# Scale-invariant Boundary Harnack Principle in Inner Uniform Domains

@article{Lierl2011ScaleinvariantBH, title={Scale-invariant Boundary Harnack Principle in Inner Uniform Domains}, author={Janna Lierl and Laurent Saloff-Coste}, journal={arXiv: Probability}, year={2011} }

We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of local regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.

## 18 Citations

### Boundary Harnack principle and elliptic Harnack inequality

- MathematicsJournal of the Mathematical Society of Japan
- 2019

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes…

### Scale-invariant Boundary Harnack Principle on Inner Uniform Domains in Fractal-type Spaces

- MathematicsPotential Analysis
- 2015

We prove a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that…

### Scale-invariant Boundary Harnack Principle on Inner Uniform Domains in Fractal-type Spaces

- Mathematics
- 2012

We prove a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that…

### Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2020

### The Dirichlet Heat Kernel in Inner Uniform Domains in Fractal-Type Spaces

- MathematicsPotential Analysis
- 2021

This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincaré inequality, and…

### Heat kernel estimates on manifolds with ends with mixed boundary condition

- Mathematics
- 2021

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend…

### Some remarks on uniform boundary Harnack Principles

- Mathematics
- 2015

We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain.

### The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms

- Mathematics
- 2012

### Some Boundary Harnack Principles with Uniform Constants

- MathematicsPotential Analysis
- 2021

We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain by using probabilistic methods.

### Powers Of Generators On Dirichlet Spaces And Applications To Harnack Principles

- Mathematics
- 2020

We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific…

## References

SHOWING 1-10 OF 40 REFERENCES

### Scale-invariant Boundary Harnack Principle on Inner Uniform Domains in Fractal-type Spaces

- Mathematics
- 2012

We prove a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that…

### A boundary Harnack principle in twisted Holder domains

- Mathematics
- 1991

The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in twisted Holder domains of order a for a E (1/2, 1]. For each a E (0, 1/2), there exists a twisted…

### The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms

- Mathematics
- 2012

### ON THE GEOMETRY DEFINED BY DIRICHLET FORMS

- Mathematics
- 1995

Every regular, local Dirichlet form on a locally compact, separable space X defines in an intrinsic way a pseudo metric ρ on the state space. Assuming that this is actually a complete metric…

### Relatively and inner uniform domains

- Mathematics
- 1998

We generalize the concept of a uniform domain in Banach spaces into two directions. (1) The ordinary metric d of a domain is replaced by a metric e ≥ d, in particular, by the inner metric of the…

### Neumann and Dirichlet Heat Kernels in Inner Uniform Domains

- Mathematics
- 2011

— This monograph focuses on the heat equation with either the Neumann or the Dirichlet boundary condition in unbounded domains in Euclidean space, Rie-mannian manifolds, and in the more general…

### Composite media and asymptotic dirichlet forms

- Mathematics
- 1994

We introduce a metric topology on the space of Dirichlet forms and study compactness and closure properties of families of local and non-local forms.

### Potential‐Theoretic Characterizations of Nonsmooth Domains

- Mathematics
- 2004

This paper describes–in terms of potential‐theoretic properties–the necessary and sufficient conditions required for a bounded domain satisfying the capacity density condition to be, respectively: a…

### Aspects of Sobolev-type inequalities

- Mathematics
- 2001

Preface Introduction 1. Sobolev inequalities in Rn 2. Moser's elliptic Harnack Inequality 3. Sobolev inequalities on manifolds 4. Two applications 5. Parabolic Harnack inequalities.

### Martin Boundary of a Fractal Domain

- Mathematics
- 2003

A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack…