JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 155.69.24.ABSTRACT. The first eigenvalue A = RI for the equation div(jVulp 2Vu) + Alulp 2u = 0 is simple in any bounded domain. (Through the nonlinear counterpart to the Rayleigh quotient A, is related to the Poincare inequality.)

For a function ϕ in L 2 (0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates ϕ(nx), n = 1, 2, 3, . . . , constitutes a Riesz basis or a complete sequence in L 2 (0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f (s) = n a n n −s , where the coefficients a n are square summable. It proves useful to model H as the H 2 space of the infinite-dimensional polydisk, or, which is the same, the H 2 space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given f in H and characters χ, f χ (s) = n a n χ(n)n −s is a vertical limit function of f . We study certain probabilistic properties of these vertical limit functions.

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Abstract. We discuss and compare various notions of weak solution for the p-Laplace equationand its parabolic counterpartIn addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise derivatives (jets). Our main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.

We study the non-local eigenvalue problemfor large values of p and derive the limit equation as p → ∞. Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.