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''(dual 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
+
An inequality that gives an estimate from above and an estimate from below for the ratio $  u( x)/u( y) $
 +
of two values of a positive harmonic function; obtained by A. Harnack [[#References|[1]]]. Let $  u \geq  0 $
 +
be a harmonic function in a domain $  G $
 +
of an $  n $-
 +
dimensional Euclidean space; let $  E _ {r} ( y) $
 +
be the ball $  \{ {x } : {| x - y | < r } \} $
 +
with radius $  r $
 +
and centre at the point $  y $.  
 +
If the closure $  \overline{ {E _ {r} ( y) }}\; \subset  G $,  
 +
then the Harnack inequality
 +
 
 +
$$ \tag{1 }
 +
\left ( {
 +
\frac{r}{r + \rho }
 +
} \right ) ^ {n - 2 }
  
<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>
+
\frac{r - \rho }{r + \rho }
 +
 
 +
u ( y)  \leq  u ( x)  \leq  \
 +
\left ( {
 +
\frac{r}{r - \rho }
 +
} \right ) ^ {n - 2 }
 +
 
 +
\frac{r + \rho }{r - \rho }
 +
u ( y) ,
 +
$$
  
 
or
 
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>
+
$$
 +
\max _ {x \in E _  \rho  ( y) } \
 +
u ( x)  \leq  \left (
  
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
+
\frac{r + \rho }{r - \rho }
 +
\right )  ^ {n} \
 +
\min _ {x \in E _  \rho  ( y) }  u ( x),
 +
$$
  
<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>
+
is valid for all  $  x \in E _  \rho  ( y) $,
 +
$  0 \leq  \rho < r $.
 +
If  $  g $
 +
is a compactum,  $  \overline{g}\; \subset  G $,
 +
then there exists a number  $  M = M( G, g) $
 +
such that
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660018.png" />. In particular,
+
$$ \tag{2 }
 +
M  ^ {-} 1 u ( y)  \leq  u ( x)  \leq  Mu ( y)
 +
$$
  
<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>
+
for any  $  x, y \in \overline{g}\; $.  
 +
In particular,
 +
 
 +
$$
 +
\max _ {x \in g }  u ( x)  \leq  \
 +
M  \min _ {x \in g }  u ( x).
 +
$$
  
 
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
 
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>
+
$$
 +
Lu  \equiv \
 +
\sum _ {i, j = 1 } ^ { n }
 +
{
 +
\frac \partial {\partial  x  ^ {i} }
 +
}
 +
\left ( a  ^ {ij} ( x)
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {j} }
 +
 
 +
\right ) +
 +
\sum _ {i = 1 } ^ { n }
 +
b  ^ {i} ( x)
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
+
 +
c ( x) u  = 0
 +
$$
  
with a uniformly positive-definite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660021.png" />:
+
with a uniformly positive-definite matrix $  \| a  ^ {ij} \| $:
  
<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>
+
$$
 +
\lambda \sum _ {i = 1 } ^ { n }
 +
\xi _ {i}  ^ {2}  \leq  \
 +
\sum _ {i, j = 1 } ^ { n }
 +
a  ^ {ij} ( x) \xi _ {i} \xi _ {j}  \leq  \
 +
\Lambda \sum _ {i = 1 } ^ { n }  \xi _ {i}  ^ {2} ,
 +
$$
  
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" />.
+
where $  \Lambda \geq  \lambda > 0 $
 +
are numbers, $  \xi = ( \xi _ {1} \dots \xi _ {n} ) $
 +
is any $  n $-
 +
dimensional vector and $  x \in G $.  
 +
The constant $  M $
 +
in inequality (2) depends only on $  \lambda $,  
 +
$  \Lambda $,  
 +
certain norms of the lower coefficients of the operator $  L $,  
 +
and the distance between the boundaries of $  G $
 +
and of $  g $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046600a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046600a.gif" />
Line 31: Line 116:
 
Figure: h046600a
 
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
+
The analogue of Harnack's inequality is also applicable [[#References|[5]]] to non-negative solutions $  u( x, t) $
 +
of uniformly-parabolic equations of the form $  u _ {t} + Lu= 0 $(
 +
the coefficients of the operator $  L $
 +
may also depend on $  t $).  
 +
In such a case only a one-sided inequality
 +
 
 +
$$
 +
u ( x, t)  \leq  Mu ( y, \tau )
 +
$$
  
<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  $  ( x, t) $
 +
lying inside the paraboloid
  
is possible for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046600/h04660038.png" /> lying inside the paraboloid
+
$$
 +
\{ {( x, t) } : {| x - y |  ^ {2} \leq
 +
\mu  ^ {2} ( \tau - t),\
 +
\tau - v  ^ {2} \leq  t \leq  \tau } \}
 +
,
 +
$$
  
<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  $  ( y, \tau ) $(
 +
Fig., left part). Here  $  M $
 +
depends on  $  y $,
 +
$  \tau $,
 +
$  \lambda $,
 +
$  \Lambda $,
 +
$  \mu $,
 +
$  \nu $,
 +
on certain norms of the lower coefficients of the operator  $  L $,
 +
and on the distance between the boundary of the paraboloid and the boundary of the domain on which  $  u \geq  0 $.  
 +
If, for instance,  $  u \geq  0 $
 +
in the cylinder
  
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
+
$$
 +
= G \times ( a, b],\ \
 +
\overline{g}\;  \subset  G,
 +
$$
  
<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  $  \partial  G $
 +
and  $  \partial  g $
 +
is at least  $  d > 0 $
 +
and if  $  d $
 +
is sufficiently small, then the inequality [[#References|[5]]]
  
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]]]
+
$$
 +
\mathop{\rm ln} 
 +
\frac{u ( x, t) }{u ( y, \tau ) }
  
<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>
+
\leq  M \left (
  
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
+
\frac{| x - y |  ^ {2} }{\tau - t }
 +
+
  
<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>
+
\frac{\tau - t }{d  ^ {2} }
 +
+
 +
1 \right )
 +
$$
 +
 
 +
is valid in  $  g \times ( a - d  ^ {2} , b ] $.
 +
In particular, if  $  u \geq  0 $
 +
in  $  Q $(
 +
Fig., right part), if the compacta  $  Q _ {1} $
 +
and  $  Q _ {2} $
 +
are situated in  $  Q $
 +
and if
 +
 
 +
$$
 +
\delta  = \
 +
\min _ {\begin{array}{c}
 +
( x, t) \in Q _ {1} , \\
 +
( y, \tau ) \in Q _ {2}
 +
\end{array}
 +
} \
 +
( t - \tau )  > 0,
 +
$$
  
 
then
 
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>
+
$$
 +
\max _ {( x, t) \in Q _ {2} }  u ( x, t)  \leq  M \
 +
\min _ {( x, t) \in Q _ {1} }  u ( x, t),
 +
$$
  
 
where
 
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>
+
$$
 +
= M ( \delta , Q, Q _ {1} , Q _ {2} , L).
 +
$$
  
 
The example of the function
 
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>
+
$$
 +
u ( x, t)  =   \mathop{\rm exp}  \left (
 +
\sum _ {i = 1 } ^ { n }
 +
k _ {i} x  ^ {i} + t
 +
\sum _ {i = 1 } ^ { n }
 +
k _ {i}  ^ {2} \right ) ,
 +
$$
  
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.
+
which is a solution of the heat equation $  u _ {t} - \Delta u = 0 $
 +
for any $  k _ {1} \dots k _ {n} $,  
 +
shows that in the parabolic case it is impossible to have two-sided estimates.
  
 
====References====
 
====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)</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)</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</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</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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</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)</TD></TR></table>
+
<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====
 
====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" />.
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See [[#References|[a2]]] for a Harnack inequality up to the boundary of $  G $.
  
 
====References====
 
====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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.L. Helms,  "Introduction to potential theory" , Wiley (Interscience)  (1969)</TD></TR></table>
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<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>

Latest revision as of 19:43, 5 June 2020


(dual Harnack inequality)

An inequality that gives an estimate from above and an estimate from below for the ratio $ u( x)/u( y) $ of two values of a positive harmonic function; obtained by A. Harnack [1]. Let $ u \geq 0 $ be a harmonic function in a domain $ G $ of an $ n $- dimensional Euclidean space; let $ E _ {r} ( y) $ be the ball $ \{ {x } : {| x - y | < r } \} $ with radius $ r $ and centre at the point $ y $. If the closure $ \overline{ {E _ {r} ( y) }}\; \subset G $, then the Harnack inequality

$$ \tag{1 } \left ( { \frac{r}{r + \rho } } \right ) ^ {n - 2 } \frac{r - \rho }{r + \rho } u ( y) \leq u ( x) \leq \ \left ( { \frac{r}{r - \rho } } \right ) ^ {n - 2 } \frac{r + \rho }{r - \rho } u ( y) , $$

or

$$ \max _ {x \in E _ \rho ( y) } \ u ( x) \leq \left ( \frac{r + \rho }{r - \rho } \right ) ^ {n} \ \min _ {x \in E _ \rho ( y) } u ( x), $$

is valid for all $ x \in E _ \rho ( y) $, $ 0 \leq \rho < r $. If $ g $ is a compactum, $ \overline{g}\; \subset G $, then there exists a number $ M = M( G, g) $ such that

$$ \tag{2 } M ^ {-} 1 u ( y) \leq u ( x) \leq Mu ( y) $$

for any $ x, y \in \overline{g}\; $. In particular,

$$ \max _ {x \in g } u ( x) \leq \ M \min _ {x \in g } u ( x). $$

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

$$ Lu \equiv \ \sum _ {i, j = 1 } ^ { n } { \frac \partial {\partial x ^ {i} } } \left ( a ^ {ij} ( x) \frac{\partial u }{\partial x ^ {j} } \right ) + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u = 0 $$

with a uniformly positive-definite matrix $ \| a ^ {ij} \| $:

$$ \lambda \sum _ {i = 1 } ^ { n } \xi _ {i} ^ {2} \leq \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} ( x) \xi _ {i} \xi _ {j} \leq \ \Lambda \sum _ {i = 1 } ^ { n } \xi _ {i} ^ {2} , $$

where $ \Lambda \geq \lambda > 0 $ are numbers, $ \xi = ( \xi _ {1} \dots \xi _ {n} ) $ is any $ n $- dimensional vector and $ x \in G $. The constant $ M $ in inequality (2) depends only on $ \lambda $, $ \Lambda $, certain norms of the lower coefficients of the operator $ L $, and the distance between the boundaries of $ G $ and of $ g $.

Figure: h046600a

The analogue of Harnack's inequality is also applicable [5] to non-negative solutions $ u( x, t) $ of uniformly-parabolic equations of the form $ u _ {t} + Lu= 0 $( the coefficients of the operator $ L $ may also depend on $ t $). In such a case only a one-sided inequality

$$ u ( x, t) \leq Mu ( y, \tau ) $$

is possible for points $ ( x, t) $ lying inside the paraboloid

$$ \{ {( x, t) } : {| x - y | ^ {2} \leq \mu ^ {2} ( \tau - t),\ \tau - v ^ {2} \leq t \leq \tau } \} , $$

which is concave downwards with apex at $ ( y, \tau ) $( Fig., left part). Here $ M $ depends on $ y $, $ \tau $, $ \lambda $, $ \Lambda $, $ \mu $, $ \nu $, on certain norms of the lower coefficients of the operator $ L $, and on the distance between the boundary of the paraboloid and the boundary of the domain on which $ u \geq 0 $. If, for instance, $ u \geq 0 $ in the cylinder

$$ Q = G \times ( a, b],\ \ \overline{g}\; \subset G, $$

if the distance between $ \partial G $ and $ \partial g $ is at least $ d > 0 $ and if $ d $ is sufficiently small, then the inequality [5]

$$ \mathop{\rm ln} \frac{u ( x, t) }{u ( y, \tau ) } \leq M \left ( \frac{| x - y | ^ {2} }{\tau - t } + \frac{\tau - t }{d ^ {2} } + 1 \right ) $$

is valid in $ g \times ( a - d ^ {2} , b ] $. In particular, if $ u \geq 0 $ in $ Q $( Fig., right part), if the compacta $ Q _ {1} $ and $ Q _ {2} $ are situated in $ Q $ and if

$$ \delta = \ \min _ {\begin{array}{c} ( x, t) \in Q _ {1} , \\ ( y, \tau ) \in Q _ {2} \end{array} } \ ( t - \tau ) > 0, $$

then

$$ \max _ {( x, t) \in Q _ {2} } u ( x, t) \leq M \ \min _ {( x, t) \in Q _ {1} } u ( x, t), $$

where

$$ M = M ( \delta , Q, Q _ {1} , Q _ {2} , L). $$

The example of the function

$$ u ( x, t) = \mathop{\rm exp} \left ( \sum _ {i = 1 } ^ { n } k _ {i} x ^ {i} + t \sum _ {i = 1 } ^ { n } k _ {i} ^ {2} \right ) , $$

which is a solution of the heat equation $ u _ {t} - \Delta u = 0 $ for any $ k _ {1} \dots k _ {n} $, shows that in the parabolic case it is impossible to have two-sided estimates.

References

[1] A. Harnack, "Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene" , Leipzig (1887)
[2] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[3] J. Serrin, "On the Harnack inequality for linear elliptic equations" J. d'Anal. Math. , 4 : 2 (1955–1956) pp. 292–308 MR0081415 Zbl 0070.32302
[4] J. Moser, "On Harnack's theorem for elliptic differential equations" Comm. Pure Appl. Math. , 14 (1961) pp. 577–591
[5] J. Moser, "On Harnack's theorem for parabolic differential equations" Comm. Pure Appl. Math. , 17 (1964) pp. 101–134
[6] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) MR0181836 Zbl 0144.34903
[7] E.M. Landis, "Second-order equations of elliptic and parabolic type" , Moscow (1971) (In Russian) MR0320507 Zbl 0226.35001

Comments

See [a2] for a Harnack inequality up to the boundary of $ G $.

References

[a1] N. Boboc, P. Mustaţă, "Espaces harmoniques associés aux opérateurs différentiels linéaires du second order de type elliptique" , Springer (1968)
[a2] L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)
How to Cite This Entry:
Harnack inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harnack_inequality&oldid=19293
This article was adapted from an original article by L.I. KamyninL.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article