Extremal problems for logarithmic capacity and extremal length.

Abstract

We solve extremal problems for two conformal invariants in the complex plane, logarithmic capacity and extremal length. For a finitely-connected domain $\Omega$ containing $\infty ,$ with boundary $\Gamma ,$ the logarithmic capacity $d(\Gamma )$ is invariant under normalized conformal maps of $\Omega .$ But the capacity of a subset $A \subset \Gamma$ may be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of capacity is the so-called "Robin capacity" of A with respect to $\Omega .$ We find the sharp upper bound, or the "maximal capacity" of A, in terms of conformal invariants of $\Omega :$ harmonic measures of the boundary components of $\Omega$ and periods of their harmonic conjugates (the Riemann matrix), and the capacity of $\Gamma .$ Extremal configurations are described explicitly for certain special cases. "Minimal extremal distance" is defined similarly. Let G be an arbitrary finitely-connected domain (not necessarily containing the point at infinity), whose boundary S is partitioned into subsets X, Y, and Z. For each conformal map f of G define $f(G)$ to be the domain containing $f(G)$ which is bounded only by $f(X)$ and $f(Z).$ Then the extremal distance $\lambda\sb{f}$ between $f(X)$ and $f(Z)$ in $f(G)$ depends on the map f and so we define the "minimal extremal distance" between X and Z in G to be the infimum over all conformal maps f of $\lambda\sb{f}.$ We calculate minimal extremal distance and characterize extremal domains. We then show that minimal extremal distance equals "bridged extremal distance". This is the extremal length of the family of curves connecting X and Z which are allowed to stop at a component of the "bridge" Y = $\partial G/(X \cup Z)$ and re-emerge from any other point of that component. We use bridged extremal distance to give an extremal length interpretation of maximal capacity, via "reduced extremal distance.".