Durch Doppelreihenentwicklung nach Legendreschen Polynomen wird die Lösung des Dirichletschen Problems mittels Picard-Iteration für einen beliebigen symmetrischen oder unsymmetrischen Rand numerisch durchgeführt. Für die erste Näherung der Lösung werden die Randwerte auf eine die Randkurve möglichst eng umschliessende Ellipse übertragen. Diese erste Näherung wird korrigiert durch Addition von Lösungen der homogen gemachten Differentialgleichung (Laplacesche Differentialgleichung), die den Fehler längs der Randkurve nach der Methode der kleinsten Quadrate zueinem Minimum machen. Die Berechnung dieser Korrektur wird besonders einfach im Falle eines achsensymmetrischen Randes. Ein Beispiel erläutert das Vorgehen.

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Given a closed positive current$T$on a bounded Runge open subset Ω of$C$^{$n$}, we study sufficient conditions for the existence of a global extension of$T$to$C$^{$n$}. When$T$has a sufficiently low density, we show that the extension is possible and that there is no propagation of singularities, i.e.$T$may be extended by a closed positive$C$^{∞}-form outside $$\bar \Omega $$ . Conversely, using recent results of$H.$Skoda and$H.$El Mir, we give examples of non extendable currents showing that the above sufficient conditions are optimal in bidegree (1, 1).

The dynamics of dilute electrons can be modelled by the fundamental VlasovPoissonBoltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.

In this paper, we present a new algorithm to solve algebraically the following lattice-related problems: 1) the small integer solution (SIS) problem under the condition: if the solution is bounded by an integer β in l_∞ norm, which we call a bounded SIS (BSIS) problem, and if the difference between the row dimension n and the column dimension m of the corresponding basis matrix is relatively small with respect the row dimension m; 2) the learning with errors (LWE) problems under the condition: if the errors are bounded - the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D < q), which we call a learning with bounded errors (LWBE) problem. We will show that we can solve these problems with polynomial complexity.