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.     

Tensor analyzing power T 20 of the dd → 3Hen and dd → 3Hp reactions at zero angle for energies 140, 200, and 270 MeV

RIKEN Accelerator Research Facility data on the tensor analyzing power T ₂₀ of the dd → ³Hen and dd → ³Hp reactions at zero angle for deuteron kinetic energies of 140, 200, and 270 MeV are reported. The observed positive sign of T ₂₀ clearly demonstrates the sensitivity to the D/S-wave ratio in the ³He and ³H wave functions in the energy range of the experiment. Data on T ₂₀ for the ³Hen channel are in agreement with those for the ³Hp channel within the experimental errors.     

Copolymers containing a spiro orthoester moiety that undergo no shrinkage during cationic crosslinking

A spiro orthoester with an exomethylene group (exoSOE) was radically copolymerized with acrylonitrile or vinyl acetate at several feed ratios to obtain the corresponding copolymers having spiro orthoester moieties in the side chain. The obtained copolymers could be crosslinked via the double ring‐opening polymerization of the spiro orthoester moieties in their side chain by a treatment with BF₃OEt₂. The volume changes upon the crosslinking of the copolymers were evaluated by density measurements with a micromeritics gas pycnometer. The copolymers experienced less than 1% volume expansion instead of volume shrinkage during typical cationic crosslinking, regardless of the copolymer compositions. Negligible shrinkage was observed during the thermal cationic crosslinking of a film cast from a nitrobenzene solution of the copolymers containing a benzylthiophenium salt as a thermally latent cationic initiator. The constantly low volume changes during the crosslinking of the copolymers from exoSOE probably depended on the almost zero volume change during the cationic polymerizations of spiro orthoester derivatives. This indicates that exoSOE is an effective monomer for crosslinkable polymers without volume changes. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 3666–3673, 2006     

Preparation and properties of isobutylene–isoprene rubber–clay nanocomposites

Isobutylene isoprene rubber (IIR)‐clay nanocomposites have been prepared successfully by melt intercalation with maleic anhydride‐grafted IIR (Ma‐g‐IIR) and organophilic clay. In IIR‐clay nanocomposites, the silicate layers of the clay were exfoliated and dispersed into the monolayer. The nanocomposites exhibited greater gas barrier properties compared with those of Ma‐g‐IIR. When 15 phr clay was added, gas barrier properties were 2.5 times greater than those of Ma‐g‐IIR. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1182–1188, 2006     

Synthesis of copolymers containing a spiro orthocarbonate moiety and evaluation of the volume change during their cationic crosslinking

Spiro orthocarbonate (SOC) monomers having either an exomethylene group {3,3‐dimethyl‐9‐methylene‐1,5,7,11‐tetraoxaspiro[5.5]undecane (ExoSOC)} or an allyl group {9‐allyl‐3,3‐dimethyl‐1,5,7,11‐tetraoxaspiro[5.5]undecane (AllylSOC)} were radically copolymerized with vinyl monomers at several feed ratios to obtain the corresponding copolymers having SOC moieties in the side chain. The obtained copolymers were crosslinked via the double ring‐opening polymerization of the SOC moieties by a treatment with boron trifluoride etherate. The volume changes during the crosslinking of the copolymers were evaluated by density measurements with a gas pycnometer. As the SOC moiety composition increased, the volume shrinkage during the crosslinking was suppressed, and that finally changed into volume expansion. The volume changes during the crosslinking of the copolymers from AllylSOC were slightly larger than those of the copolymers from ExoSOC. The higher volume expansions in the crosslinking of AllylSOC‐based copolymers were ascribable to the lower steric hindrance around the SOC moieties. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 7040–7053, 2006     

Preparation and certification of the new reference materials; plastics (disk form, JSAC 0621-0625) for determination of mercury using x-ray fluorescent analysis.

The Japan Society for Analytical Chemistry has developed some new plastic certified reference materials (CRMs) for the analysis of mercury in polyester disks using XRF analysis. These CRMs (named as JSAC 0621-0625) were prepared by casting polyesters including a toluene solution of organometallic compounds as a standard. Concentrations of the five levels of mercury ranged from 0 to 250 mg/kg. Homogeneity tests of prepared disks had shown excellent results. Interlaboratory comparison study for the certification was performed by 15 laboratory participants. The z-scores in robust statistical method was applied for the evaluation of outliers. The certified values were assigned after discarding outliers. The uncertainties of certified values were determined as the confidence levels of 95%.     

A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term. The method of proof is based on expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self‐similar solutions to nonlinear wave equations are proved on up to five space dimensions. © 2006 Wiley Periodicals, Inc.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.