On variational principles of conformal mappings

Abstract

Abstract. Refinements and generalizations of the classical variational principles of conformal mappings are presented; mainly, they follow from potential theory and symmetrization. Part of the results can be viewed as properties of Robin functions and Robin capacities, and also as distortion theorems for univalent functions in finitely connected domains. §1. Introduction and main definitionsThe principles mentioned in the title are of major importance in the theory of functions of a complex variable and in the mechanics of solids [1,2]. In the present paper, we are concerned with qualitative variational principles, i.e., statements enabling us to judge the variation of a conformal map by the variation of the boundary of a domain. For instance, a classical qualitative variational principle says that if a function w = f (z) maps a domain D, ∞ ∈ D ⊂ ∆ z := {z : |z| > 1}, conformally and univalently onto the exterior ∆ w of the unit disk in such a way that f (∞) = ∞, then |f (∞)| ≤ 1, at an arbitrary point z ∈ (∂D) ∩ (∂∆ z ) we have |f (z)| ≤ 1 (if the derivative exists), and for every ρ > 1 the level curve |f (z)| = ρ is included in |z| ≥ ρ. In other words, when ∆ z is deformed to D, the modulus of the derivative decays at infinity and at the fixed points on the boundary, and the level curves expand. Similar principles are valid for functions realizing conformal mapping onto other types of standard domains: the half-plane H w := {w : Im w > 0} and the strip S w := {w : 0 < Im w < 1}; see [2, Subsection 61] (in contrast to [2], f is compared with the identity mapping; this does not lead to any loss of generality). The principles mentioned above can be regarded as properties of the complex potentials of stationary parallel plane vector fields. They follow, for example, from the Schwarz lemma or the maximum principle for harmonic functions. Various extensions and refinements of these principles have been known for some time. Moreover, modern methods of the geometric theory of functions of a complex variable (see [3]) make it possible to obtain new variational theorems that take into account additional normalization of a mapping or the nature of the deformation of the domain in question. However, these facts have not been reflected in the literature in due form. Our purpose is to fill this gap, at least partly. In the paper, we present variational principles that are derived from potential theory and symmetrization in a uniform way. For a canonical domain K z , we study the properties of a map f : D → K w in their dependence on the deformation (D\K z ) ∪ (K z \D). If D ⊂ K z , we use the term inner deformation, and if D ⊃ K z , we use the term outer 2000 Mathematics Subject Classification. Primary 30C70, 30C85.

Variational formulae for conformal capacity

“…), the function u was defined in Theorem 1, and ∂/∂l denotes differentiation along the tangent to γ . Formulae (1.2) and (1.3) refine quantitatively some variational principles for conformal mappings (for instance, see [2], [3] and [12]). It is also known that the Dirichlet integral has various physical interpretations.…”

“…In addition, [9] contains applications of the results obtained there to estimates for variations of the aerodynamical lift of an airfoil. In the general case the quadratic forms in Theorem 3 were also considered in [4] and [12]. In [17] the reader can find general inequalities for such quadratic forms, which contain reduced moduli of strips and half-strips, with applications to problems of extremal partition and related problems in geometric function theory.…”

О Квадратичных Формах, Порожденных Функциями Грина И Робена

“…, n. Ранее изучалась монотонность квадратичной формы (1) при деформации множеств B и Γ в ряде частных случаев (см. [3]- [5], [7]) и при поляризации (см. [7]).…”

“…[3]- [5], [7]) и при поляризации (см. [7]). Наиболее полные утверждения о монотонности вытекают из свойств емкостей обобщенных конденсаторов (см.…”

“…Неравенства для квадратичных форм (1) приводят к теоремам искажения в теории однолистных функций (см. [7], [20], [21], [28]). Разбиения на полосы и полуполосы эффективно даже в случае n = 1 и Γ = ∂B.…”