We prove the equivalence of Hardyand Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a...

Figalli–Kim–McCann proved in [14] the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent [15, 16], they extend their results to the assumption (A3w) of Trudinger-Wang [34], and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and...

In this paper, we study Hölder-continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant subbundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if th...

Rough Path theory is currently formulated in pvariation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Hölder metric ρ can be used. Based on fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Itô-map (between Rough Path spaces equipped with ρ). We then consider a number of approximations to Browni...

We formulate indefinite integration with respect to an irregular functionas an algebraic problem and provide a criterion for the existence and uniqueness of asolution. This allows us to define a good notion of integral with respect to irregular pathswith Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study theproblem of the existence, uniqueness and conti...

It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Hölder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case (x ∈ R1), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations Di(a 1 ijDju) + ...

In multifractal denoising techniques, the acuracy of the Hölder exponents estimations is crucial for the quality of the outputs. In continuity with the method described in [1], where a wavelet decomposition was used, we investigate the use of another Hölder exponent estimation technique, based on the analysis of the local “oscillations” of the signal. The associated inverse problem to be solved...

We prove the local boundedness and Hölder continuity of weak solutions to nonlocal equations with variable orders exponents under sharp assumptions.

|ξ| ≤ f(x, u, ξ) ≤ L(1 + |ξ|). A function u ∈ W 1,p loc (Ω) is a local minimizer of F in Ω if F (u; spt (v − u)) ≤ F (v; spt (v − u)) , for every v ∈ W 1,p loc (Ω) such that spt (v − u) ⊂⊂ Ω. Well known results due to Giaquinta and Giusti [13, 15] ensure that local minimizers of F are locally α-Hölder continuous for some α < 1. According to Meyers’ example in [19], when f is not continuous in Ω...

The aim of this paper is to show the global existence of weak solutions for a moving boundary problem arising in the non-isothermal crystallization of polymers. The main features of our works are (i) the moving interface is shown to be of co-dimension one; (ii)finite Hölder continuous propagation speed yields an intrinsic estimate of finite co-dimension one Hausdorff measure of the moving inter...