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Harnack's first theorem: If a sequence of functions which are harmonic in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466201.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466202.png" /> converges uniformly on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466203.png" />, then it also converges uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466204.png" /> to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,
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Harnack's first theorem: If a sequence of functions which are harmonic in a bounded domain $G$ and continuous on $\overline G$ converges uniformly on the boundary $\partial G$, then it also converges uniformly in $G$ to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,
  
<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/h046620/h0466205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial y_i}+\sum_{i=1}^na_i(x)\frac{\partial u}{\partial x_i}+a(x)u=0,\label{*}\tag{*}$$
  
which has a unique solution of the [[Dirichlet problem|Dirichlet problem]] for any continuous boundary function . If the sequence of solutions of equation (*) converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466206.png" />, then it also converges uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466207.png" /> to a solution of equation (*).
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which has a unique solution of the [[Dirichlet problem|Dirichlet problem]] for any continuous boundary function . If the sequence of solutions of equation \eqref{*} converges uniformly on $\partial G$, then it also converges uniformly in $G$ to a solution of equation \eqref{*}.
  
Harnack's second theorem, the Harnack principle: If a monotone sequence of harmonic functions in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466208.png" /> converges at some point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h0466209.png" />, then it converges at all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662010.png" /> to a harmonic function, and this convergence is uniform on any closed subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662011.png" />. Harnack's second theorem can be generalized to monotone sequences of solutions of the elliptic equation .
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Harnack's second theorem, the Harnack principle: If a monotone sequence of harmonic functions in a bounded domain $G$ converges at some point in $G$, then it converges at all points of $G$ to a harmonic function, and this convergence is uniform on any closed subdomain of $G$. Harnack's second theorem can be generalized to monotone sequences of solutions of the elliptic equation .
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
In the axiomatic theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]) the first Harnack theorem is known as the Bauer convergence property and the second Harnack theorem as the Brélot convergence property, see [[#References|[a3]]] and [[#References|[a1]]]. The following properties are equivalent to the Brélot convergence property (see [[#References|[a4]]]): 1) each positive harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662012.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662013.png" /> is either strictly positive or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662014.png" />. Moreover, the set of positive harmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662015.png" />, equal to 1 in a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662016.png" />, is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]); and 2) for any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662017.png" /> and any compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662019.png" /> there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662021.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662022.png" /> and any positive harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046620/h04662024.png" /> (the [[Harnack inequality|Harnack inequality]]).
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In the axiomatic theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]) the first Harnack theorem is known as the Bauer convergence property and the second Harnack theorem as the Brélot convergence property, see [[#References|[a3]]] and [[#References|[a1]]]. The following properties are equivalent to the Brélot convergence property (see [[#References|[a4]]]): 1) each positive harmonic function $u$ on a domain $U$ is either strictly positive or $u=0$. Moreover, the set of positive harmonic functions on $U$, equal to 1 in a given point $u\in U$, is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]); and 2) for any domain $U$ and any compact subset $K$ of $U$ there exists a constant $c>0$ such that $u(x)\leq cu(y)$ for any $x,y\in K$ and any positive harmonic function $u$ on $U$ (the [[Harnack inequality|Harnack inequality]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-M. Bony,  "Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel"  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)  pp. 69–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Loeb,  B. Walsh,  "The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brélot"  ''Ann. Inst. Fourier'' , '''15''' :  2  (1965)  pp. 597–600</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-M. Bony,  "Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel"  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)  pp. 69–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Loeb,  B. Walsh,  "The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brélot"  ''Ann. Inst. Fourier'' , '''15''' :  2  (1965)  pp. 597–600</TD></TR></table>

Latest revision as of 17:09, 14 February 2020

Harnack's first theorem: If a sequence of functions which are harmonic in a bounded domain $G$ and continuous on $\overline G$ converges uniformly on the boundary $\partial G$, then it also converges uniformly in $G$ to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,

$$\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial y_i}+\sum_{i=1}^na_i(x)\frac{\partial u}{\partial x_i}+a(x)u=0,\label{*}\tag{*}$$

which has a unique solution of the Dirichlet problem for any continuous boundary function . If the sequence of solutions of equation \eqref{*} converges uniformly on $\partial G$, then it also converges uniformly in $G$ to a solution of equation \eqref{*}.

Harnack's second theorem, the Harnack principle: If a monotone sequence of harmonic functions in a bounded domain $G$ converges at some point in $G$, then it converges at all points of $G$ to a harmonic function, and this convergence is uniform on any closed subdomain of $G$. Harnack's second theorem can be generalized to monotone sequences of solutions of the elliptic equation .

References

[1] I.G. [I.G. Petrovskii] Petrowski, "Vorlesungen über partielle Differentialgleichungen" , Teubner (1965) (Translated from Russian)
[2] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)


Comments

In the axiomatic theory of harmonic spaces (cf. Harmonic space) the first Harnack theorem is known as the Bauer convergence property and the second Harnack theorem as the Brélot convergence property, see [a3] and [a1]. The following properties are equivalent to the Brélot convergence property (see [a4]): 1) each positive harmonic function $u$ on a domain $U$ is either strictly positive or $u=0$. Moreover, the set of positive harmonic functions on $U$, equal to 1 in a given point $u\in U$, is equicontinuous (cf. Equicontinuity); and 2) for any domain $U$ and any compact subset $K$ of $U$ there exists a constant $c>0$ such that $u(x)\leq cu(y)$ for any $x,y\in K$ and any positive harmonic function $u$ on $U$ (the Harnack inequality).

References

[a1] J.-M. Bony, "Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel" M. Brelot (ed.) H. Bauer (ed.) J.-M. Bony (ed.) J. Deny (ed.) G. Mokobodzki (ed.) , Potential theory (CIME, Stresa, 1969) , Cremonese (1970) pp. 69–119
[a2] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[a3] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a4] P. Loeb, B. Walsh, "The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brélot" Ann. Inst. Fourier , 15 : 2 (1965) pp. 597–600
How to Cite This Entry:
Harnack theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harnack_theorem&oldid=17463
This article was adapted from an original article by L.I. Kamynin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article