For an essentially arbitrary convex function Psi : [1, infinity) -> [1, infinity), we consider uniqueness in two related extremal problems for integral(X)psi |k(f)(z,f))d sigma (z), K-f(z,f) = |f(z)|(2)+ |f(z & strns;)|(2) / |f(z)|(2)- |f(z & strns;)|(2) where K-f (z, f) is the pointwise distortion of a finite distortion function f. The first case is the boundary value problem on the unit disk X = D and d sigma(z) = lambda(z) dz for a weight lambda >= 1 and boundary data f(0) : & sect; -> & sect;. The second case is to find the extremal in the homotopy class [f(0)] of a homeomorphism between a Riemann surface X with the hyperbolic (area) metric d sigma(z). The main new tools developed here are generalisations of the classical Reich-Strebel inequalities to this variational setting. These give uniqueness theorems for our recent studies uniting different approaches to the Teichmu & uml;ller theory of Riemann surfaces. We are then able to give a much stronger version of uniqueness for Teichmu & uml;ller-type mappings to the setting of mappings of finite distortion.
We address the compressible Euler equations in a domain with a free elastic boundary, evolving according to a damped fourth-order plate equation forced by the fluid pressure. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data in H-3. The main new device introduced is a variable coefficients space tangential-time differential operator Q of order 1, which is of transport type and allows the logarithm of the density g to evolve according to a wave-type equation Q(2)g - div(a )(f g) = F, where f is related to the speed of the waves and a is the change of variable. This operator captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics.
A compact set K is an element of C-n is said to be rationally convex if every point p outside of K admits a holomorphic polynomial whose zero locus passes through p but does not intersect K. There are two main generalizations of this to a general Stein manifold X: one where the polynomials are replaced with entire functions, and another where the zero locus of the polynomial is replaced by a complex hypersurface. We show that the latter is precisely the notion of convexity with respect to meromorphic functions, while the former is precisely the notion of convexity with respect to strong meromorphic functions. Various approximation results and a Duval-Sibony-type theorem are shown for each notion of convexity. Other generalizations of rational convexity to Stein manifolds are discussed.
This study establishes the existence of inertial manifolds for the hyperviscous Navier-Stokes equations (HNSE) on a two-dimensional periodic domain: partial derivative(t)u + nu(-triangle)(beta)u + (u center dot del)u + del p = f, on T-2, with del center dot u = 0, for any beta > 17/12. In the context of the 2D HNSE, the exponent beta = 3/2 is identified as the "critical" value for the inertial manifold problem, below which the spectral gap condition is no longer satisfied. A key contribution of this work is that it extends the theory to the "supercritical" regime, where beta < 3/2. An important component of the argument involves a refined analysis of the sparse distribution of lattice points in annular regions.
We confirm, in dimension two, B & lstrok;ocki's conjectures on sharp lower bounds for Bergman kernels of tube domains. To that end, we verify a broader class of Lp-Mahler conjectures due to Berndtsson and the authors, where p = 1 are B & lstrok;ocki's conjecture, and p = infinity are Mahler's conjectures. The proofs are technically challenging as the Lp-Mahler volume is considerably harder to deal with analytically compared to Mahler's volume, and furthermore invariance under duality is lost. In addition, unlike in the classical Mahler setting, the non-symmetric setting is considerably more involved than the symmetric one. The proofs involve studying the effect of Mahler's classical sliding of vertices on two-dimensional polytopes on the Lp-polar body (no longer a polytope). Some arguments are inspired by works of CampiGronchi and Meyer-Reisner on volumes of classical polar bodies of shadow systems. In passing, we also explore how Mahler's sliding affects the isotropic constant. This leads to an elementary proof of Bourgain's strong hyperplane conjectures in dimension two, originally due to Bisztriczky-Bo & uml;oro & uml;czky, Campi-ColesantiGronchi, and Meckes. Specifically, we show that, as a function of the sliding parameter, the isotropic constant raised to an appropriate power is a convex quadratic polynomial.
