Sharp Nash inequalities on manifolds with boundary in the presence of symmetries

In this paper we establish the best constant $\widetilde A_{opt}(\bar{M})$ for the Trace Nash inequality on a $n-$dimensional compact Riemannian manifold in the presence of symmetries, which is an improvement over the classical case due to the symmetries which arise and reflect the geometry of manifold. This is particularly true when the data of the problem is invariant under the action of an arbitrary compact subgroup $G$ of the isometry group $Is(M,g)$, where all the orbits have infinite cardinal

Exponential elliptic boundary value problems on a solid torus in the critical of supercritical case

In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are invariant under the group $G=O(2)\times I \subset O(3)$. The model problems of interest are stated below: {ll} {\bf(P_1)} & \displaystyle \Delta\upsilon+\gamma=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T, \quad\upsilon |_{_{\partial T}}=0. and {ll}\bf{(P_2)} & \displaystyle \Delta\upsilon+a+fe^\upsilon=0, \upsilon>0\quad \mathrm{on}\quad T, [1.3ex] &\displaystyle \frac{\partial \upsilon}{\partial n}+b+ge^\upsilon=0\quad \mathrm{on} \quad{\partial T}. We prove that exist solutions which are $G-$invariant and these exhibit no radial symmetries. In order to solve the above problems we need to find the best constants in the Sobolev inequalities in the exceptional case

A Neumann problem with the qq-Laplacian on a solid torus in the critical of supercritical case

Following the work of Ding [21] we study the existence of a nontrivial positive solution to the nonlinear Neumann problem $$displaylines{ Delta_qu+a(x)u^{q-1}=lambda f(x)u^{p-1}, quad u>0quad hbox{on } T,cr abla u|^{q-2}frac{partial u}{partial u}+b(x) u^{q-1} =lambda g(x)u^{ilde{p}-1} quadhbox{on }{partial T},cr p =frac{2q}{2-q}>6,quad ilde{p}=frac{q}{2-q}>4,quad frac{3}{2}<q<2, }$$ on a solid torus of $mathbb{R}^3$. When data are invariant under the group $G=O(2)imes I subset O(3)$, we find solutions that exhibit no radial symmetries. First we find the best constants in the Sobolev inequalities for the supercritical case (the critical of supercritical)

Characterization of a novel large deletion and single point mutations in the <it>BRCA1 </it>gene in a Greek cohort of families with suspected hereditary breast cancer

Abstract Background Germline mutations in BRCA1 and BRCA2 predispose to breast and ovarian cancer. A multitude of mutations have been described and are found to be scattered throughout these two large genes. We describe analysis of BRCA1 in 25 individuals from 18 families from a Greek cohort. Methods The approach used is based on dHPLC mutation screening of the BRCA1 gene, followed by sequencing of fragments suspected to carry a mutation including intron – exon boundaries. In patients with a strong family history but for whom no mutations were detected, analysis was extended to exons 10 and 11 of the BRCA2 gene, followed by MLPA analysis for screening for large genomic rearrangements. Results A pathogenic mutation in BRCA1 was identified in 5/18 (27.7 %) families, where four distinct mutations have been observed. Single base putative pathogenic mutations were identified by dHPLC and confirmed by sequence analysis in 4 families: 5382insC (in two families), G1738R, and 5586G > A (in one family each). In addition, 18 unclassified variants and silent polymorphisms were detected including a novel silent polymorphism in exon 11 of the BRCA1 gene. Finally, MLPA revealed deletion of exon 20 of the BRCA1 gene in one family, a deletion that encompasses 3.2 kb of the gene starting 21 bases into exon 20 and extending 3.2 kb into intron 20 and leads to skipping of the entire exon 20. The 3' breakpoint lies within an AluSp repeat but there are no recognizable repeat motifs at the 5' breakpoint implicating a mechanism different to Alu-mediated recombination, responsible for the majority of rearrangements in the BRCA1 gene. Conclusions We conclude that a combination of techniques capable of detecting both single base mutations and small insertions / deletions and large genomic rearrangements is necessary in order to accurately analyze the BRCA1 gene in patients at high risk of carrying a germline mutation as determined by their family history. Furthermore, our results suggest that in those families with strong evidence of linkage to the BRCA1 locus in whom no point mutation has been identified re-examination should be carried out searching specifically for genomic rearrangements.</p