View source for Harnack inequality

''(dual Harnack inequality)''

An inequality that gives an estimate from above and an estimate from below for the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466001.png" /> of two values of a positive harmonic function; obtained by A. Harnack [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466002.png" /> be a harmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466003.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466004.png" />-dimensional Euclidean space; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466005.png" /> be the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466006.png" /> with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466007.png" /> and centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466008.png" />. If the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h0466009.png" />, then the Harnack inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660011.png" /></td> </tr></table>

is valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660014.png" /> is a compactum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660015.png" />, then there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660016.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660018.png" />. In particular,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660019.png" /></td> </tr></table>

Harnack's inequality has the following corollaries: the strong [[Maximum principle|maximum principle]], the [[Harnack theorem|Harnack theorem]] on sequences of harmonic functions, compactness theorems for families of harmonic functions, the Liouville theorem (cf. [[Liouville theorems|Liouville theorems]]), and other facts. Harnack's inequality can be generalized [[#References|[3]]], [[#References|[4]]] to non-negative solutions of a wide class of linear elliptic equations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660020.png" /></td> </tr></table>

with a uniformly positive-definite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660021.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660023.png" /> are numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660024.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660025.png" />-dimensional vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660026.png" />. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660027.png" /> in inequality (2) depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660029.png" />, certain norms of the lower coefficients of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660030.png" />, and the distance between the boundaries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660031.png" /> and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660032.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046600a.gif" />

Figure: h046600a

The analogue of Harnack's inequality is also applicable [[#References|[5]]] to non-negative solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660033.png" /> of uniformly-parabolic equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660034.png" /> (the coefficients of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660035.png" /> may also depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660036.png" />). In such a case only a one-sided inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660037.png" /></td> </tr></table>

is possible for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660038.png" /> lying inside the paraboloid

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660039.png" /></td> </tr></table>

which is concave downwards with apex at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660040.png" /> (Fig., left part). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660041.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660047.png" />, on certain norms of the lower coefficients of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660048.png" />, and on the distance between the boundary of the paraboloid and the boundary of the domain on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660049.png" />. If, for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660050.png" /> in the cylinder

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660051.png" /></td> </tr></table>

if the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660053.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660054.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660055.png" /> is sufficiently small, then the inequality [[#References|[5]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660056.png" /></td> </tr></table>

is valid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660057.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660059.png" /> (Fig., right part), if the compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660061.png" /> are situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660062.png" /> and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660063.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660064.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660065.png" /></td> </tr></table>

The example of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660066.png" /></td> </tr></table>

which is a solution of the heat equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660067.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660068.png" />, shows that in the parabolic case it is impossible to have two-sided estimates.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Harnack,   "Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene" , Leipzig  (1887)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Serrin,   "On the Harnack inequality for linear elliptic equations"  ''J. d'Anal. Math.'' , '''4''' :  2  (1955–1956)  pp. 292–308  {{MR|0081415}} {{ZBL|0070.32302}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Moser,   "On Harnack's theorem for elliptic differential equations"  ''Comm. Pure Appl. Math.'' , '''14'''  (1961)  pp. 577–591  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Moser,   "On Harnack's theorem for parabolic differential equations"  ''Comm. Pure Appl. Math.'' , '''17'''  (1964)  pp. 101–134  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)  {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.M. Landis,   "Second-order equations of elliptic and parabolic type" , Moscow  (1971)  (In Russian)  {{MR|0320507}} {{ZBL|0226.35001}} </TD></TR></table>



====Comments====
See [[#References|[a2]]] for a Harnack inequality up to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660069.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Boboc,   P. Mustaţă,   "Espaces harmoniques associés aux opérateurs différentiels linéaires du second order de type elliptique" , Springer  (1968)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.L. Helms,   "Introduction to potential theory" , Wiley (Interscience)  (1969)</TD></TR></table>