A Hölder estimate for non-uniform elliptic equations in a random medium

2017 ◽  
Vol 148 ◽  
pp. 61-87
Author(s):  
Shiah-Sen Wang ◽  
Li-Ming Yeh
Keyword(s):  
Elliptic Equations ◽  
Random Medium ◽  
Hölder Estimate

scholarly journals A sharp Hölder estimate for elliptic equations in two variables

2005 ◽  
Vol 135 (1) ◽  
pp. 165-173 ◽  
Author(s):  
Tonia Ricciardi
Keyword(s):  
Elliptic Equations ◽  
Type Inequality ◽  
Divergence Form ◽  
Two Dimensional ◽  
Measurable Coefficients ◽  
The Matrix ◽  
Hölder Estimate

We prove a sharp Hölder estimate for solutions of linear, two-dimensional, divergence-form, elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has unit determinant. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.

scholarly journals Asymptotic Hölder regularity for the ellipsoid process

2020 ◽  
Vol 26 ◽  
pp. 112 ◽  
Author(s):  
Ángel Arroyo ◽  
Mikko Parviainen
Keyword(s):  
Stochastic Process ◽  
Dynamic Programming ◽  
Random Walk ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Dynamic Programming Principle ◽  
Step Size ◽  
Measurable Coefficients ◽  
Continuity Assumption ◽  
Hölder Estimate

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

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