Hölder regularity of the SLE trace

S. Rohde and O. Schramm have recently shown that the SLE trace is Hölder continuous (2005). However, their results are not optimal for all values of and only yield a Hölder exponent near for near 0. In this paper, we give improved lower bounds on the optimal Hölder exponent for two natural parametrizations of the SLE trace. Our estimates give a Hölder exponent near 1 for near 0, as expected. The work of I. Binder and B. Duplantier (2002) suggests that our results are optimal for the two parametrizations considered.

     

Global Hölder regularity for discontinuous elliptic equations in the plane

-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane.

In particular, we deal with the Dirichlet boundary condition

where , 2$">, or with the following normal derivative boundary conditions:

where , 2$">, 0$"> and is the unit outward normal to the boundary .

     

Hölder estimates of solutions to a degenerate diffusion equation

This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations

with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

     

Linear bijections preserving the Hölder seminorm

Let be a compact metric space and let be a real number with The aim of this paper is to solve a linear preserver problem on the Banach algebra of Hölder functions of order from into We show that each linear bijection having the property that for every where

is of the form for every where with is a surjective isometry and is a linear functional.

     

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain in with smooth boundary, the cut locus is the closure of the set of nondifferentiability points of the distance from the boundary of . The normal distance to the cut locus, , is the map which measures the length of the line segment joining to the cut locus along the normal direction , whenever . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of is of class . Our main result is the global Hölder regularity of in the case of a domain with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain . The above regularity result for is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

     

On Hölder continuous Riemannian and Finsler metrics

We discuss smoothness of geodesics in Riemannian and Finsler metrics.

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

We introduce a new method for proving the estimate

where solves the equation . The method can be applied to the Laplacian on . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

     

On the best Hölder exponent for two dimensional elliptic equations in divergence form

We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form:

where is a bounded open subset of and, for every , is a symmetric matrix with bounded measurable coefficients. Such an estimate ``interpolates' between the well-known estimate of Piccinini and Spagnolo in the isotropic case , where is a bounded measurable function, and our previous result in the unit determinant case . Furthermore, we show that our estimate is sharp. Indeed, for every we construct coefficient matrices such that is isotropic and has unit determinant, and such that our estimate for reduces to an equality, for every .

     

Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

We consider the operator

acting on functions in . We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on and . In contrast to previous work, the need only be nonnegative on the boundary rather than strictly positive, at the expense of the and being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.

     

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation

We establish that the initial value problem for the quadratic non-linear Schrödinger equation

where , is locally well-posed in when . The critical exponent for this problem is , and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for .

     

Hörmander index in finite-dimensional case

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.     

Functional Calculus in Hölder-Zygmund Spaces

In this paper we characterize those functions of the real line to itself such that the nonlinear superposition operator defined by maps the Hölder-Zygmund space to itself, is continuous, and is times continuously differentiable. Our characterizations cover all cases in which is real and 0$">, and seem to be novel when 0$"> is an integer.

     

Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions

The well-known Banach-Mazur theorem says that every separable Banach space can be isometrically embedded into . We prove that this embedding can have the property that the image of each nonzero element is a nowhere approximatively differentiable and nowhere Hölder function. It improves a recent result of L. Rodriguez-Piazza where the images are nowhere differentiable functions.

     

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

Foundations of Computational Mathematics - We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and...     

Equilibriums of some non-Hölder potentials

We consider one-sided subshifts with some potential functions which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures and invariant measures absolutely continuous with respect to , where may be finite or infinite. We show that the systems are exact, and are weak Gibbs measures and equilibriums for . We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.

     

Link of groups and homogeneous Hörmander operators

We study a notion of link of Lie groups suggested by the structure of the partial differential operators of Kolmogorov type. As an application of our link procedure we construct explicit examples of stratified Lie groups, with dimension and step arbitrarily large. We also give a set of examples of hypoelliptic second-order operators which are left translation invariant and homogeneous of degree two on the previous groups.

     

Box-counting by Hölder’s traveling salesman

We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a Hölder curve. This implies in particular that if the upper box-counting dimension is less than $$d \ge 1$$, then it can be covered by an $$\frac{1}{d}$$-Hölder curve. On the other hand, for each $$1\le d <2$$ we give an example of a compact set in the plane with lower box-counting dimension equal to zero and upper box-counting dimension equal to d, just failing the above Dini-type condition, that can not be covered by a countable collection of $$\frac{1}{d}$$-Hölder curves.