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STRING THEORY EXTENSIONS OF EINSTEIN–MAXWELL FIELDS: THE STATIC CASE
Preview AbstractWe present a new approach for generating solutions in both the four–dimensional heterotic string theory with one vector field and the five–dimensional bosonic string theory, starting from static Einstein–Maxwell fields. Our approach allows one to construct classes of solutions which are invariant with respect to the total subgroup of three-dimensional charging symmetries of these string theories. The new solution-generating procedure leads to the extremal Israel–Wilson–Perjes subclass of string theory solutions in a special case and provides its natural continuous extension to the realm of nonextremal solutions. We explicitly calculate all string theory solutions related to three-dimensional gravity coupled to an effective dilaton field which arises after an appropriate charging symmetry invariant reduction of the static Einstein–Maxwell system.
ON THE CONNECTION BETWEEN TWO QUASILINEAR ELLIPTIC PROBLEMS WITH SOURCE TERMS OF ORDER 0 OR 1
Preview AbstractWe establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ℝN. The first one, of the form
involves a source gradient term with natural growth, where β is non-negative, λ > 0, f(x) ≧ 0, and α is a non-negative measure. The second one, of the form
presents a source term of order 0, where g is non-decreasing, and μ is a non-negative measure. Here β and g can present an asymptote. The correlation gives new results of existence, non-existence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.
Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions
Preview AbstractConsider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions
where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.