neural crest cells and the vertebrate evolution from the evo-devo viewpoint. In the text it is claimed that nbural crest cells, a population of highly migrated embryonic source for the majority of shared derived features of vertebrates. Consequently, after discussing some topics of the germ-layer theory, it is argued that neural crest cells might be understand as the forth germ-layer of vertebrates (after the ecto-, ento-, and mesoderm) and vertebrates can be seen as tetrablastic, not triploblastic animals.
We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.