H?lder estimates for parabolic p(x, t)-Laplacian systems

In this paper we establish the local H?lder continuity of the spatial gradient of weak solutions to the parabolic p(x, t)-Laplacian system $$\begin{array}{lll}\partial_{t}u - {\rm div} \left( a(x, t)|Du|^{p(x, t)-2}Du \right) = 0.\end{array}$$ More precisely, we prove that $$\begin{array}{lll}Du \in C_{\rm loc}^{0;\alpha,\alpha/2} \quad {\rm for\; some} \; \alpha \in (0, 1],\end{array}$$ provided p(·) and a(·) are H?lder-continuous.     

Existence and regularity of solutions for a class of singular (p(x), q(x))-Laplacian systems

In this paper, we study the existence of positive smooth solutions for a class of singular (p(x), q(x))-Laplacian systems using sub and supersolution methods.     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Local Hölder regularity for set-indexed processes

In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff–Merzbach. The first key result is a Kolmogorov-like Hölder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as X_U ? X_V, hence we considered different notions of Hölder-continuity. Then, the localization of these properties leads to various definitions of Hölder exponents, which we compare to one another.     

Interior hölder estimates for solutions of schrödinger equations and the regularity of nodal sets

Abstract

Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.     

Hölder regularity for nondivergence nonlocal parabolic equations

This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first proves Hölder continuity for viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative.     

Hölder regularity for equations of prescribed anisotropic mean curvature type

In this note, we prove Hölder regularity for equations of prescribed anisotropic mean curvature type. As an application, we obtain the regularity of weak surfaces with prescribed anisotropic mean curvature.     

Hölder regularity of optimal mappings in optimal transportation

It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C ^1,α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper.     

Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient

The aim of this work is to prove \({\mathcal{C}^{1,\gamma}}\) regularity up to the boundary for solutions of some fully nonlinear degenerate elliptic equations with a “sublinear” Hamiltonian term.     

Partial Hölder regularity for elliptic systems with non-standard growth

We study partial Hölder regularity for elliptic systems with non-standard growth. We consider general systems with Orlicz growth and discontinuous coefficient factor, for which we prove that their weak solutions are partially Hölder continuous for any Hölder exponents. In addition, we also obtain a similar result for systems with double phase growth.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).