Volume 28, Issue 1

In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional J , which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that J is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional J . Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of J or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying .

Consider the Navier–Stokes equations with the initial data a ∈ L 2 σ (R d ). Let u and v be two weak solutions with the same initial value a . If u satisfies the usual energy inequality and if where X 1 (R d ) is the multiplier space (see the definition in the text), then we have u = v .

Let ƒ be a nonconstant entire function and Q be a nonconstant polynomial. Let M be a positive number and k ≥ 2 be an integer. We prove that if ƒ and ƒ ' share Q CM and | ƒ ( k ) ( z )| ≤ M · (1 + | Q ( z )|) whenever ƒ ( z ) = Q ( z ), then is constant. Furthermore, we prove the corresponding normality criterion.

We consider a hypersurface in R n parametrized by a diffeomorphism Φ o of the unit sphere in R n into R n , and we take a point w in the domain I[ Φ o ] enclosed by the image of Φ o , and we consider the ‘hole’ R[ w + εξ ] enclosed by the image of the hypersurface w + εξ , where ξ is a diffeomorism as Φ o with 0 ∈ I[ ξ ] and ε is a small positive real parameter. Then we consider the Dirichlet problem for the Laplace equation in the perforated domain I[ Φ o ] with the hole I[ w + εξ ] removed and show real analytic continuation properties of the solution u and of the corresponding energy integral as functionals of the sextuple of w , ε , ξ , Φ o , and of the Dirichlet data in the interior and exterior boundaries of the perforated domain, which we think of as a point in an appropriate Banach space, around a degenerate sextuple with ε = 0.

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

This article studies diffusion in solids in the case of two phases under isothermal conditions where due to plastic effects the number of vacancies changes when crossing a transition layer, i.e. a reconstitutive phase transition. A segregation model is derived and the equations are studied in the limit of a sharp interface. A Gibbs–Thomson law is derived and it is shown that the vacancy component of the chemical potential jumps across the transition layer thereby explaining recent experimental observations. The thermodynamic correctness of the model is shown as well as the existence of weak solutions with logarithmic free energies.