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If the so-called Collatz method is applied to get twosided estimates of the first eigenvalue $λ_{1}$ , the sequences of the so-called Schwarz quatients (which are upper bounds for $λ_{1}$ ) and of the so-called Temple quotients (which are lower bounds) are constructed. While monotony of the first sequence was proved many years ago, monotony of the second one has been proved only recently by F. goerisch and J. Albrecht in their common paper “Die Monotonie der Templeschen Quotienten” (ZAMM, in print). In the present...

A propagation theorem for a class of microfunctions

Andrea D'Agnolo, Giuseppe Zampieri (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let $A$ be a closed set of $M ≃ R^{n}$ , whose conormai cones $x + y_{x}^{*} (A)$ , $x \in A$ , have locally empty intersection. We first show in §1 that $dist (x, A)$ , $x \in M ∖ A$ is a $C^{1}$ function. We then represent the n microfunctions of $C_{A | X}$ , $X ≃ C^{n}$ , using cohomology groups of $O_{X}$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${C_{A | X}|}_{{\dot{π}}^{- 1} (x_{0})}$ , $x_{0} \in \partial A$ , satisfy the principle of the analytic continuation in the complex integral manifolds of ${\{H (ϕ_{i}^{C})\}}_{i = 1, \dots, m}$ , $\{ϕ_{i}\}$ being a base for the linear hull of $γ_{x_{0}}^{*} (A)$ in $T_{x_{0}}^{*} M$ ; in particular we get ${Γ_{A \times_{M} T^{*}_{M} X} (C_{A | X})|}_{\partial A \times_{M} \dot{T}^{*}_{M} X} = 0$ . When $A$ is a half space with $C^{ω}$ -boundary,...

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A priori regularity for weak solutions of some nonlinear elliptic equations

A Priori Schranken für die Ableitungen der Lösungen gewisser elliptischer Differentialgleichungssysteme.

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen.

A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation.

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

A problem in Rayleigh-Taylor instability

A problem of thermal shock with thermal relaxation.

A problem of transversal anisotropic ellipticity

A problem related to the Hall effect in a semiconductor with an electrode of an arbitrary shape.

A problem with internal conditions for a pseudoparabolic equation.

A problem with nonlocal boundary conditions for a quasilinear parabolic equation.

A proof of monotony of the Temple quotients in eigenvalue problems

A propagation theorem for a class of microfunctions

A Property of Biharmonic Functions with Dirichlet Finite Laplacians.

A Property of Measures in Rn and an Application to Unique Continuation.

A property of the defining equations for the Lie algebra in the group classification problem for wave equations.

A property of the $H$ -convergence for elasticity in perforated domains.

A Pseudo-Algebra of Observables for the Dirac Equation.

A pseudoconformal compactification of the nonlinear Schrödinger equation and applications.

A pseudoconvex domain with bounded solutions for ..., but not admitting ...-estimates.

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