The Harnack Inequality for Singular Equations

2011 ◽  
pp. 137-185
Author(s):  
Emmanuele DiBenedetto ◽  
Ugo Gianazza ◽  
Vincenzo Vespri
Keyword(s):  
Harnack Inequality ◽  
Singular Equations

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

scholarly journals On the parabolic Harnack inequality for non-local diffusion equations

2019 ◽  
Vol 295 (3-4) ◽  
pp. 1751-1769 ◽  
Author(s):  
Dominik Dier ◽  
Jukka Kemppainen ◽  
Juhana Siljander ◽  
Rico Zacher
Keyword(s):  
Harnack Inequality ◽  
Diffusion Equations ◽  
Parabolic Harnack Inequality ◽  
Non Local ◽  
Local Diffusion

Boundary behaviour for solutions of elliptic singular equations with a gradient term

2008 ◽  
Vol 69 (12) ◽  
pp. 4550-4566 ◽  
Author(s):  
F. Cuccu ◽  
E. Giarrusso ◽  
G. Porru
Keyword(s):  
Gradient Term ◽  
Boundary Behaviour ◽  
Singular Equations

Harnack inequality under the change of metric

2015 ◽  
Vol 115 ◽  
pp. 89-102
Author(s):  
Santi Tasena ◽  
Laurent Saloff-Coste ◽  
Sompong Dhompongsa
Keyword(s):  
Harnack Inequality

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.

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