Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, f/x and ² f/x ² are continuous on ⁿ × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.     

Structural features of aliphatic N-nitrosamines of 7-azabicyclo[2.2.1]heptanes that facilitate N-NO bond cleavage

N-Nitrosamines can be considered as potential nitric oxide (NO)/nitrosonium ion (NO(+)) donors. However, the relation of the structures of N-nitrosamines, in particular of aliphatic N-nitrosamines, to the characteristics of release of NO or NO(+) remains unclear. Here we show that aliphatic N-nitrosoamines of 7-azabicyclo[2.2.1]heptanes can undergo heterolytic N-NO bond cleavage. On the basis of the observation of reduced rotational barriers of the N-NO bonds in solution and nitrogen-pyramidal structures of the N-nitroso group in the solid state, we postulate that N-NO bond cleavage of N-nitrosamines is enhanced by a reduction of the resonance in the N-NO group. Computational studies suggest that these structural features of the N-nitrosamines of 7-azabicyclo[2.2.1]heptane are derived from angle strain imposed on the CNC angles.     

Time-resolved fluorescence spectroscopic and scanning near-field optical microscopic studies of rhodamine dye adsorbed in cationic Langmuir-Blodgett films

The fluorescence lifetimes decays and picosecond time-resolved fluorescence spectra were measured to investigate the dynamics of the excited state of sulforhodamine B (SRB) molecules adsorbed in the mono- and multilayered Langmuir-Blodgett (LB) films of octadecylamine. Steady-state and time-resolved fluorescence spectroscopy reveals that the fluorescence lifetimes and contents of the monomer and dimers in the molecular organizates depend upon the concentration of the dye in the solution and the adsorption process. SRB dye molecules adsorbed in LB films have been imaged with scanning near-field optical microscopy (SNOM). This information is exploited to map the distribution with molecular spatial resolution. SNOM provide the visual evidence of the monomers and dimers of SRB in cationic LB films.     

Condensed thienopyrimidines. II. Synthesis and gastric antisecretory activity of thiazole and polymethylene condensed thienopyrimidine derivatives

Thiazolo[3,2-a]thieno[3,2-d]-, [3,4-d]- and [2,3-d]pyrimidin-5-one derivatives (6, 11 and 16), and polymethylene condensed thieno[3,2-d]-, [3,4-d]- and [2,3-d]pyrimidin-5-one derivatives (19-21), in which the oxygen atom of the oxazolidine moiety in 3 was replaced by a sulfur atom or methylene groups, were synthesized and evaluated for gastric antisecretory activity in pylorus-ligated rats. The structure-activity relationships of these compounds are discussed.