Kernel Functions and Elliptic Differential Equations in Mathematical Physics

By Stefan Bergman and Menahem Schiffer

This text focuses on the theory of boundary value problems in partial differential equations, which plays a central role in various fields of pure and applied mathematics, theoretical physics, and engineering. Geared toward upper-level undergraduates and graduate students, it discusses a portion of the theory from a unifying point of view and provides a systematic and self-contained introduction to each branch of the applications it employs.
The two-part treatment begins with a survey of boundary value problems occurring in certain branches of theoretical physics. It introduces fundamental solutions in a heuristic way and examines their physical significance. Many concepts can be unified by concentrating upon these particular kernels, and the text explains the common mathematical background of widely varying theories, such as those of heat conduction, hydrodynamics, electrostatics, magnetostatics, and elasticity. In addition to its intrinsic interest, this material provides illustrations and exact mathematical formulation of the problems and the methods. The second part is confined to a rather special type of partial differential equation, which is dealt with in the greatest detail so that students can make applications and generalizations to similar problems.

• Language: English
• Publisher: Dover Publications
• Release date: Jan 23, 2013
• ISBN: 9780486154657

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Index

PART A

Boundary Value Problems for Partial Differential Equations of Elliptic Type

In Part A, various problems of theoretical physics which give rise to boundary value problems in partial differential equations will be considered. These considerations will motivate the detailed theory of boundary value problems which will be carried out in Part B, and will provide intuitive formulations of various questions in the abstract mathematical theory in terms of physical concepts. The insight into the physical significance of the equations considered will lead us to a correct and useful formulation of the mathematical problems and will frequently yield heuristic proofs of existence and uniqueness theorems which the exact mathematical treatment will establish only after rather lengthy investigation. We are dealing with a field of applied mathematics where the interaction between mathematics and physics is highly stimulating and fruitful for both sciences.

CHAPTER I

THEORY OF HEAT CONDUCTION

1. The differential equation of heat transfer: In (x, y, z)-space let us consider a body B with the boundary surface S. Let T (x, y, z, t) denote the temperature in the body at the point (x, y, z) and at the time t. The difference in temperature at various points of B creates a flow of heat represented by the vector

(1.1)

Here x = x (x, y, z) is the coefficient of heat conductivity at the point (x, y, z), and through a surface element dσ at (x, y, z) with unit normal vector ν the amount of heat

(1.2)

flows in the time interval Δ t. Let us denote the specific heat of the material of B at the point (x, y, z) by c (x, y, z). If we consider a portion B1 of the body B which is bounded by a surface S1 in B, the heat content of B1 is given by the integral

(1.3)

The heat content of B1 can be changed only by a flow of heat through the boundary surface S1. This flow can be computed by means of (1.2). We see that in the time interval Δ t we have a total change of heat content,

(1.4)

We denote this change by + Δ QB1 if v is understood to be the interior normal of S1 with respect to B1 and the integral (1.4) measures, therefore, the inflow of heat. Comparing (1.3) and (1.4), we obtain

(1.5)

The right-hand integral in (1.5) can be transformed into a volume integral by means of Green’s identity

(1.6)

which is valid for any two functions U and V which are twice continuously differentiable in the closed region B1 + S1, and where ν denotes the interior normal vector to S1 with respect to B1. Thus, (1.5) may be written in the form

(1.7)

Since this formula must hold for an arbitrary choice of the portion B1 of the body B, we have the identity

(1.8)

valid throughout B and for all times t. This is a partial differential equation for the change of the temperature T (x, y, z, t) in space and time.

The above equation is a homogeneous linear partial differential equation since the dependent function T (x, y, z, t) occurs in it homogeneously and linearly. Within the general theory of partial differential equations, the linear differential equations play an important role. Their theory is much easier and much further developed than that of general differential equations. The main reason for this is the fact that the knowledge of two solutions T1 and T2 of a linear homogeneous equation immediately leads to infinitely many additional solutions α1 T1 + α2 T2 with arbitrary constant factors α1, α2. Thus, the important role of special solutions in the theory of linear partial differential equations becomes obvious. A large part of this book will be devoted to the question of how a set of special solutions may be utilized to solve the problems connected with the linear partial differential equations.

It should be pointed out that the conductivity x and the specific heat c frequently depend on the temperature T of the material considered. In this case, the differential equation (1.8), is again obtained but now the dependent function T also occurs in k and c and the equation becomes non-linear. The theory of linear partial differential equations can be used to obtain results even in such non-linear problems.

Equation (1.8) becomes particularly simple in the case of a homogeneous isotropic body where x and c are independent of (x, y, z), i. e., x and c are constants. In this case, we have

(1.9)

This is a well-known partial differential equation and plays a central role in the theory of diffusion as well as in heat conduction.

Bateman [2], Frank-Mises [14], Riemann-Weber [66], Sommerfeld [71], Webster [89], Webster-Szego [90].

2. The special case of steady flow: An important special case in the problem of heat transfer is that of steady flow. Here the temperature T and heat flow q are stationary, i. e., independent of time. In this case, we have the simple partial differential equation for T (x, y, z)

(2.1)

which in the homogeneous case becomes Laplace’s equation

(2.2)

We shall give an important normal form for equation (2.1) which leads to an easier comparison of the general case (2.1) with the homogeneous one (2.2). We start with the identity

(2.3)

which holds for each pair of twice differentiable functions U and V. We write (2.1) in the form

(2.4)

Dividing (2.4) by x½ and using (2.3), we obtain

(2.5)

We then introduce as a new dependent function T*= x½ T and find that it satisfies the simple differential equation

(2.6)

In particular, if x½ is a harmonic function in B, equation (2.1) can be reduced to Laplace’s equation by a simple change of the dependent variable.

A steady flow through a body B can, of course, persist only if there are sources from which the heat is coming and sinks which will absorb the flow. These sources and sinks will be situated on the boundary S of B and will be the points of inflow and outflow of heat. We can realize such a steady flow through B by keeping the boundary points at fixed temperatures and thus create a flow through B from points on S of higher temperature to points on S of lower temperature. Mathematically, this means that we are looking for a solution of the steady heat conduction equation (2.1) which has prescribed values on the boundary S. By physical intuition it seems obvious that to any boundary values of T on S there should exist a solution of the partial differential equation (2.1) which assumes these values on S; it also seems obvious that the boundary values will determine the solution T in a unique way.

While the exact proof of the existence of a solution of (2.1) with given boundary values on S requires an involved argument, we can easily establish the uniqueness of such a solution. Let, in fact, T1 (x, y, z) and T2 (x, y, z) be two solutions of equation (2.1) which on S have the same values. We shall show that their difference

(2.7)

vanishes identically in B. We remark that in view of the linear and homogeneous character of the partial differential equation (2.1) the new function d (x, y, z) is itself a solution of (2.1) and, by construction, vanishes identically on S. The proof of the general uniqueness theorem has thus been reduced to proving the following particular statement: The only solution of (2.1) which vanishes everywhere on S is T (x, y, z) ≡0.

We may justify this result heuristically by the following physical interpretation. Let us immerse the body B in a medium which is kept at the constant temperature T = 0, so that we always have T = 0 on S. Let us wait until a steady temperature distribution throughout B has been established. This distribution T (x, y, z) will satisfy the equation (2.1) and have the boundary value zero on S. But it is also quite intuitive that it must be identically zero since there is no source or sink in B which might emit or absorb a flow of heat.

The mathematical proof of the uniqueness theorem is based on the identity

(2.8)

which is an immediate consequence of Green’s identity and the partial differential equation (2.1) satisfied by T. If there were a solution T (x, y, z) of (2.1) which vanished on S, we would have by virtue of (2.8)

(2.9)

i. e., T = const., and since T = 0 on S we conclude T ≡ 0 in B. This proves the uniqueness theorem.

The preceding physical and mathematical reasoning also suggests other uniqueness theorems. We easily recognize that if two solutions T1 and T2 of (2.1) have equal normal derivatives on S they can differ only by a constant. Here we have to show that the only solution of (2.1) with vanishing normal derivative on S is T ≡ const. This follows immediately from (2.8) and (2.9). The result is also intuitive from the physical interpretation. In fact, let the body B be isolated thermally from its surroundings; this means that we isolate the boundary S and prevent the flow of heat across it, i. e., .-- x ∂T/∂ν = 0 on S. Clearly, the only steady state of the isolated body is a constant temperature distribution over it. We shall later prove the existence theorem: There always exists a solution of (2.1) with prescribed normal derivatives on S. This means that we can always obtain a steady temperature distribution T in B with a prescribed heat flow—q · v across the boundary S. There is only one restriction on the values of the normal derivatives on S; namely, since no sources or sinks exist in B, the total flow of heat through S must be zero, i. e.,

(2.10)

This follows immediately from (2.1) by integration over B and the application of Green’s identity. But except for the restriction (2.10) the values of ∂T/∂ν can be prescribed arbitrarily on S.

Between the two extreme cases in which the body B is isolated from its surroundings or is in perfect heat contact with it, there is the case in which the body is immersed in a thermostatic fluid of temperature 0 and the difference of temperature between the surroundings and the boundary points of S creates a flow of temperature through the boundary which is proportional to this difference. Hence, we have the boundary condition:

(2.11)

where the factor μ may in turn depend on the boundary point. In this case, too, it seems clear that the only steady state attainable is the state T ≡ 0. We are thus led to the following uniqueness theorem:

Let λ be a positive function on S. If for two solutions Ti (x, y, z) of (2.1) in B the expression has the same values on S, then both solutions are identical in B.

The mathematical proof of this statement follows again from (2.8). It is sufficient to show that if for a solution T on S, then necessarily T ≡ 0. In fact, from (2.8) and the above condition we deduce

(2.12)

Since the right-hand side is non-positive and the left-hand side non-negative, we are led to grad T ≡ 0 in B and T ≡ 0 on S. Combined, these two results imply T ≡ 0.

The preceding uniqueness theorem can again be extended to an existence theorem: There exists exactly one solution of (2.1) with prescribed values for

Courant-Hilbert [13] vol. 1, Frank-Mises [14], Jeffreys [25], Kellogg [30], Murnaghan [53].

3. Point sources and fundamental singularities: Let us consider the case of a steady flow in a homogeneous body B. Here the temperature T (x, y, z) satisfies Laplace’s equation

(3.1)

Let P ≡ (x, y, z) and Q ≡ (ξ, η, ζ) be two variable points in B and let r (P, Q) = [(x—ξ)² + (y—η)² + (z—ζ)²]½ be the distance between them. Then, for fixed Q ∈ S, the function

(3.2)

is a solution of Laplace’s equation and may be interpreted as a temperature distribution over B. However, this solution becomes infinite at the point Q and we must study its singular character there.

Let us draw a sphere Sε of radius ε around Q and observe that on Sε.

(3.3)

where ν is the exterior normal on Sε, i. e., directed away from the center. Thus, if we assume the constant value of the coefficient x to be one, by virtue of (1.2), during each second the amount of heat:

(3.4)

enters B through Sε. This leads to an interpretation of U (x, y, z) as the temperature distribution in B due to a source of heat at the point Q which emits one calory of heat per second. We shall say that the source at Q has strength one.

The knowledge of the particular solution (3.2) with a point singularity at Q now permits the construction of various other types of sources. Thus, the function

(3.5)

will represent a steady temperature distribution in B due to a distribution of heat sources over B with strength density p. By means of the Laplace-Poisson equation¹ we then derive the inhomogeneous partial differential equation:

(3.6)

for the steady heat distribution V in a body with given source density p.

The solution

(3.7)

of (3.1) represents the temperature field due to a source at Q2 and a sink at Q1 of equal strength λ. Let Q1 ≡ (ξ, η, ζ), Q2 = (ξ + Δ ξ, η + Δη, ζ + Δ ζ) and ε = [(Δ ξ)² + (Δ ζ)² + (Δ ζ)²] ½ ; then by Taylor’s theorem, we may write

(3.8)

where o (x. The vector Q1Q2 with components Δ ξ, Δ η, Δ ξ can also be represented in the form ε e where e measures its length, and e is a unit vector describing its direction. Keeping e fixed, we let ε → 0 but at the same time increase the strength λ of the point sources in such a manner that λ ε = α remains constant. Thus in the limit as ε → 0 we obtain a new solution of Laplace’s equation

(3.9)

We could have deduced directly from (3.2) that W solves Laplace’s equation by differentiating the original equation with respect to the parameter point Q, but our construction shows that W is the solution due to a source and sink of equal strength which have combined without cancelling each other. Such a singularity is called a dipole. e is called the axis of the dipole and α its strength.

We could construct additional singular solutions of (3.1) by similar limit processes and linear operations upon the identity

(3.10)

¹Kellogg [30], p. 156. with respect to the parameter point Q. But we shall now show that the solutions (3.2) and (3.9), i. e., the point sources and the dipoles, play a particularly important role in the theory of the Laplace equation. Let T (x, y, x) be an arbitrary solution of (3.1); we consider the domain Bε* which is obtained by deleting from B the interior of the sphere Sε of radius ε around Q. We apply Green’s identity with respect to Bε* in the following form:

(3.11)

This takes into account that both T and 1/r are continuously differentiable solutions of (3.1) in the domain Bε*. We observe that

(3.12)

where dω is the solid angle subtended at Q by the surface element dσ. Letting e → 0, we clearly obtain from (3.11):

(3.13)

This shows that the solution T (Q) of Laplace’s equation is uniquely determined in B by its boundary values and the values of its normal derivative. Moreover, the formal aspect of (3.13) shows that the field T(Q) may be conceived as created by a distribution of sources on S with strength density —∂T/∂ν and of dipoles with their axes in the normal direction and strength density T. Thus, we have proved: The general solution of (3.1) may be created by an appropriate distribution of sources and dipoles on the boundary of the domain considered.

, we recognize the central role of this function in the theory of Laplace’s equation. It is called the fundamental singularity of this equation. There arises the problem of generalizing this concept to the case of the more general partial differential equation of steady temperature distribution:

(3.14)

Here, in analogy to the preceding special case, we may define the fundamental singularity S (P, Q) as follows:

S (P, Q) is, for fixed Q ∈ B, a solution of (3.14) and is continuous throughout B except at the point Q, where it becomes infinite. At the point Q we shall require that

(3.15)

where Sε. is a sphere of radius ε around Q.

S (P, Q) can be interpreted as the field due to a source of strength one at the point Q. It is not uniquely determined by our requirements since we may add to it an arbitrary solution of (3.14) without changing its characteristic properties. The existence of fundamental singularities is heuristically clear; for we can realize experimentally temperature distributions due to a point source. The exact mathematical proof for the existence is by no means simple and will be given in the second part of the book. We may expect to have

(3.16)

where A(P, Q) and B(P, Q) are twice continuously differentiable functions everywhere in B and

(3.17)

In fact, if (3.16) and (3.17) are satisfied, then clearly (3.15) will also hold. The asymptotic behavior at Q of S (P, Q) as described by (3.16) and (3.17) can be established by the general theory which will be developed later.

Sommerfeld [73].

4. Fundamental solutions: Among all possible fundamental singularities connected with the partial differential equation of the steady temperature distribution certain particular ones can be distinguished by their physical significance as well as by their useful mathematical properties. We shall call them the fundamental solutions of the equation considered and define them intuitively as follows.

Consider the body B with a point source of strength one at the point Q ∈ B and suppose that it is immersed in a thermostatic fluid of constant temperature T = 0. The temperature discontinuity at the boundary S of the body is assumed to create an outflow of heat from the body to the exterior which is proportional to the discontinuity, i. e., we shall assume that the steady temperature finally attained satisfies the boundary condition

(4.1)

where μ is a positive function of the position on S. In this way, we are led to a particular fundamental singularity S (P, Q) of our partial differential equation with the boundary condition (4.1). To each choice of μ such a solution can be found. We define:

Robin’s function Rλ (P, Q) is that fundamental singularity of the equation

(4.2)

which satisfies on S the boundary condition

(4.3)

By our preceding uniqueness theorem it is clear that Robin’s function is defined in a unique way. The most important consequence of our choice of the fundamental singularity is the fact that Robin’s function is symmetric in its argument and parameter points and therefore satisfies for fixed P ∈ B the partial differential equation (4.2) as a function of Q.

In order to prove this fact, we start with Green’s identity

(4.4)

which simplifies to

(4.5)

if v is a solution of (4.2). If both u and v satisfy (4.2), then by symmetry we derive from (4.5)

(4.6)

We apply this result to u = Rλ (P, Q) and v = Rλ (P, O). Since these functions are not continuous in B, we must at first delete from B two spheres of radius ε around the points Q and O and apply (4.6) to the remaining (4.3) domain. Then letting ε → 0 we obtain by a reasoning analogous to that which led to (3.13)

(4.7)

But now we can make use of the boundary conditions (4.3) which show that the left-hand integral vanishes. Thus, we obtain the symmetry law:

(4.8)

as was asserted.

If u is a regular solution of (4.2) in B and v = Rλ (P, Q), we may apply the identity (4.6) to the domain Bε* from which the interior of the sphere Sε of radius e around Q has been deleted. Letting ε → 0 and using the limit relation (3.15) which is valid for every fundamental singularity, we find quite readily that

(4.9)

We may simplify this result by applying (4.3) and obtain

(4.10)

This tells us that the knowledge of Robin’s function Rλ (P, Q) enables us to determine the solution of (4.2) whose boundary combination (∂u ∂v)—λu is known.

A limiting case of the above Robin’s functions is the Green’s function which corresponds to the value λ = ∞. From (4.1) we recognize that Green’s function G (P, Q) corresponds to the field of a unit source at the point Q in case of perfect heat contact with the thermostatic fluid at the boundary. In this case, we clearly have:

(4.11)

Green’s function serves for the representation of a solution of (4.2) by means of its boundary values on S. In fact, applying (4.6) to a regular solution u and for v = G (P, Q), we derive by the above limiting procedure:

(4.12)

It is easily seen that Green’s function and all Robin’s functions are positive in B. Since a larger value of λ means a better transfer of heat out of the body, we may expect an inequality

(4.13)

and indeed such an inequality can be derived by the methods of Section B. V. 1. If we let λ → 0, the Robin’s functions will increase beyond bounds and there cannot exist a finite limit function R0 (P, Q). In fact, such a function would represent the steady temperature distribution in a body B which is isolated from its surroundings and contains a source of heat with strength one. Clearly such a body would be heated to infinity and, hence, R0 (P, Q) could not have finite values.

The same result can also be derived by calculation. The function R0 (P, Q) would be, by definition, a solution of (4.2) with singularity at Q and the boundary condition ∂R0/∂v = 0 on S. Let us now apply Green’s identity with respect to the domain Bε* obtained from B by deleting the interior of the sphere Sε with radius ε around Q. We find

(4.14)

since R0 satisfies (4.2) in Bε*. By the boundary condition on S, we thus obtain

(4.15)

But this result contradicts the asymptotic formula (3.15) which R0 must satisfy as a fundamental singularity. Thus, the assumption of a finite R0 (P, Q) leads to a contradiction.

Our physical interpretation of the fundamental solutions Rλ (P, Q) suggests the introduction of a fundamental singularity which is as near as possible to the definition of R0. We define Neumann’s function N (P, Q) as that fundamental singularity in B with unit source at Q which on S has the value of its normal derivative proportional to x—1. Since N(P, Q) is determined by this requirement only up to an additive constant, we complete the definition by requiring the normalization,

(4.16)

The existence of Neumann’s function appears certain from physical considerations. It is the steady temperature distribution due to a unit source at Q with constant outflow through the boundary S. The factor of proportionality can be chosen in such a way that the loss of heat through S equals exactly the heat created at Q at each moment. Thus, we have the requirement

(4.17)

and since x∂N/∂v = const. on S,

(4.18)

on S is independent of the parameter point Q.

If we use the normalization (4.16) and the same reasoning which proved the symmetry of the Robin’s functions, we can prove the symmetry of the Neumann’s function

(4.19)

Neumann’s function plays an important role in the representation of solutions of (4.2) by means of the values of their normal derivatives on S. We have already remarked in Section 2 that we cannot prescribe these values completely arbitrarily on S but for any solution u (P) of (4.2) must require

(4.20)

Furthermore, the solution u (P) will not be determined uniquely by prescribing its normal derivative on S; let us normalize it, therefore, by the additional requirement

(4.21)

If we prescribe ∂u/∂v arbitrarily on S except for the condition (4.20) we may always find a function u (P) which satisfies (4.2), has the normalization (4.21), and has the prescribed values of the normal derivative. The representation of this function by means of the Neumann’s function is easily obtained as follows. We apply the identity (4.6) to the solution u considered and to v (P) = N (P, Q) with respect to the domain Bε*. By the usual limit procedure, we then derive:

(4.22)

Since x is constant on S and u is normalized by (4.21), we obtain

(4.23)

which determines the solution u (Q) by means of its normal derivative on S.

Green’s, Neumann’s and the Robin’s functions will be called the fundamental solutions of (4.2) for the domain B. Using them, we are now able to solve the principal boundary value problems connected with (4.2) and the domain B. Since every solution of (4.2) has a simple integral representation in terms of these fundamental solutions the entire analysis of the regular solutions of (4.2) can be reduced to a detailed study of these fundamental solutions.

Goursat [16], Gunther [17], Poincaré [63]. Smirnoff [69].

5. Discontinuities: Let us suppose that the body B consists of two homogeneous parts B1 and B2 in which the coefficient of thermal conductivity has the constant values x1, and x2, respectively. Clearly, in each component of B a steady temperature distribution T (x, y, z) must satisfy Laplace’s equation. For physical reasons, the field of temperature T must be continuous in B, and the normal component of the heat flow vector must vary continuously across the surface ∑ which separates the two homogeneous components. For otherwise an indefinite accumulation of heat on ∑ would take place leading to infinite values of T along this separating surface. Thus, we find along ∑ the condition

(5.1)

where v1, and v2 denote the interior normals on ∑ of B1 and B2, respectively.

Figure 1

If we prescribe the values of the solution T of the steady temperature problem on the boundary S of B, we are led to a boundary value problem of the following type: Find a solution of Laplace’s equation in B with prescribed boundary values on S which is continuous in B but whose derivatives have a jump across a prescribed surface ∑ of B described by (5.1).

Similarly, we may seek solutions of Laplace’s equation with a prescribed discontinuity (5.1) and given values of (∂u/∂v)—λu, λ>0 on S. We shall show, at first, that all these boundary value Problems have at most one solution. In fact, suppose that there were two functions u and v which on S have the same values of (∂u/∂v)—λu, and on ∑ satisfy the jump condition (5.1). Their difference w = u—v would still satisfy (5.1) on ∑ and, on S would satisfy

(5.2)

Consider then the integral

(5.3)

Since w is harmonic in B1 and in B2, we obtain by Green’s identity

(5.4)

where S1 and S2 designate the boundary surfaces of B1 and B2, respectively. Let us subdivide S1 = C1 + ∑, S2 = C2 + ∑ so that C1 + C2 constitutes the boundary surface S of B. Since w is continuous on ∑ and its normal derivative satisfies the discontinuity condition (5.1) there, we see that the integrals extended over ∑ in (5.4) cancel each other. On C1 and C2 we have the relation (5.2), so that (5.4) may be put in the form:

(5.5)

From definition (5.3) it is obvious that D {w} is non-negative while (5.5) shows that the same expression is non-positive. This leads clearly to

(5.6)

and therefore

(5.7)

which proves the asserted uniqueness theorem.

The mathematical tools for dealing with solutions of a given partial differential equation with prescribed discontinuities along an interior surface ∑ will be developed in Sections B. III. 1—2. In this case, the functions required can also be described in terms of their boundary values and the values of their normal derivatives on the boundary by the use of the fundamental solutions for B and certain functions closely related to them.

Riemann-Weber [66].

6. Dirichlet’s principle: In order to give uniqueness proofs for boundary value problems connected with the partial differential equation

(6.1)

we have used the integral

(6.2)

We shall now show the close formal relation between this integral expression and the partial differential equation (6.1). For this purpose we introduce the bilinear integral

(6.3)

and notice that by Green’s identity we have

(6.4)

If, in particular, u satisfies (6.1) we obtain

(6.5)

i. e., for an arbitrary function v in B the expression D {u, v} will depend only on the boundary values of v on S. If v = 0 on S, we have

(6.6)

The bilinear integral always vanishes if one argument function satisfies (6.1) and the other vanishes on S.

The connection between D {u} and D {u, v} is established by the obvious identity

(6.7)

which is an immediate consequence of the homogeneous quadratic dependence of D {u} upon u. Since D {u + tv} ≧ 0, we obtain Schwarz’ inequality

(6.8)

by setting t =—D {u, v}/D {v}. Equality is possible only for u (P) + + tv (P) ≡ const. in B.

We may consider D {u} as a function of the function u; it is usual to call a number which depends upon the infinitude of values of a function u in a domain B a functional of u. Let us now study the change of the functional D {u} under a change δu = ε v(P) of its argument function. We have by (6.7)

(6.9)

and using (6.4)

(6.10)

If we consider a function F (x1,..., xn) depending on n variables xi, we have the identity

(6.11)

and we call ∂F/∂xi the partial derivative of F with respect to the variable xi-Analogously, let us write:

(6.12)

and denote—2 div (x grad u) as the functional derivative of D {u} with respect to the interior values u (P), and— 2 x (∂u/∂v) the functional derivative of D {u} with respect to the boundary values u (P) on S.

The close relation between the differential expression div (x grad u) and the quadratic integral D {u} now becomes clear. The differential expression is essentially the functional derivative of the quadratic functional D {u}.

In the theory of functions of a finite number of variables, the extremal points of the function are characterized by the vanishing of all partial derivatives ∂F/∂xi. Analogously, we may expect some extremum property of the functional D {u} for an argument function u in B for which all the functional derivatives in B vanish, i. e., which satisfies the partial differential equation (6.1). In fact, we have the theorem: Each solution u of the Partial differential equation (6.1) leads to the minimum value of the integral (6.2) among all functions in B with the same boundary values on S as u. In order to prove this statement let w be an arbitrary function in B which on S has the same values as u. The difference v = w - u then has the boundary values zero on S. Using (6.7), we have

(6.13)

Since u satisfies (6.1) and v vanishes on S, we obtain from (6.6)

(6.14)

Equality can hold only if D {v} = 0, w ≡ u in B. Thus, the extremal property of the solution u of (6.1) among all functions with the same boundary values on S has been established.

One can now try to reverse this reasoning as follows. There are surely an infinity of functions with prescribed fixed continuous boundary values on S. Within this class of functions one can ask for that particular function which yields the minimum value of the integral (6.2). It is easy to show that if such a function exists it must necessarily satisfy the partial differential equation (6.1). This line of reasoning has, in fact, been used in order to establish the existence of a solution of a partial differential equation of type (6.1) with prescribed boundary values. The main difficulty in this approach is the proof that there in fact exists a continuously differentiable function u (P) in B with prescribed values on S which yields the minimum value for D {u}. In the last century, Lord Kelvin and Riemann attempted the first existence proof for the boundary value problem along these lines. They took for granted the existence of a minimum function, postulating its existence in the so-called Dirichlet principle. Weierstrass objected to this reasoning and gave examples of minimum problems in the calculus of variations where no continuous function exists which solves the problem. The method of the Dirichlet principle was rehabilitated, however, by Hilbert who constructed minimum sequences and proved that uniformly convergent subsequences could be selected which converge towards a continuously differentiable limit function. This limit function was proved by Hilbert to satisfy the differential equations and the boundary conditions required. In the second part of this book, we shall base the mathematical theory of the boundary value problem upon a detailed study of one integral of the type D {u} and also base an existence proof upon the method of orthogonal projection which is closely related to Dirichlet’s principle. We shall call an integral D {u} the Dirichlet integral of a given partial differential equation if it is semi-definite and if it has the partial differential equation as its functional derivative.

As we shall see on the following pages, the Dirichlet integral has physical significance in various boundary value problems. It is slightly more difficult to provide such interpretation in the case of heat conduction. But even here, some interesting observations can be made about D {u}. Consider a body B whose boundary S is kept at a fixed temperature, say T = 0. Let T (x, y, z, t) be the temperature distribution inside B at the time t; then T will satisfy the partial differential equation (1.8) of heat conduction. Now, the integral

(6.15)

is a function of time and we may calculate its derivative with respect to t. We find:

(6.16)

Using Green’s identity, the fact that T and, hence ∂T/∂t vanish on S, and (1.8) we obtain

(6.17)

This result shows that during the equalization of temperature in B, the Dirichlet integral D is decreasing monotonically.

Courant [12], Courant-Hilbert [13], Levy [38], Volterra [86], [87], Volterra-Pérès [88].

7. A modified heat equation: We now consider a particular problem in heat conduction which leads to a partial differential equation which will occupy our attention throughout a large part of the mathematical section of the book. Consider a body B with a coefficient of thermal conductivity x = 1; at each point of B an endothermal chemical process will be assumed to take place at