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''method of auxiliary functions''

A method which is employed in the theory of linear and non-linear partial differential equations. Bernstein's method consists in introducing certain new (auxiliary) functions, which depend on the solution being sought, and which make it possible to establish a priori estimates of the maximum modulus of the derivatives of this solution of the required order.

A simple example of the application of Bernstein's method is the a priori estimate of the modulus of the derivatives of the solution of the Dirichlet problem for the non-linear (quasi-linear) elliptic equation

$$ \tag{* }
\left .

\begin{array}{r}
{
\frac{\partial   ^ {2} z }{\partial  x  ^ {2} }
 +

\frac{\partial   ^ {2} z }{\partial  y  ^ {2} }
  =  f
\left ( x, y, z, 
\frac{\partial z}{\partial x}
 , 
\frac{\partial z}{\partial y}
 \right )  \equiv  }  \\
{\equiv  a \left ( 
\frac{\partial z}{\partial x}
 \right )  ^ {2} + 2b

\frac{\partial z}{\partial x}
 
\frac{\partial z}{\partial y}
 + c
\left ( 
\frac{\partial z}{\partial y}
 \right )  ^ {2} + }  \\
{+ 2d 
\frac{\partial z}{\partial x}
 + 2e 
\frac{\partial z}{\partial y}
 +
g \   \mathop{\rm in}  \textrm{ the }  \textrm{ disc }  D, }  \\
\end{array}
  \right \}
$$

$$ 
z \mid  _ {C}  =  0,
$$

where  $  a, b, c, d, e, g $
are smooth functions of  $  x, y, z $; 
$  C $
is the circle, the boundary of the disc  $  D $
with radius  $  R $(
the assumption to the effect that  $  D $
is a disc and  $  z\mid  _ {C} = 0 $
is immaterial, since the general case of an arbitrary-connected domain and inhomogeneous boundary condition is readily reduced to the case under consideration by a change of the function and a conformal transformation of the domain).

If  $  f _ {z} ^ { \prime } \geq  0 $, 
then the estimated maximum modulus  $  n $

$$ 
n  =  \max _ {(x,y) \in D + C } \ 
| z (x, y) |
$$

of the solution of problem (*) is immediately obtained from the maximum principle.

In order to prove that a regular solution of problem (*) exists it is sufficient to have a priori estimates of the maximum modulus of the derivatives of the solution up to the third order (cf. [[Continuation method (to a parametrized family)|Continuation method (to a parametrized family)]]). To estimate  $  \max _ {C}  | \partial  z/ \partial  x | $
and  $  \max _ {C}  | \partial  z/ \partial  y | $, 
it is sufficient to estimate  $  \max _ {C}  | \partial  z/ \partial  \rho | $(
since  $  z \mid  _ {C} = 0 $), 
where  $  \rho , \theta $
are polar coordinates in the disc  $  D $. 
Now introduce a new (auxiliary) function  $  u $, 
given by the formula

$$ 
z  =  \phi _ {1} (u)  =  -n- \alpha + \alpha   \mathop{\rm ln}  u ,
$$

where  $  \alpha > 0 $
will be selected later. The function  $  u = u(x, y) $
varies from  $  e $
to  $  e ^ {(2n + \alpha )/ \alpha } $
in the same direction as  $  z(x, y) $(
$  -n \leq  z \leq  n $). 
Since

$$ 

\frac{\partial z}{\partial x}
  =  
\frac \alpha {u}

\frac{\partial u}{\partial x}
 ,
$$

$$ 

\frac{\partial   ^ {2} z }{\partial x  ^ {2} }
  = \ 

\frac \alpha {u}

\frac{\partial   ^ {2} u }{\partial x  ^ {2} }
 -

\frac \alpha {u  ^ {2} }
 \left ( 
\frac{\partial u}{\partial x}
 \right )  ^ {2} ,
$$

and similar results for derivatives with respect to  $  y $, 
it follows that  $  u $
satisfies the equation

$$ 

\frac{\partial   ^ {2} u }{\partial  x  ^ {2} }
 +

\frac{\partial   ^ {2} u }{\partial  y  ^ {2} }
  = \ 

\frac{1}{u}

\left [  \left ( 
\frac{\partial u}{\partial x}
 \right )  ^ {2} +
\left ( 
\frac{\partial u}{\partial y}
 \right )  ^ {2} \ 
\right ] +
$$

$$ 
+ 

\frac \alpha {u}
 \left [ a \left ( 
\frac{\partial u}{\partial x}
 \right )  ^ {2} + 2b 
\frac{\partial u}{\partial x}

\frac{\partial u}{\partial y}
 + c
\left ( 
\frac{\partial u}{\partial y}
 \right )  ^ {2}  \right ] +
$$

$$ 
+ 
2d 
\frac{\partial u}{\partial x}
 + 2e 
\frac{\partial u}{\partial y}
 + g 
\frac{u} \alpha 
  \equiv  Q.
$$

Let  $  M $
be the upper bound of  $  | a |, | b |, | c | $
in  $  D $, 
and let  $  \alpha = 1/8 M $. 
If  $  \partial  u/ \partial  x $
and  $  \partial  u/ \partial  y $
are considered as current coordinates in the plane, and  $  x, y, z $
as parameters, the equation  $  Q = 0 $
is the equation of an ellipse, since the determinant  $  a _ {1} c _ {1} - b _ {1}  ^ {2} > 2 u  ^ {2} /3 $, 
where

$$ 
a _ {1}  =  
\frac{1}{u}
 \left ( 1+ 
\frac{a}{8M}
 \right ) ,\ \ 
b _ {1}  =  
\frac{b}{8Mu}
 ,\ \ 
c _ {1}  =  
\frac{1}{u}

\left ( 1 + 
\frac{t}{8M}
 \right ) .
$$

Thus, for any  $  \partial  u/ \partial  x $
and  $  \partial  u/ \partial  y $, 
$  Q $
will not be smaller than a certain negative number  $  -P $, 
$  Q \geq  -P $(
the number  $  P $
is readily obtained in explicit form). If one introduces the function  $  u _ {1} $
given by the formula

$$ 
u _ {1}  =  u + 
\frac{P}{4}
 (x  ^ {2} +y  ^ {2} ),
$$

one obtains

$$ 

\frac{\partial   ^ {2} u _ {1} }{\partial  x  ^ {2} }

+ 
\frac{\partial   ^ {2} u _ {1} }{\partial y  ^ {2} }

 =  Q + P  \geq   0,
$$

and  $  u _ {1} $
attains its maximum on the boundary  $  C $
of the domain  $  D $
and, since  $  u _ {1} $
is constant on  $  C $, 
one has

$$ 

\frac{\partial  u _ {1} }{\partial  \rho }
  \geq   0 \ \ 
\textrm{ and } \ \ 

\frac{\partial u}{\partial \rho}
  \geq  \ 
- 
\frac{1}{2}
 PR ,
$$

where  $  R $
is the radius of the circle  $  C $. 
Hence it is possible to find a negative lower bound for  $  \partial  z/ \partial  \rho $:

$$ 

\frac{\partial z}{\partial \rho}
  = \ 

\frac \alpha {u}

\frac{\partial u}{\partial \rho}
  \geq  \ 
- 
\frac{\alpha PR }{2e ^ {(2n + \alpha )/ \alpha } }
 .
$$

If the same reasoning is applied to a second auxiliary function  $  u $

$$ 
z  =  \phi _ {2} (u)  = \ 
-n- \alpha + \alpha
  \mathop{\rm ln}  
\frac{1}{1-u}
 ,
$$

one obtains an estimate from above

$$ 

\frac{\partial z}{\partial \rho}
  \leq  \ 

\frac{\alpha P _ {1} R }{2}

e ^ {(n+ \alpha )/2 } .
$$

Thus,  $  \max _ {C}  | \partial  z/ \partial  \rho | $
is estimated, which means that  $  \max _ {C}  | \partial  z/ \partial  x | $
and  $  \max _ {C}  | \partial  z/ \partial  y | $
are estimated as well. The estimate of the maximum modulus of the first derivatives inside the domain  $  D $
is performed in a similar manner: introduce an auxiliary function  $  u $
given by the formula

$$ 
z  =  \phi _ {3} (u)  =  -n + \alpha \ 
 \mathop{\rm ln}   \mathop{\rm ln}  u .
$$

The function  $  u $
varies in the same direction as  $  z $, 
from  $  e $
to  $  e ^ {e ^ {2n/ \alpha } } $. 
In view of (*), on may write the following expression for  $  u $

$$ 

\frac{\partial   ^ {2} u }{\partial x  ^ {2} }

+

\frac{\partial   ^ {2} u }{\partial y  ^ {2} }
  = \ 

\frac{1}{u   \mathop{\rm ln}  u }

\left [
(1+  \mathop{\rm ln}  u + \alpha a)
\left ( 
\frac{\partial u}{\partial x}
 \right )  ^ {2} +2ab

\frac{\partial u}{\partial x}

\frac{\partial u}{\partial y} +\right. 
$$

$$ 
+ \left .
(1 +  \mathop{\rm ln}  u + \alpha c) \left ( 
\frac{\partial u}{\partial y}
 \right )  ^ {2}  \right ] + 2d

\frac{\partial u}{\partial x}
 + 2e

\frac{\partial u}{\partial y}
 + g

\frac{u   \mathop{\rm ln}  u } \alpha 
  \equiv  Q _ {1} .
$$

Considerations similar to those given above show that if the function

$$ 
w  =  \left ( 
\frac{\partial u}{\partial x}
 \right )  ^ {2} +
\left ( 
\frac{\partial u}{\partial y}
 \right )  ^ {2}
$$

attains a maximum in the domain  $  D $, 
this maximum does not exceed some number, the value of which depends solely on  $  n $
and  $  M $. 
This yields the required estimates of  $  \max _ {D}  | \partial  z/ \partial  x | $
and  $  \max _ {D}  | \partial  z/ \partial  y | $.

Bernstein's method may also be used to estimate, in a similar manner, the maximum modulus in the domain  $  D + C $
of all highest derivatives of the solution (the only other operation which is required is the differentiation of the initial equation).

The method was first utilized by S.N. Bernstein [e theory of functions','../c/c025430.htm','Continuation method (to a parametrized family)','../c/c025520.htm','Euler–Lagrange equation','../e/e036510.htm','Fourier series','../f/f041090.htm','Functions of a real variable, theory of','../f/f042130.htm','Hilbert problems','../h/h120080.htm','Jackson inequality','../j/j054000.htm','Laplace theorem','../l/l057530.htm','Lebesgue constants','../l/l057800.htm','Limit theorems','../l/l058920.htm','Linear elliptic partial differential equation and system','../l/l059180.htm','Lyapunov theorem','../l/l061200.htm','Mathematical statistics','../m/m062710.htm','Minimal surface','../m/m063920.htm','Markov–Bernstein-type inequalities','../m/m110060.htm','Ornstein–Uhlenbeck process','../o/o070240.htm','Orthogonal polynomials','../o/o070340.htm','Plateau problem, multi-dimensional','../p/p072850.htm','Quasi-analytic class','../q/q076370.htm')" style="background-color:yellow;">S.N. Bernshtein] . The method was subsequently extended and was systematically utilized in the study of various problems for elliptic and parabolic differential operators [[#References|[3]]], [[#References|[4]]], [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S.N. [S.N. Bernshtein] Bernstein,   "Sur la généralisation du problème de Dirichlet (première partie)"  ''Math. Ann.'' , '''62'''  (1906)  pp. 253–271</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S.N. [S.N. Bernshtein] Bernstein,   "Sur la généralisation du problème de Dirichlet (deuxième partie)"  ''Math. Ann.'' , '''69'''  (1910)  pp. 82–136</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''3''' , Moscow  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Pogorelov,   "Die Verbiegung konvexer Flächen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.A. Oleinik,   S.N. Kruzhkov,   "Quasi-linear parabolic equations of second order in several independent variables"  ''Russian Math. Surveys'' , '''16''' :  2  (1961)  pp. 105–146  ''Uspekhi Mat. Nauk'' , '''16''' :  5  (1961)  pp. 115–155</TD></TR></table>