scholarly journals Monotonicity of Harnack Inequality for Positive Invariant Harmonic Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Yifei Pan ◽  
Mei Wang
Keyword(s):  
Positive Solutions ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Monotonicity Property ◽  
Weinstein Equation ◽  
Refined Estimate

A monotonicity property and a refined estimate of Harnack inequality are derived for positive solutions of the Weinstein equation.

scholarly journals Integral Harnack inequality

1985 ◽  
Vol 26 (2) ◽  
pp. 115-120 ◽  
Author(s):  
Murali Rao
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Convex Domains ◽  
Dimensional Hausdorff Measure ◽  
Image Position ◽  
Continuous Boundary ◽  
Integral Version

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals A matrix Harnack inequality for semilinear heat equations

2022 ◽  
Vol 5 (1) ◽  
pp. 1-15
Author(s):  
Giacomo Ascione ◽  
◽  
Daniele Castorina ◽  
Giovanni Catino ◽  
Carlo Mantegazza ◽  
...  
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Harnack Inequality ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Heat Equations ◽  
Standard Heat ◽  
Matrix Version ◽  
Sectional Curvatures ◽  
Parallel Ricci Tensor

<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

Gaussian Estimates for the Heat Kernel of the Weighted Laplacian and Fractal Measures

1992 ◽  
Vol 44 (5) ◽  
pp. 1061-1078 ◽  
Author(s):  
Alberto G. Setti
Keyword(s):  
Riemannian Manifold ◽  
Heat Kernel ◽  
Positive Solutions ◽  
Harnack Inequality ◽  
Smooth Function ◽  
Complete Riemannian Manifold ◽  
Parabolic Harnack Inequality ◽  
Fractal Measures ◽  
Weighted Laplacian ◽  
Gaussian Estimates

AbstractLet 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

Gradient Estimates on Rd

1994 ◽  
Vol 37 (4) ◽  
pp. 560-570 ◽  
Author(s):  
Feng-Yu Wang
Keyword(s):  
Maximum Principle ◽  
Positive Solutions ◽  
Harnack Inequality ◽  
Upper Bounds ◽  
Positive Definite ◽  
Coupling Method ◽  
Gradient Estimates ◽  
The Maximum Principle ◽  
Image Position

AbstractThis paper uses both the maximum principle and coupling method to study gradient estimates of positive solutions to Lu = 0 on Rd, wherewith (aij) uniformly positive definite and aij,bi € C1(Rd). We obtain some upper bounds of |∇u|/u and ∥∇u∥∞/∥u∥∞, which imply a Harnack inequality and improve the corresponding results proved in Cranston [4]. Besides, two examples show that our estimates can be sharp.

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