Bernstein method

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)). 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 [3], [4], [5].

References

[1a]	S.N. [S.N. Bernshtein] Bernstein, "Sur la généralisation du problème de Dirichlet (première partie)" Math. Ann. , 62 (1906) pp. 253–271
[1b]	S.N. [S.N. Bernshtein] Bernstein, "Sur la généralisation du problème de Dirichlet (deuxième partie)" Math. Ann. , 69 (1910) pp. 82–136
[2]	S.N. Bernshtein, , Collected works , 3 , Moscow (1960)
[3]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)
[4]	A.V. Pogorelov, "Die Verbiegung konvexer Flächen" , Akademie Verlag (1957) (Translated from Russian)
[5]	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