Outer Measures on a Commutative Ring Induced by Measures on Its Spectrum

On a commutative ring R we study outer measures induced by measures on Spec(R). The focus is on examples of such outer measures and on subsets of R that satisfy the Carathéodory condition.

A Semilinear Dirichlet Problem

Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem(I)where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem(II)has nontrivial weak solutions.