Harnack Inequality

Harnack Inequality - If Function $U(M)=U(x_{1},...,x_{k})$ ${\ displaystyle U (M) = U (x_ {1}, ..., x_ {k})}$ $U (M) = U (x_ {1}, ..., x_ {k})$ harmonic in $k$ ${\ displaystyle k}$ $k$ -dimensional ball $Q_{R}$ ${\ displaystyle Q_ {R}}$ $Q_ {R}$ radius $R$ ${\ displaystyle R}$ $R$ centered at some point $M_{0}$ ${\ displaystyle M_ {0}}$ $M_ {0}$ is non-negative in this ball, then for its values at points $M$ ${\ displaystyle M}$ $M$ the following inequalities are valid inside the ball in question: $R^{k-2}{\frac {R-r}{(R+r)^{k-1}}}U(M_{0})\leqslant U(M)\leqslant R^{k-2}{\frac {R+r}{(R-r)^{k-1}}}U(M_{0})$ ${\ displaystyle R ^ {k-2} {\ frac {Rr} {(R + r) ^ {k-1}}} U (M_ {0}) \ leqslant U (M) \ leqslant R ^ {k- 2} {\ frac {R + r} {(Rr) ^ {k-1}}} U (M_ {0})}$ $R ^ {{k-2}} {\ frac {Rr} {(R + r) ^ {{k-1}}}} U (M_ {0}) \ leqslant U (M) \ leqslant R ^ {{ k-2}} {\ frac {R + r} {(Rr) ^ {{k-1}}}} U (M_ {0})$ where $r=\rho (M_{0},M)<R$ ${\ displaystyle r = \ rho (M_ {0}, M) <R}$ $r = \ rho (M_ {0}, M) <R$ .

Proof

By the Poisson formula for points $M$ ${\ displaystyle M}$ $M$ inside the ball $Q_{R'}(R'<R)$ ${\ displaystyle Q_ {R '} (R' <R)}$ $Q_{{R'}}(R'<R)$ we have $U(M)={\frac {\Gamma (k/2)}{2\pi ^{k/2}R'}}\int \limits _{\gamma }(Q_{R'})U(N){\frac {{R'}^{2}-r^{2}}{({R'}^{2}+r^{2}-2R'r\cos \theta )^{r/2}}}d\sigma$ ${\ displaystyle U (M) = {\ frac {\ Gamma (k / 2)} {2 \ pi ^ {k / 2} R '}} \ int \ limits _ {\ gamma} (Q_ {R'}) U (N) {\ frac {{R '} ^ {2} -r ^ {2}} {({R'} ^ {2} + r ^ {2} -2R'r \ cos \ theta) ^ { r / 2}}} d \ sigma}$ $U(M)={\frac {\Gamma (k/2)}{2\pi ^{{k/2}}R'}}\int \limits _{{\gamma }}(Q_{{R'}})U(N){\frac {{R'}^{{2}}-r^{2}}{({R'}^{2}+r^{2}-2R'r\cos \theta )^{{r/2}}}}d\sigma$ . Given the inequalities $(R'-r)^{2}\leqslant {R'}^{2}+r^{2}-2R'r\cos \theta \leqslant (R'+r)^{2}$ ${\ displaystyle (R'-r) ^ {2} \ leqslant {R '} ^ {2} + r ^ {2} -2R'r \ cos \ theta \ leqslant (R' + r) ^ {2}}$ $(R'-r)^{2}\leqslant {R'}^{2}+r^{2}-2R'r\cos \theta \leqslant (R'+r)^{2}$ due to the condition $U(N)\geqslant 0$ ${\ displaystyle U (N) \ geqslant 0}$ $U(N)\geqslant 0$ get from here that ${R'}^{k-2}{\frac {{R'}^{2}-r^{2}}{(R'+r)^{k}}}{\frac {\Gamma (k/2)}{2\pi ^{k/2}{R'}^{k-1}}}\int \limits _{\gamma }(Q_{R'})U(N)d\sigma \leqslant U(M)\leqslant {R'}^{k-2}{\frac {{R'}^{2}-r^{2}}{(R'-r)^{k}}}{\frac {\Gamma (k/2)}{2\pi ^{k/2}{R'}^{k-1}}}\int \limits _{\gamma }(Q_{R'})U(N)d\sigma$ ${\ displaystyle {R '} ^ {k-2} {\ frac {{R'} ^ {2} -r ^ {2}} {(R '+ r) ^ {k}}} {\ frac {\ Gamma (k / 2)} {2 \ pi ^ {k / 2} {R '} ^ {k-1}}} \ int \ limits _ {\ gamma} (Q_ {R'}) U (N) d \ sigma \ leqslant U (M) \ leqslant {R '} ^ {k-2} {\ frac {{R'} ^ {2} -r ^ {2}} {(R'-r) ^ {k} }} {\ frac {\ Gamma (k / 2)} {2 \ pi ^ {k / 2} {R '} ^ {k-1}}} \ int \ limits _ {\ gamma} (Q_ {R' }) U (N) d \ sigma}$ ${R'}^{{k-2}}{\frac {{R'}^{2}-r^{2}}{(R'+r)^{k}}}{\frac {\Gamma (k/2)}{2\pi ^{{k/2}}{R'}^{{k-1}}}}\int \limits _{\gamma }(Q_{{R'}})U(N)d\sigma \leqslant U(M)\leqslant {R'}^{{k-2}}{\frac {{R'}^{2}-r^{2}}{(R'-r)^{k}}}{\frac {\Gamma (k/2)}{2\pi ^{{k/2}}{R'}^{{k-1}}}}\int \limits _{\gamma }(Q_{{R'}})U(N)d\sigma$ , or, applying the Gauss theorem ${R'}^{k-2}{\frac {{R'}^{2}-r^{2}}{(R'+r)^{k}}}U(M_{0})\leqslant U(M)\leqslant {R'}^{k-2}{\frac {{R'}^{2}-r^{2}}{(R'-r)^{k}}}U(M_{0})$ ${\ displaystyle {R '} ^ {k-2} {\ frac {{R'} ^ {2} -r ^ {2}} {(R '+ r) ^ {k}}} U (M_ {0 }) \ leqslant U (M) \ leqslant {R '} ^ {k-2} {\ frac {{R'} ^ {2} -r ^ {2}} {(R'-r) ^ {k} }} U (M_ {0})}$ ${R'}^{{k-2}}{\frac {{R'}^{2}-r^{2}}{(R'+r)^{k}}}U(M_{0})\leqslant U(M)\leqslant {R'}^{{k-2}}{\frac {{R'}^{2}-r^{2}}{(R'-r)^{k}}}U(M_{0})$ . Thus, passing to the limit at $R'\rightarrow R$ ${\ displaystyle R '\ rightarrow R}$ $R'\rightarrow R$ , we obtain the Harnack inequality $R^{k-2}{\frac {R-r}{(R+r)^{k-1}}}U(M_{0})\leqslant U(M)\leqslant R^{k-2}{\frac {R+r}{(R-r)^{k-1}}}U(M_{0})$ ${\ displaystyle R ^ {k-2} {\ frac {Rr} {(R + r) ^ {k-1}}} U (M_ {0}) \ leqslant U (M) \ leqslant R ^ {k- 2} {\ frac {R + r} {(Rr) ^ {k-1}}} U (M_ {0})}$ $R^{{k-2}}{\frac {R-r}{(R+r)^{{k-1}}}}U(M_{0})\leqslant U(M)\leqslant R^{{k-2}}{\frac {R+r}{(R-r)^{{k-1}}}}U(M_{0})$ .

Literature

Timan A.F., Trofimov V.N. Introduction to the theory of harmonic functions, M., Nauka , 1968, 206 pp., Shooting gallery 39500 copies.

Harnack Inequality

Proof

Literature

More articles: