Characterization of function spaces via low regularity mollifiers

Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\ast\rho_\varepsilon$ to $f$, where $\rho$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $\rho$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $\rho$ is a characteristic function

Asymptotic behavior of critical points of an energy involving a loop-well potential

We describe the asymptotic behavior of critical points of $\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry on $W$ or $\Gamma$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest

Density in Ws,p(Ω;N)W^{s,p}(\Omega ; N)

Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$, 0\textless{}s\textless{}\infty and 1\le p\textless{}\infty. We prove that $C^\infty(\overline\Omega\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$ except when 1\le sp\textless{}2 and $n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when s\textless{}1. With 0\textless{}s\textless{}1, 1\le p\textless{}\infty and sp\textless{}n, $\Omega$ a ball, and $N$ a general compact connected manifold, we prove that $C^\infty(\overline\Omega \, ; N)$ is dense in $W^{s,p}(\Omega \, ; N)$ if and only if $\pi\_{[sp]}(N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\ldots$ [General domains $\Omega$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to s\textless{}1.] The case where s\textgreater{}1, $s\not\in{\mathbb N}$, is still open.Comment: To appear in J. Funct. Anal. 49

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let $\Omega$ be a smooth bounded simply connected domain in $\mathbb{R}^2$. We investigate the existence of critical points of the energy $E_\varepsilon (u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\Omega$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in "most" of the domains

Design of a knowledge and management system for starch bioconversion

AbstractIn this paper a knowledge acquisition and management system (KAMS) which allows the collection, analysis, ordering and storage of informations generated at starch liquefaction was developed. KAMS was structured on three levels: PostgreSQL as a backend, D2RQ as middle tier and Seaside as frontend. The system was used to store knowledge about the liquefaction process with the goal to be used as a decision support system in chosing the condition for this operation. The tests had shown that the implemented KAMS provides support for: Distributed acquisition of the scientific data generated by the researchers; Structured data storage; Support for the generation and storage of knowledge on the starch bioconversion