Prompted by recent work of Xiuxiong Chen, a unified approach to the Hamilton-Ricci and Calabi flows on a closed, compact surface is presented, recovering global existence and exponentially fast asymptotic convergence from concentration-compactness results for conformal metrics.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺.

Curved triangular $C^m$-elements which can be pieced together with the generalized Bell’s $C^m$-elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order $2(m+1)$ in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the...

We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension $n-3$.

For a principal type pseudodifferential operator, we prove that condition $\left(\psi \right)$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $ϵ+3/2$ for any $ϵ\>0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $\left(\psi \right)$ doesnotimply local solvability with a loss of 1 derivative,...

L’article reprend systématiquement la théorie des cycles proches pour les $𝒟$-modules holonomes. La théorie est étendue aux complexes, et l’on obtient une équivalence de catégories entre complexes monodromiques et complexes spécialisables (ces derniers, sur le complété de $𝒟$ pour la $V$ filtration). On obtient en particulier les théorèmes de commutation par rapport à la dualité, aux images inverses lisses et aux images directes propres qu’il était naturel d’espérer.

The Special Issue of Kybernetika is devoted to the publication of selected peer-reviewed articles submitted by the participants of the Czech-Japanese Seminar in Applied Mathematics 2008 which took place on September 1-7, 2008 in Takachi-ho and Miyazaki, Japan. The Czech-Japanese Seminar in Applied Mathematics 2008 was organized by the Department of Applied Physics, Faculty of Engineering, University of Miyazaki. It was the fourth meeting in the series of the Czech-Japanese Seminars in Applied Mathematics....