Holder Estimates Research Articles

Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space

In this article we investigate the asymptotic behavior at infinity of viscosity solutions to degenerate elliptic equations. We obtain Holder estimates, up to the flat boundary, by using the rescaling method. Also as a byproduct we obtain a Liouville type result on Baouendi-Grushin type operators.

Global regularity for a class of Monge-Ampère type equations

In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampere type equations. By employing the concept of (a, η) type domain, we emphasize that the boundary regularity depends on the convexity of the domain in nature. The key idea of our proof is to provide more effective global Holder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.

Schauder-type estimates for higher-order parabolic SPDEs

In this paper, we consider the Cauchy problem for 2m-order stochastic partial differential equations of parabolic type in a class of stochastic Holder spaces. The Holder estimates of solutions and their spatial derivatives up to order 2m are obtained, based on which the existence and uniqueness of solution are proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when m is odd or even: the condition for odd m coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even m it depends on the integrability index p. The sharpness of the new-found coercivity condition is demonstrated by an example.

An $$L^2$$ to $$L^\infty $$ Framework for the Landau Equation

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $$L^{2}\ \text{to}\ L^{\infty }$$ framework to construct global unique solutions near Maxwellian with small $$L^{\infty }$$ norm. The first step is to establish global $$L^{2}$$ estimates with strong velocity weight and time decay, under the assumption of $$L^{\infty }$$ bound, which is further controlled by such $$L^{2}$$ estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv:1505.04608 (2015)). The second step is to employ estimates in $$S_{p}$$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv:1506.01908 (2015), Imbert and Mouhot in arXiv:1505.04608 (2015)).

Pseudo-Differential Operators in Hölder Spaces Revisited: Weyl–Hörmander Calculus and Ruzhansky–Turunen Classes

In this work, we obtain continuity results on Holder spaces for operators belonging to a Weyl–Hormander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With our results we recover some historical Holder boundedness theorems (see Beals in Annales de l’institut Fourier 29(3):239–260, 1979; Lp and Holder estimates for pseudodifferential operators: necessary conditions. In: Harmonic analysis in Euclidean spaces (proceedings of symposia in pure mathematics, Williams College, Williamstown, MA, 1978). American Mathematical Society, Providence, pp 153–157, 1979). The action of (periodic) Ruzhansky–Turunen classes of pseudo-differential operators on Holder spaces also will be investigated.

A Weak Reverse Hölder Inequality for Caloric Measure

Following a result of Bennewitz–Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set $$\Omega \subset \mathbb {R}^{n+1}$$, assuming as a background hypothesis only that the essential boundary of $$\Omega $$ satisfies an appropriate parabolic version of Ahlfors–David regularity (which entails some backwards in time thickness). We also show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with “lateral” data in $$L^p$$, for some $$p<\infty $$, in this setting.

Global estimates for generalized Forchheimer flows of slightly compressible fluids

This paper is focused on the generalized Forchheimer flows of slightly compressible fluids in porous media. They are reformulated as a degenerate parabolic equation for the pressure. The initial boundary value problem is studied with time-dependent Dirichlet boundary data. The estimates up to the boundary and for all time are derived for the L∞-norm of the pressure, its gradient and time derivative. Large-time estimates are established to be independent of the initial data. Particularly, thanks to the special structure of the pressure’s nonlinear equation, the global gradient estimates are obtained in a relatively simple way, avoiding complicated calculations and a prior requirement of Holder estimates.

Periodic homogenization of elliptic systems with stratified structure

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp \begin{document}$ O(\varepsilon) $\end{document} -convergence rate in \begin{document}$ L^{p_0}(\Omega) $\end{document} with \begin{document}$ p_0 = \frac{2d}{d-1} $\end{document} is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an \begin{document}$ O(\varepsilon^\sigma) $\end{document} -convergence rate is also derived for some \begin{document}$ \sigma by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior \begin{document}$ W^{1, p} $\end{document} and Holder estimates are also obtained by the real variable method.

Hölder metric subregularity for constraint systems in Asplund spaces

This paper deals with the Holder metric subregularity property of a certain constraint system in Asplund space. Using the techniques of variational analysis, its main part is devoted to establish new sufficient conditions in dual spaces for Holder metric subregularity and estimate its modulus, which are derived in terms of coderivatives and normal cones. As an application, the results are applied to study the relationship between higher order growth condition of an unconstraint minimization problem and Holder metric subregularity property of the related constraint system.

A note on Holder estimates of solutions to a degenerate diffusion equation

A note on Holder estimates of solutions to a degenerate diffusion equation

Tangential Cauchy–Riemann Equations on Pseudoconvex Boundaries of Finite and Infinite Type in $$\mathbb {C}^2$$ C 2

In this paper, we will present what we think is a natural extension of the finite type condition in the sense of Range on pseudoconvex domains. This type condition generalizes the notion of finite type in the original theory as well as consists many cases of infinite type in the sense of Range. Following the integral representation by Henkin, we also provide \(L^p\) and “new” Holder estimates for solutions of \(\bar{\partial }_b\)-equations on the boundaries of domains which emphasize the infinite type condition.

Hölder estimates for viscosity solutions of equations of fractional p-Laplace type

We prove Holder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation $$PV {\int_{\mathbb{R}^{n}}\frac{|u(x)-u(x + y)|^{p-2}(u(x) - u(x + y))}{|y|^{n+sp}}\,dy = 0},$$ where \({s \in (0,1)}\) and \({p > 2}\) or \({1/(1-s) < p < 2}\). Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Holder estimates for weak solutions.

ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains

We prove global Hölder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions à la Cabré.

Wiener-type tests from a two-sided Gaussian bound

In this paper, we are concerned with hypoelliptic diffusion operators \(\mathcal {H}\). Our main aim is to show, with an axiomatic approach, that a Wiener-type test of \(\mathcal {H}\)-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of \(\mathcal {H}\) with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Holder estimates of the solution under a suitable exterior cone condition.

Schauder estimates for singular parabolic and elliptic equations of Keldysh type

We show that solutions of equations of the form \[ -u_t+D_{11}u+(x^1)D_{22}u = f \] (and also more general equations in any number of dimensions) satisfy simple Holder estimates involving their derivatives. We also examine some pointwise properties for these solutions. Our results generalize those of Daskalopoulos and Lee, and Hong and Huang.

Hölder Continuity of Solutions to Nonlinear Parabolic Equations Degenerated on a Part of the Domain

We study the regularity of solutions to parabolic p-Laplace type equations degenerating uniformly with respect to a small parameter e on a part of the domain. We prove e-uniform estimates for the maximum of modulus, and Holder estimates for the modulus of continuity of the solution. We also prove the Harnack inequality of a special form.

On the Hölder continuity of solutions of the Venttsel’ elliptic problem

For solutions of the Venttsel’ problem in divergence form, a local Holder estimate is obtained. It is shown that Harnack’s inequality in classical form does not hold.

Hölder stability for Serrin’s overdetermined problem

In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–932, 1999) for Serrin’s overdetermined problem. It also extends to a large class of semilinear equations the Holder estimate obtained in Brandolini et al. (J Differ Equ 245:1566–1583, 2008) for the case of torsional rigidity (\(f\equiv 1\)) by means of integral identities. The proof hinges on ideas contained in Aftalion et al. (1999) and uses Carleson-type estimates and improved Harnack inequalities in cones.

A compactness result for Fano manifolds and Kähler Ricci flows

We obtain a compactness result for Fano manifolds and Kahler Ricci flows. Comparing to the more general Riemannian versions in Anderson (Invent Math 102(2):429–445, 1990) and Hamilton (Am J Math 117:545–572, 1995), in this Fano case, the curvature assumption is much weaker and is preserved by the Kahler Ricci flows. One assumption is the $$C^1$$ boundedness of the Ricci potential and the other is the smallness of Perelman’s entropy. As one application, we obtain a new local regularity criteria and structure result for Kahler Ricci flows. The proof is based on a Holder estimate for the gradient of harmonic functions and mixed derivative of Green’s function.

Hölder Estimates for Nonlocal-Diffusion Equations with Drifts

We study a class of nonlocal-diffusion equations with drifts, and derive a priori \(\Phi \)-Holder estimate for the solutions by using a purely probabilistic argument, where \(\Phi \) is an intrinsic scaling function for the equation.