View source for Linear elliptic partial differential equation and system

A partial differential equation (system) of the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591802.png" /> is the linear elliptic operator

<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/l/l059/l059180/l0591803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

The operator (1) with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591804.png" /> is elliptic at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591805.png" /> if the characteristic form

<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/l/l059/l059180/l0591806.png" /></td> </tr></table>

is definite at this point. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591808.png" /> is a multi-index (a set of non-negative integers), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l0591809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918012.png" />. In particular, the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918013.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918014.png" /> must be even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918015.png" />. Up to sign the condition for definiteness of forms is written as

<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/l/l059/l059180/l05918016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918017.png" /> is elliptic in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918018.png" /> if it is elliptic at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918019.png" />, and it is uniformly elliptic in this region if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918020.png" /> in (2) that does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918021.png" />.

In the case of an equation of the second order,

<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/l/l059/l059180/l05918022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

this definition can be restated as follows. Equation (3) is elliptic in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918023.png" /> if at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918024.png" /> by a change of independent variables it can be reduced to the canonical form

<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/l/l059/l059180/l05918025.png" /></td> </tr></table>

with the [[Laplace operator|Laplace operator]]

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in the principal part. In the case of an elliptic partial differential equation in the plane, under very general assumptions about the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918027.png" /> such a transformation is possible not only at a point but also in the whole region (see [[#References|[1]]]).

The simplest elliptic partial differential equation is the [[Laplace equation|Laplace equation]], and its solutions are called harmonic functions (cf. [[Harmonic function|Harmonic function]]). Solutions of a linear elliptic partial differential equation can be characterized by the fact that they have many properties in common with harmonic functions. In the planar case every harmonic function is the real part of an analytic function; it is a real-analytic function of two variables. The solutions of a general linear elliptic partial differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918028.png" /> have a similar property. If the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918030.png" />, and the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918031.png" /> are analytic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918032.png" /> in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918033.png" />, then any solution of this equation is also analytic.

There are other assertions of similar type. For example, if the coefficients and the right-hand side of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918034.png" /> are continuously differentiable up to (and including) order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918035.png" /> and their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918036.png" />-th derivatives satisfy a [[Hölder condition|Hölder condition]] with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918038.png" />, then any solution has derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918039.png" /> satisfying a Hölder condition with the same exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918040.png" />. Here the fact of belonging to a Hölder class is essential. If the coefficients and the right-hand side are only continuous, then the solution need not have continuous derivatives of order equal to the order of the equation. This is true even for the simplest linear elliptic partial differential equation, the [[Poisson equation|Poisson equation]]

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The above-said refers to classical solutions, that is, solutions that have continuous derivatives up to the order of the equation. There are various generalizations of the concept of a solution.

For example, if the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918042.png" /> are sufficiently smooth, then for the operator (1) one can define the Lagrange adjoint operator

<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/l/l059/l059180/l05918043.png" /></td> </tr></table>

A locally integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918044.png" /> is called a weak solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918045.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918046.png" /> (infinitely-differentiable functions with compact support) one has

<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/l/l059/l059180/l05918047.png" /></td> </tr></table>

Then, if the coefficients and the right-hand side of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918048.png" /> are Hölder continuous, every weak solution is a classical solution.

For the Laplace equation the simplest well-posed problem is the [[Dirichlet problem|Dirichlet problem]]. In the general case of an equation with operator (1) the boundary value problem consists in finding in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918049.png" /> a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918050.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918051.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918052.png" /> boundary conditions of the form

<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/l/l059/l059180/l05918053.png" /></td> </tr></table>

To the [[Neumann problem]] correspond the boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918055.png" /> denotes differentiation in the direction of the outward normal.

For a boundary value problem to be Noetherian (cf. [[Noetherian operator|Noetherian operator]]) the boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918056.png" /> must satisfy the Shapiro–Lopatinskii complementation condition (see [[#References|[2]]]) — an algebraic condition relating the polynomials

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at points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918058.png" />. The boundary operators of the Dirichlet problem satisfy this condition with respect to any elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918059.png" />.

If the coefficients of the differential operator and the solution are considered in the class of complex functions, then the fact that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918060.png" /> in (1) is elliptic is determined by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918062.png" />. This definition allows of elliptic operators of odd order, as the example of the Cauchy–Riemann operator shows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918063.png" />. In addition, properties of operators of even order can change. For example, for the [[Bitsadze equation|Bitsadze equation]] (see [[#References|[3]]])

<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/l/l059/l059180/l05918064.png" /></td> </tr></table>

the Dirichlet problem is not well-posed: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918065.png" /> is the unit disc, then functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918066.png" /> are solutions of the homogeneous Dirichlet problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918067.png" /> for any analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918068.png" />.

This example leads to the necessity of distinguishing classes of elliptic operators for which the property that the Dirichlet problem be Noetherian is preserved. In this way the concept of a strongly elliptic operator arises (cf. [[#References|[4]]]). The operator (1) is strongly elliptic if for some complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918069.png" /> the condition

<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/l/l059/l059180/l05918070.png" /></td> </tr></table>

is satisfied. In particular, the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918071.png" /> is necessarily even.

The next, wider, concept is that of proper (regular) ellipticity. The operator (1) is properly elliptic if its order is even and if for any pair of linearly independent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918073.png" /> the polynomial (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918074.png" />)

<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/l/l059/l059180/l05918075.png" /></td> </tr></table>

has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918076.png" /> roots with negative imaginary part and the same number of roots with positive imaginary part. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918077.png" /> any elliptic operator is properly elliptic, so the definition essentially refers only to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918078.png" />.

In the theory of linear elliptic partial differential equations a significant role is played by a priori estimates of the norms of solutions in terms of the norms of the right-hand sides of the equation and the boundary conditions. S.N. Bernshtein (see [[#References|[6]]]) began to use these estimates systematically, and a more recent development is due to J. Schauder (see [[#References|[7]]]). The Schauder estimates refer to solutions of a linear elliptic partial differential equation of the second order in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918079.png" /> with Hölder-continuous coefficients, and two forms occur. Estimates of the first form (estimates  "from inside" ) consist in the fact that on any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918080.png" /> the derivatives up to the second order inclusive and their Hölder constants are estimated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918081.png" /> and in terms of the modulus and the Hölder constant of the right-hand side of the equation. Estimates of the second form (estimates  "up to the boundary" ) refer to boundary value problems. Here the same quantities are estimated, but in the closure of the region in question, and the norms of the right-hand sides of the boundary conditions occur in the estimate.

Schauder estimates have been extended further to general linear elliptic partial differential equations and boundary value problems (see [[#References|[7]]]). The derivation of these estimates is based on potential theory. By means of a partition of unity a local character is given to them, and the matter reduces to estimating the norms of singular integral operators that represent a convolution with functions connected with the fundamental solutions (estimates  "from inside" ) or with the Green functions of the corresponding boundary value problem in some standard region (estimates  "up to the boundary" ). These estimates, obtained originally in the metric of the Hölder spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918082.png" />, have been extended to the Sobolev spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918083.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918084.png" />-estimates) and refer to generalized solutions.

For strongly elliptic operators there is an priori estimate, called the [[Gårding inequality|Gårding inequality]], which is obtained by other methods. It lies at the heart of a fundamental approach to the investigation of boundary value problems (Hilbert space methods).

In the theory of linear elliptic partial differential equations an important place is taken by fundamental solutions. For an operator (1) with sufficiently smooth coefficients a fundamental solution is defined as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918085.png" /> that satisfies the condition

<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/l/l059/l059180/l05918086.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918087.png" />. From the point of view of the theory of generalized functions this implies that

<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/l/l059/l059180/l05918088.png" /></td> </tr></table>

where the right-hand side is Dirac's delta-function.

Fundamental solutions of linear elliptic partial differential equations exist for equations with analytic coefficients (and then they themselves are analytic), for equations with infinitely-differentiable coefficients (and they then belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918089.png" />) and for a number of other equations with weaker restrictions on the coefficients. For an elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918090.png" /> with constant coefficients, consisting of terms of highest order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918091.png" />, a fundamental solution depends only on the difference of the arguments and has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918092.png" />:

<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/l/l059/l059180/l05918093.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918094.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918095.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918096.png" /> even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918097.png" />; in the remaining cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l05918099.png" /> is analytic on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180100.png" /> (see [[#References|[8]]]).

In particular, for the Laplace operator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180101.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180103.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180106.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180107.png" />.

Fundamental solutions make it possible to construct various explicit representations for solutions of linear elliptic partial differential equations. They are a necessary apparatus for studying boundary value problems by means of integral equations. For an equation of the second order this method is classical and gives the sharpest results (see [[#References|[9]]]).

The [[Maximum principle|maximum principle]] has had numerous applications in the theory of second-order linear elliptic partial differential equations. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180110.png" /> are assumed to be continuous and the operator (3) is assumed to be uniformly elliptic in some region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180111.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180112.png" /> is taken to be continuous in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180113.png" /> and to belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180114.png" />.

The maximum principle in its strong form consists in the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180115.png" /> be the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180116.png" /> in (3) in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180117.png" />.

a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180119.png" /> attains its maximum at an interior point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180120.png" /> is constant.

b) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180121.png" /> and the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180122.png" /> is attained at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180123.png" /> that lies on the surface of some ball entirely contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180124.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180125.png" /> is constant or the derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180126.png" /> in the direction of the outward normal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180127.png" />, is positive.

Similar assertions hold for an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180128.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059180/l059180129.png" /> if in a) and b) the maximum is understood as a positive maximum. The maximum principle is an essential element in the proofs of uniqueness theorems for solutions of a number of boundary value problems. It also has some analogues in the case of an equation of higher order.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,   "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)  {{MR|0152665}} {{MR|0150320}} {{MR|0138774}} {{ZBL|0127.03505}} {{ZBL|0100.07603}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)  {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Agmon,   A. Douglis,   L. Nirenberg,   "Estimates close to the boundary of solutions of elliptic partial differential equations under general boundary conditions"  ''Comm. Pure Appl. Math.'' , '''12'''  (1959)  pp. 623–727  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. John,   "Plane waves and spherical means: applied to partial differential equations" , Interscience  (1955)  {{MR|0075429}} {{ZBL|0067.32101}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)  {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.N. Bernshtein,   "Collected works" , '''3''' , Moscow  (1960)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Schauder,   "Ueber lineare elliptische Differentialgleichungen zweiter Ordnung"  ''Math. Z.'' , '''38'''  (1934)  pp. 257–282  {{MR|}} {{ZBL|0008.25502}}  {{ZBL|60.0422.01}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Ya.B. Lopatinskii,   "On a method for reducing boundary value problems for a system of differential equations of elliptic type to regular integral equations"  ''Ukrain. Mat. Zh.'' , '''5''' :  2  (1953)  pp. 123–151  (In Russian)  {{MR|73828}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.I. Vishik,   "On strongly elliptic systems of differential equations"  ''Mat. Sb.'' , '''29''' :  3  (1951)  pp. 615–676  {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====
For the reduction of equation (3) to canonical form, [[#References|[a1]]] has been fundamental.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,   "Conformality with respect to Riemannian metrics"  ''Ann. Acad. Sci. Fenn. Ser. A1'' , '''206'''  (1955)  pp. 1–22  {{MR|0074855}} {{ZBL|0067.30702}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Equations aux dérivées partielles de type elliptique" , Dunod  (1969)  (Translated from Russian)  {{MR|}} {{ZBL|0164.13001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gilbarg,   N.S. Trudinger,   "Elliptic partial differential equations of second order" , Springer  (1977)  {{MR|0473443}} {{ZBL|0361.35003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.H. Potter,   H.F. Weinberger,   "Maximum principles in differential equations" , Prentice-Hall  (1967)  {{MR|219861}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)  {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR></table>