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.
We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge $\Theta$ for which we introduce a suitable notion of Legendre transform $\Theta^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $\Theta^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.
A holomorphic function f on the unit disc D belongs to the class U-A(D) of Abel universal functions if the family {f(r) : 0 <= r < 1} of its dilates f(r) (z) := f (rz) is dense in the Banach space of all continuous functions on K, endowed with the supremum norm, for any proper compact subset K of the unit circle. We prove that this property is invariant under composition from the left with any non-constant entire function. As an application, we show that U-A(D) is strongly-algebrable. Furthermore, we prove that Abel universality is invariant under composition from the right with an automorphism Phi of D if and only if Phi a rotation. On the other hand, we establish the existence of a subset of U-A(D) which is residual in the space of holomorphic functions on D and is invariant under composition from the right with any automorphism of D.
The goal of this note is to demonstrate that as soon as the hyper-diffusion exponent is greater than one, a class of finite time blow-up scenarios consistent with the analytic structure of the flow (prior to the possible blow-up time) can be ruled out. The argument is self-contained, in spirit of the regularity theory of the hyper-dissipative Navier-Stokes system in turbulent regime developed by Grujic and Xu.
We give a necessary and sufficient condition for a holomorphic self-map phi of the tridisc to induce a bounded composition operator on the associated Hardy space. This condition depends on the behaviour of the first and the second derivative of the symbol at boundary points. We also discuss compactness of composition operators on the bidisc and the tridisc.
By means of hypercyclic operator theory, we complement our previous results on hypercyclic holomorphic maps between complex Euclidean spaces having slow growth rates,by showing {\it abstract abundance} rather than {\it explicit existence}. Next, we establish that, in the space of holomorphic maps from $\mathbb{C}^n$ to any connected Oka manifold $Y$, equipped with the compact-open topology, there exists a {\em dense} subset consisting of common {\em frequently hypercyclic} elements for all nontrivial translation operators. To our knowledge, this is new even for $n=1$ and $Y=\mathbb{C}$.
Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay-a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from co-homology to asymptotic depth. In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, that is, hypersurfaces whose polar maps are birational.
We prove that if f is an element of L-p(R-k) with p < (k(2) + k + 2)/2 satisfies that f is supported on a small perturbation of the moment curve in R-k, then f is identically zero. This improves the more general result in [AN04], and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.
In this paper we prove global well-posedness and scattering for the defocusing, intercritical nonlinear wave equation in dimensions d >= 4 with radial initial data. We prove this for sharp initial data.
We prove that every K3 surface with automorphism group (Z/2Z)(2) admits an explicit birational model as a double sextic surface. This model is canonical for Picard number greater than 10. For Picard number greater than 9, the K3 surfaces in question possess a second birational model, in the form of a projective quartic hypersurface, generalizing the Inose quartic.
In this paper, we show that the rate of convergence in periodic homogenization of convex Hamilton-Jacobi equations is always O(epsilon), which is optimal. This is a natural extension of a result concerning stable norms in metric geometry [6] that is essentially equivalent to the homogenization of convex static Hamilton-Jacobi equations. Another extremely interesting question in this direction is whether the O(epsilon) rate holds in the nonconvex setting, where novel approaches are required. We present a special nonconvex case with O(epsilon) convergence rate, which relies on first identifying the shape of the effective Hamiltonian and then designing suitable strategies in the corresponding game theory interpretation formulas.
. In this paper, we are interested in studying multiplicity of solutions for nonlinear elliptic equations with perturbed symmetry. Existence of infinitely many solutions of superlinear problems with perturbed symmetry was considered by several mathematicians in the 1980s under some restrictive growth conditions on both the unperturbed nonlinear term and the perturbing term which is fixed. While we have a small parameter epsilon to drive the perturbing term, we only need very weak growth conditions on the unperturbed nonlinear term and the perturbing term. We prove that the equations have as many solutions as prescribed when epsilon is suitably small. We give a further extension of the classical result of Berestycki and Lions [ARMA, 82 (1983), 347-375], and we also improve the famous symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [JFA, 14 (1973), 349-381]. A key ingredient of our proof is to find an infinite number of nondegenerate critical values of the unperturbed energy functionals.
We study asymptotic behavior of maximizers for the critical Trudinger-Moser inequalities with a scaling parameter. In particular, we show the point condensation of the maximizers. We also clarify the location of the peak of maximizers in the critical case, as well as in the subcritical case. The location of the peak of maximizer depends on geometric properties of a bounded domain.
