Linear elliptic partial differential equation and system

A partial differential equation (system) of the form

$$ L u = f , $$

where $ L $ is the linear elliptic operator

$$ \tag{1 } L u = \ \sum _ {| \alpha | \leq m } a _ \alpha ( x) D ^ \alpha u ( x) . $$

The operator (1) with real coefficients $ a _ \alpha ( x) $ is elliptic at a point $ x $ if the characteristic form

$$ \omega ( x , \xi ) = \ \sum _ {| \alpha | = m } a _ \alpha ( x) \xi ^ \alpha $$

is definite at this point. Here $ x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n} $, $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $ is a multi-index (a set of non-negative integers), $ | \alpha | = \sum {\alpha _ {i} } $, $ \xi = ( \xi _ {1} \dots \xi _ {n} ) \in \mathbf R ^ {n} $, $ D ^ \alpha = D _ {1} ^ {\alpha _ {1} } \dots D _ {n} ^ {\alpha _ {n} } $ and $ \xi ^ \alpha = \xi _ {n} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $. In particular, the order $ m $ of the operator $ L $ must be even, $ m = 2 m ^ \prime $. Up to sign the condition for definiteness of forms is written as

$$ \tag{2 } (- 1) ^ {m ^ \prime } \omega ( x , \xi ) \geq \delta | \xi | ^ {m} ,\ \ \delta = \delta ( x) > 0 . $$

The operator $ L $ is elliptic in a region $ D $ if it is elliptic at every point $ x \in D $, and it is uniformly elliptic in this region if there is a $ \delta > 0 $ in (2) that does not depend on $ x $.

In the case of an equation of the second order,

$$ \tag{3 } \sum _ {i , j = 1 } a _ {ij} ( x) \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum _ { i= } 1 ^ { n } a _ {i} ( x) \frac{\partial u }{\partial x _ {i} } + a ( x) u = f ( x) , $$

this definition can be restated as follows. Equation (3) is elliptic in a region $ D $ if at every point of $ D $ by a change of independent variables it can be reduced to the canonical form

$$ \Delta u + \sum _ {i = 1 } ^ { n } a _ {i} ( x) \frac{\partial u }{\partial x _ {i} } + a ( x) u = f ( x) $$

with the Laplace operator

$$ \Delta = \ \frac{\partial ^ {2} }{\partial x _ {1} ^ {2} } + \dots + \frac{\partial ^ {2} }{\partial x _ {n} ^ {2} } $$

in the principal part. In the case of an elliptic partial differential equation in the plane, under very general assumptions about the coefficients $ a _ {ij} $ such a transformation is possible not only at a point but also in the whole region (see [1]).

The simplest elliptic partial differential equation is the Laplace equation, and its solutions are called harmonic functions (cf. 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 $ L u = f $ have a similar property. If the coefficients $ a _ \alpha ( x) $, $ | \alpha | \leq m $, and the right-hand side $ f ( x) $ are analytic with respect to $ x = ( x _ {1} \dots x _ {n} ) $ in a region $ D $, 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 $ L u = f $ are continuously differentiable up to (and including) order $ k $ and their $ k $- th derivatives satisfy a Hölder condition with exponent $ \alpha $, $ 0 < \alpha < 1 $, then any solution has derivatives up to order $ k + m $ satisfying a Hölder condition with the same exponent $ \alpha $. 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

$$ \Delta u = f . $$

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 $ a _ \alpha ( x) $ are sufficiently smooth, then for the operator (1) one can define the Lagrange adjoint operator

$$ L ^ {*} u = \ \sum _ {| \alpha | \leq m } (- 1) ^ {| \alpha | } D ^ \alpha ( a _ \alpha u ) . $$

A locally integrable function $ u ( x) $ is called a weak solution of the equation $ L u = f $ if for all $ \phi \in C _ {0} ^ \infty $( infinitely-differentiable functions with compact support) one has

$$ \int\limits f ( x) \phi ( x) \ d x = \int\limits u ( x) L ^ {*} \phi ( x) d x . $$

Then, if the coefficients and the right-hand side of the equation $ L u = f $ are Hölder continuous, every weak solution is a classical solution.

For the Laplace equation the simplest well-posed problem is the Dirichlet problem. In the general case of an equation with operator (1) the boundary value problem consists in finding in a region $ D $ a solution $ u ( x) $ of the equation $ L u = f $ satisfying $ m ^ \prime = m / 2 $ boundary conditions of the form

$$ ( B _ {j} u ) ( x) = \ \sum _ {| \alpha | < m } b _ {\alpha j } ( x) D ^ \alpha u ( x) = \ \phi _ {j} ( x) ,\ \ x \in \partial D ,\ 1 \leq j \leq m ^ \prime . $$

To the Neumann problem correspond the boundary operators $ b _ {j} = ( \partial / \partial v ) ^ {j} $, where $ \partial / \partial v $ denotes differentiation in the direction of the outward normal.

For a boundary value problem to be Noetherian (cf. Noetherian operator) the boundary operators $ B _ {j} $ must satisfy the Shapiro–Lopatinskii complementation condition (see [2]) — an algebraic condition relating the polynomials

$$ \sum _ {| \alpha | = m _ {j} } b _ {\alpha j } ( x) \xi ^ \alpha ,\ \ 1 \leq j \leq m ^ \prime ,\ \ \sum _ {| \alpha | = m } a _ \alpha ( x) \xi ^ \alpha $$

at points of the boundary $ x \in \partial D $. The boundary operators of the Dirichlet problem satisfy this condition with respect to any elliptic operator $ L $.

If the coefficients of the differential operator and the solution are considered in the class of complex functions, then the fact that the operator $ L $ in (1) is elliptic is determined by the conditions $ \omega ( x , \xi ) \neq 0 $, $ \xi \neq 0 $. This definition allows of elliptic operators of odd order, as the example of the Cauchy–Riemann operator shows: $ \partial / \partial x _ {1} + i \partial / \partial x _ {2} $. In addition, properties of operators of even order can change. For example, for the Bitsadze equation (see [3])

$$ \frac{\partial ^ {2} u }{\partial x ^ {2} } + 2 i \frac{\partial ^ {2} u }{\partial x \partial y } + \frac{\partial ^ {2} u }{\partial y ^ {2} } = 0 $$

the Dirichlet problem is not well-posed: If $ D $ is the unit disc, then functions of the form $ u ( z) = f ( z) ( 1 - | z | ^ {2} ) $ are solutions of the homogeneous Dirichlet problem in $ D $ for any analytic function $ f ( z) $.

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. [4]). The operator (1) is strongly elliptic if for some complex function $ \gamma ( x) \neq 0 $ the condition

$$ \mathop{\rm Re} \gamma ( x) \sum _ {| \alpha | = m } a _ \alpha ( x) \xi ^ \alpha \geq \delta | \xi | ^ {m} ,\ \ \delta > 0 , $$

is satisfied. In particular, the order $ m $ 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 $ \xi $ and $ \xi ^ \prime $ the polynomial (in $ \tau $)

$$ \sum _ {| \alpha | = m } a _ \alpha ( x) ( \xi + \tau \xi ^ \prime ) ^ \alpha $$

has exactly $ m ^ \prime = m / 2 $ roots with negative imaginary part and the same number of roots with positive imaginary part. For $ n \geq 3 $ any elliptic operator is properly elliptic, so the definition essentially refers only to the case $ n = 2 $.

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 [6]) began to use these estimates systematically, and a more recent development is due to J. Schauder (see [7]). The Schauder estimates refer to solutions of a linear elliptic partial differential equation of the second order in a region $ D $ 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 $ K \subset D $ the derivatives up to the second order inclusive and their Hölder constants are estimated in terms of $ \sup | u | $ 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 [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 $ C ^ \alpha $, have been extended to the Sobolev spaces $ W _ {p} ^ {l} $( $ L _ {p} $- estimates) and refer to generalized solutions.

For strongly elliptic operators there is an priori estimate, called the 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 $ J ( x , y ) = J _ {y} ( x) $ that satisfies the condition

$$ \int\limits L ^ {*} \phi ( x) J ( x , y ) d x = \phi ( y) $$

for all $ \phi \in C _ {0} ^ \infty $. From the point of view of the theory of generalized functions this implies that

$$ J _ {y} = \delta _ {y} , $$

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 $ C ^ \infty $) and for a number of other equations with weaker restrictions on the coefficients. For an elliptic operator $ L _ {0} $ with constant coefficients, consisting of terms of highest order $ m = 2 m ^ \prime $, a fundamental solution depends only on the difference of the arguments and has the form $ ( y = 0 ) $:

$$ J ( x) = | x | ^ {m-} n \psi \left ( \frac{x}{| x | } \right ) + q ( x) \mathop{\rm ln} | x | , $$

where $ q $ is a polynomial of degree $ m - n $ with $ n $ even and $ m - n \geq 0 $; in the remaining cases $ q = 0 $ and $ \psi $ is analytic on the sphere $ | x | = 1 $( see [8]).

In particular, for the Laplace operator ( $ m = 2 $) $ q = 0 $, $ \psi = \textrm{ const } $ for $ n > 2 $ and $ q = \textrm{ const } $, $ \psi = 0 $ for $ n = 2 $.

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 [9]).

The maximum principle has had numerous applications in the theory of second-order linear elliptic partial differential equations. The functions $ a _ {ij} $, $ a _ {i} $, $ a $ are assumed to be continuous and the operator (3) is assumed to be uniformly elliptic in some region $ D $. The function $ u $ is taken to be continuous in the closure $ \overline{D}\; $ and to belong to the class $ C ^ {2} ( D) $.

The maximum principle in its strong form consists in the following. Let $ M $ be the operator $ L $ in (3) in which $ a \equiv 0 $.

a) If $ Mu \geq 0 $ and $ u $ attains its maximum at an interior point, then $ u $ is constant.

b) If $ Mu \geq 0 $ and the maximum of $ u $ is attained at a boundary point $ x _ {0} $ that lies on the surface of some ball entirely contained in $ \overline{D}\; $, then either $ u $ is constant or the derivative at $ x _ {0} $ in the direction of the outward normal, $ \partial u / \partial v $, is positive.

Similar assertions hold for an operator $ L $ with $ a \leq 0 $ 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

[1]	I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) MR0152665 MR0150320 MR0138774 Zbl 0127.03505 Zbl 0100.07603
[2]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) MR0226183 Zbl 0167.09401
[3]	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
[4]	F. John, "Plane waves and spherical means: applied to partial differential equations" , Interscience (1955) MR0075429 Zbl 0067.32101
[5]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[6]	S.N. Bernshtein, "Collected works" , 3 , Moscow (1960) (In Russian)
[7]	J. Schauder, "Ueber lineare elliptische Differentialgleichungen zweiter Ordnung" Math. Z. , 38 (1934) pp. 257–282 Zbl 0008.25502 Zbl 60.0422.01
[8]	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) MR73828
[9]	M.I. Vishik, "On strongly elliptic systems of differential equations" Mat. Sb. , 29 : 3 (1951) pp. 615–676

Comments

For the reduction of equation (3) to canonical form, [a1] has been fundamental.

References

[a1]	L.V. Ahlfors, "Conformality with respect to Riemannian metrics" Ann. Acad. Sci. Fenn. Ser. A1 , 206 (1955) pp. 1–22 MR0074855 Zbl 0067.30702
[a2]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Equations aux dérivées partielles de type elliptique" , Dunod (1969) (Translated from Russian) Zbl 0164.13001
[a3]	D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1977) MR0473443 Zbl 0361.35003
[a4]	M.H. Potter, H.F. Weinberger, "Maximum principles in differential equations" , Prentice-Hall (1967) MR219861
[a5]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501