On the SO(n + 3) to SO(n) branching multiplicity space

We study the decomposition as an $\text{SO}(3)$-module of the multiplicity space corresponding to the branching from $\text{SO}(n+3)$ to $\text{SO}(n)$. Here, $\text{SO}(n)$ (resp. $\text{SO}(3)$) is considered embedded in $\text{SO}(n+3)$ in the upper left-hand block (resp. lower right-hand block). We show that when the highest weight of the irreducible representation of $\text{SO}(n)$ interlaces the highest weight of the irreducible representation of $\text{SO}(n+3)$, then the multiplicity space decomposes as a tensor product of $?(n+2)/2?$ reducible representations of $\text{SO}(3)$.     

A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF;$t\in [0\text{,}1]$), given as a mean theoretic expression:$A?B>0\text{,}t\in [0\text{,}1]?A^{-r+t}{\#}_{\frac{1-t+r}{(p-t)s+r}}(A^t{?}_s{B}^p)?B\text{for}r?t\text{;}p\text{,}s?1\text{,}$where ${\#}_{\alpha }$ is the $\alpha$-geometric mean and ${?}_s$ ($s?[0\text{,}1]$) is a formal extension of ${\#}_{\alpha }$. It is shown that (SGF; $t\in [0\text{,}1]$) has the Löwner–Heinz property, i.e. (SGF; $t=1$) implies (SGF;t) for every $t\in [0\text{,}1]$. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Löwner–Heinz property.     

Elliptic equations with critical exponent on S3: new non-minimising solutions

Consider the problem:

$\{\begin{array}{cc}?{\Delta }_{S^3}U=\lambda U+U^5\text{,}U>0\hfill & \text{on?}{B}^{\prime }\text{,}\hfill \\ U=0\hfill & \text{on?}?B^{\prime }\text{,}\hfill \end{array}$

where $B^{\prime }$ is a ball on ${\mathbf{S}}^3$ with geodesic radius ${\theta }_1$, and ${\Delta }_{S^3}$ is the Laplace–Beltrami operator on ${\mathbf{S}}^3$. We prove that for any ${\theta }_1\in (\pi /2\text{,}\pi )$ and any $k>1$, there exist at least 2k solutions of this problem for λ sufficiently large negative. To cite this article: H. Brezis, L.A. Peletier, C. R. Acad. Sci. Paris, Ser. I 339 (2004).