Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

We consider the semilinear elliptic equation Δu=h(u) in Ω{0}, where Ω is an open subset of (N2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q(1,C_N) (that is, lim_u→∞h(λu)/h(u)=λ^q, for every λ>0), where C_N denotes either N/(N−2) if N3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=u^q.     

Conformal restriction: The chordal case

We characterize and describe all random subsets of a given simply connected planar domain (the upper half-plane , say) which satisfy the ``conformal restriction' property, i.e., connects two fixed boundary points ( and , say) and the law of conditioned to remain in a simply connected open subset of is identical to that of , where is a conformal map from onto with and . The construction of this family relies on the stochastic Loewner evolution processes with parameter and on their distortion under conformal maps. We show in particular that SLE is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE.     

Fourier restriction for affine arclength measures in the plane

We obtain an analog, uniform for a large class of curves in the plane, of the Fefferman-Zygmund theorem on restriction of the Fourier transform.

     

The Liouville equation in an annulus

We give the solutions to the Liouville equation in an annulus A of R² that satisfy a certain Neumann condition on each component of ∂A. As a consequence, we classify all the metrics of constant curvature in A that have constant geodesic curvature on ∂A.     

Korn's inequality and Donati's theorem for the conformal Killing operator on pseudo-Euclidean space

We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space Rp,q, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.     

Estimates for conformal capacity

Let a,b,c,d be distinct points on overline bf R ⁿ . By p we denote the minimal conformal capacity of all rings (E,F) with a,b ∈ E and c,d∈ F . For n=2 , we use explicit expressions of p in terms of complete elliptic integrals to prove a sharp inequality that connects p and the conformal capacity of Teichmüller's ring. We also show, by a concrete example, how we can use techniques involving polarization and hyperbolic geometry to prove estimates for the conformal capacity of rings. November 16, 1998. Date revised: March 2, 1999. Date accepted: May 7, 1999.     

On the theory and application of a domain decomposition method for computing conformal modules

We consider the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals. The method has been studied already by us and also by Gaier and Hayman. Our main purpose here is to extend its area of application and, in the same time, improve some of our earlier error estimates.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities

We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Lalín. As a corollary, we also obtain three q-series expansions for the dilogarithm.     

Estimates for an oscillatory integral operator related to restriction to space curves

We consider the oscillatory integral operator defined by

where 1$">, and is a real-valued function in . This operator may be thought of as a variable-curve version of the adjoint of the Fourier restriction operator for space curves. Under a certain nondegeneracy condition on , we obtain estimates for with a suitable bound for the operator norm . This generalizes a result of Hörmander for the plane to higher dimensions.

     

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

二阶椭圆型方程奇摄动边值问题的渐近解

研究了一类二阶线性椭圆型方程的奇摄动边值问题.利用边界层函数法构造出问题的零次形式近似,并应用椭圆型算子的最大值原理对问题的解作出渐近估计.     

Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.     

Elliptic equations for measures: Regularity and global bounds of densities

We consider elliptic equations of the form L^*μ=ν for measures on . The membership of solutions in the Sobolev classes is established under appropriate conditions on the coefficients of L. Bounds of the form (x)CΦ(x)⁻¹ for the corresponding densities are obtained.     

Periodic reiterated homogenization for elliptic functions

In this paper, we study reiterated homogenization for equations of the form . We assume that a is a Carathéodory function and satisfies some monotonicity and growth conditions and its reiterated unfolding converges almost everywhere to a Carathéodory type function. Under these assumptions, we show that the sequence of solutions converges to the solution of a limit variational problem. In particular this contains the case , where a is periodic in the second and third arguments, and continuous in each argument.     

On nonuniqueness of solutions to elliptic equations for probability measures

We consider the set of all probability measures μ on Rd satisfying an elliptic equation L^∗μ=0 in the weak worm. We give sufficient conditions in order that this set contains at least two different elements. We also construct new examples of nonuniqueness for such equations.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.