Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have
Theorem (Maximum principle). Let
satisfy the differential inequality
![\displaystyle L[u] = \sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum\limits_{k = 1}^n {{b_k}(x)\frac{{\partial u}}{{\partial {x_k}}}} \geqslant 0 \displaystyle L[u] = \sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum\limits_{k = 1}^n {{b_k}(x)\frac{{\partial u}}{{\partial {x_k}}}} \geqslant 0](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+L%5Bu%5D+%3D+%5Csum%5Climits_%7Bi%2Cj+%3D+1%7D%5En+%7B%7Ba_%7Bij%7D%7D%28x%29%5Cfrac%7B%7B%7B%5Cpartial+%5E2%7Du%7D%7D%7B%7B%5Cpartial+%7Bx_i%7D%5Cpartial+%7Bx_j%7D%7D%7D%7D+%2B+%5Csum%5Climits_%7Bk+%3D+1%7D%5En+%7B%7Bb_k%7D%28x%29%5Cfrac%7B%7B%5Cpartial+u%7D%7D%7B%7B%5Cpartial+%7Bx_k%7D%7D%7D%7D+%5Cgeqslant+0&bg=ffffff&fg=333333&s=0)
in a domain
where
is uniformly elliptic. Suppose the coefficients
and
are uniformly bounded. If
attains a maximum
at a point of
, then
in
.
In order to memorize the above result, let us think about the parabola
with
and
. In this one-dimentional case,
which confirms that
only achieves its maximum at
.
As you may know the operator
is only assumed to be strongly elliptic which only effects the coefficients
. Regarding to the coefficients
, we only assume these are uniformly bounded. However, the uniform ellipticity of the operator L and the boundedness of the coefficients are not really essential as you can check in the proof. Besides, the domain
need not be bounded in this version.
Now for operators of the form
, we still have a result analogous to the above.
Theorem (Maximum principle). Let
satisfy the differential inequality
![\displaystyle (L + h)[u] >0 \displaystyle (L + h)[u] >0](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28L+%2B+h%29%5Bu%5D+%3E0&bg=ffffff&fg=333333&s=0)
with
, with
uniformly elliptic in
, and with the coefficients of
and
bounded. If
attains a non-negative maximum
at an interior point of
, then
.
Clearly, the assumption
and
are crucial as we may face some difficulty as raised in this note. Counterexamples are easily obtained if
. For example, the function
has an absolute maximum at
.
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