Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra Ŝ of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or co...

In this work it is shown that for every integral n varifold in an open subset U of Rn+m, n,m ∈ N, of locally bounded first variation there exists a countable collection C of n dimensional submanifolds of U of class C2 such that μ(U ∼ S C) = 0 and for each member M of C ~ Hμ(x) = ~ HM (x) for μ almost all x ∈M.

where Ḣ 1 (Ω) is the closure of C ∞ 0 (Ω) in the Sobolev space H 1 1 (Ω) of functions in L(Ω) with one derivative in L. The purpose of this paper is the study of the dependence of λ1,Ω under regular perturbations by diffeomorphisms of Ω, i.e. we want to compute the first variation, the so-called shape derivative, of the functional Ω → λ1,Ω. General results about the stability of λ1,Ω under pert...

We establish the various relationships that exist among the integral transform Ᏺ α,β F , the convolution product (F * G) α , and the first variation δF for a class of functionals defined on K[0,T ], the space of complex-valued continuous functions on [0,T ] which vanish at zero. 1. Introduction and definitions. In a unifying paper [10], Lee defined an integral transform Ᏺ α,β of analytic functi...

A variational problem closely related to the bending energy of curves contained is surfaces of real 3-space forms is considered. We seek curves in a surface which are critical for the elastic energy when this is weighted by the total squared normal curvature energy, under two different sets of constraints: clamped curves and one free end curves of constant length. We start by deriving the first...

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

We consider a family of domains (ΩN )N>0 obtained by attaching an N×1 rectangle to a fixed set Ω0 = {(x, y) : 0 < y < 1, −φ(y) < x < 0}, for a Lipschitz function φ ≥ 0. We derive full asymptotic expansions, as N → ∞, for the mth Dirichlet eigenvalue (for any fixed m ∈ N) and for the associated eigenfunction on ΩN . The second term involves a scattering phase arising in the Dirichlet problem on ...