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8. The Newtonian Potential
38
I. INTRODUCTION
8. The Newtonian Potential In the space Rn,we consider the Laplacian differential operator
and note that if u(x) is a function only of radius, r = 1x1, the operator takes a particularly simple form Au(x) = f"(r)
n-1 +f'(r), r
where u(x) = f ( l x l ) . Hence, the solutions to the equation Au = 0 when n 2 3 which are functions only of r = 1x1 are necessarily of the form u(x) = A + BrZ-"; here, we use the fact that the solutions of a second-order linear differential operator form a two-dimensional vector space; we then have only to verify that u = constant and u(x) = rZ-" are a pair of linearly independent solutions to the equation. When n = 2, the solutions are of the form A + B log r but for simplicity in the sequel, we shall exclude this case, although it is actually the most interesting in view of the relation between potential theory and the theory of analytic functions. Let p be a positive Bore1 measure on Rn of finite total mass; for n 2 3 we form the Newtonian potential of p,
a function which is unambiguously defined for all x, although, perhaps, often infinite. Let F denote the support of p, that is, the smallest closed set, the complement of which has p-measure 0. The Newtonian potential of p then has the following properties: (1) u(x) is a C"-function in the complement of F and Au = 0 there. (2) u(x) is positive and lower semicontinuous. (3) u(x) is integrable over any sphere 1x1 < R, hence is infinite only on a set of Lebesgue measure 0. (4) If u(x) is bounded on F by M, then u(x) 5 2"-'M for all x in R".
PROOF: If x is considered in a neighborhood U contained in the open complement of F, the differentiation under the integral sign is legitimate; since there the function Ix - yI2-" is C" and x is bounded away from y in F,
8.
THE NEWTONIAN POTENTIAL
39
we find that u(x) is C" in the neighborhood U ;since r'-" is harmonic, we find that Au = 0 in any such neighborhood. That the function is lower semicontinuous follows immediately from the theorem of Fatou: if & converges to x,, , then u(xo) I lim inf u(xk).If we compute the integral of u(x) over a sphere of radius R we have
the interchange of integrations being permitted by the fact that the function Ix - yI2-" is a Borel function in the 2n-dimensional space and positive there. The inner integral is the potential computed with a measure v consisting of Lebesgue measure confined to the sphere 1x1 c R; since r2-" is Lebesgue' integrable this potential is a bounded function; it is surely lower semicontinuous and it obviously takes its maximum at the origin, where it is finite. Thus we have to compute the integral of a bounded Borel function relative to the measure p which has finite total mass. The integral is therefore finite. Hence u(x) is locally integrable, and therefore is infinite only on a set of Lebesgue measure 0. We suppose finally that u(x) 5 M on F. Let x be outside F, and x' the nearest point of F to x. (If there is more than one such point, x' is any convenient choice of the nearest point.) Now, for all y in F, Ix' - yI
5 21x - yl ,
whence Ix - yI2-" 5 2n-21x' - y l z - " .
From this it follows directly that u(x) 5 2"-' u(x') S 2"-'M. We use these facts to compute two interesting potentials. Let
U ( x ) = Jdw(Y)/lx - A"-'; here w is the uniform distribution of unit mass over the sphere Jyl = 1. The function is clearly harmonic outside F = [lyl = 13 and is a function only of radius, owing to the symmetry. Thus, inside the sphere, since U(0) = 1, the functionisconstant, U ( x ) = 1, and outside thesphere wehave U ( x ) = BIxJ'-", since U ( x ) must vanish as 1x1 grows large. We will show presently that U ( x )is continuous, hence B = 1. Let x vary on some line, say the positive x,-coordinate axis, and let z be the intersection of that line with the unit sphere. By the lower semicontinuity of U ( x ) , or what is the same thing, Fatou's theorem, U(z) 5 1 and hence l/(z- yl"-' is w-integrable. As x varies on the positive x,-axis, z is the nearest point to x belonging to the support of w, hence, as we have seen, Ix - yl'-" 5 2"-'1z - y12-" for all y such that Iyl = 1; from the Lebesgue convergence theorem, then, U ( x )approaches U(z) and U ( x )is continuous.
40
I. INTRODUCTION
Another interesting potential is
the measure being Lebesgue measure restricted to the unit ball IyI 5 1. As we have already observed, this function is finite at all points, since r 2 - , is locally integrable in R". We write it in spherical coordinates to obtain dw(2)
n-2
r d r , where y = rz with r = Iyl and lzl = 1 .
Thus, if U ( x ) denotes the potential which we have just studied, then 1
V ( x ) = J- rLI(z) d r w, o r
If we compute AV, which is well defined except at the surface of the unit sphere, we find AV = o,(2 - n) inside the sphere, and A V = 0 outside it. We consider a form of Green's formula: If V is an open set with a suitably regular boundary B, and u(x) and v(x) are two C2-fun$ions defined in the closure of V, then
AUV- AVUd x =
au
a0
-V - - u d S , I,an an
where dS is the element of surface area of B and du/dn is the exterior normal derivative of u(x). The formula is easily verified when V is a rectangular parallelepiped, however, we shall need it when V is a sphere. Let V be the volume between two concentric spheres in R" centered about the origin, that is, the set E c r c R; let u(x) be a Cz-function which vanishes for 1x1 >= Rand let v(x) = r2-",a function which is smooth and harmonic in V. We apply Green's formula, and note that the integrand vanishes on the outer boundary of V, whence
dx =
-J .aur ( E , 0) dw(8) W , E + ( 2 - n)w, / u ( c , 0) d w ( 0 )
8.
THE NEWTONIAN POTENTIAL
41
Since IxI2-" is integrable (we have n 2 3), the left-hand side tends with decreasing E to A U ( X ) / ~ Xd~x"; the - ~ first term on the right-hand approaches 0, since the gradient of the C2-function u(x) is uniformly bounded; the last term evidently approaches (2 n)w,u(O).Since the origin can be chosen anywhere, we have finally
s
where the integral may be taken over the whole space. Another interesting consequence of the Green's formula is obtained by applying it when v(x) = 1 and V is the sphere of radius R.We have then jvAu(x) d x
=s 7
au ( R , 0)d o ( @ R " - ' W ~ . r
We shall write F ( r ) = ju(r, 0) dw(0) and note that F(0) = u(0). Since F ( r ) is absolutely continuous and F'(r) = j(au/dr)(r, 0) do(@ we have
j
R 1 F'(R) = - lsRI
Au d x ,
SR
where SR denotes the ball 1x1 < R and IS,l its measure. Thus, finally, F(R) - u(0) =
IRnr -1 -
0
IS.1 s,
Au d x d r .
Thus, if we suppose AM2 0 inside a sphere 1x1 < R , then for all r < R we have u(0) S / u ( r , 0) d w ( 0 ) .
It follows that if u ( x ) is C2 in an open set G where AMis nonnegative, then for any x in G and any r smaller than the distance of x to the boundary, we have u(x) S
Iu ( x + r z )
dw(z)
I n this case we say that u is subharmonic in G . Similarly, if AM5 0 in G we have the reverse inequality; u(x) is superharmonic in G. Finally, for A M = 0 in G, we have equality, and we say that u(x) is harmonic in G. In the next section we introduce a seemingly more general definition of harmonic functions, while in Section 27 the subharmonic functions are considered in some detail.
I. INTRODUCTION
8. The Newtonian Potential In the space Rn,we consider the Laplacian differential operator
and note that if u(x) is a function only of radius, r = 1x1, the operator takes a particularly simple form Au(x) = f"(r)
n-1 +f'(r), r
where u(x) = f ( l x l ) . Hence, the solutions to the equation Au = 0 when n 2 3 which are functions only of r = 1x1 are necessarily of the form u(x) = A + BrZ-"; here, we use the fact that the solutions of a second-order linear differential operator form a two-dimensional vector space; we then have only to verify that u = constant and u(x) = rZ-" are a pair of linearly independent solutions to the equation. When n = 2, the solutions are of the form A + B log r but for simplicity in the sequel, we shall exclude this case, although it is actually the most interesting in view of the relation between potential theory and the theory of analytic functions. Let p be a positive Bore1 measure on Rn of finite total mass; for n 2 3 we form the Newtonian potential of p,
a function which is unambiguously defined for all x, although, perhaps, often infinite. Let F denote the support of p, that is, the smallest closed set, the complement of which has p-measure 0. The Newtonian potential of p then has the following properties: (1) u(x) is a C"-function in the complement of F and Au = 0 there. (2) u(x) is positive and lower semicontinuous. (3) u(x) is integrable over any sphere 1x1 < R, hence is infinite only on a set of Lebesgue measure 0. (4) If u(x) is bounded on F by M, then u(x) 5 2"-'M for all x in R".
PROOF: If x is considered in a neighborhood U contained in the open complement of F, the differentiation under the integral sign is legitimate; since there the function Ix - yI2-" is C" and x is bounded away from y in F,
8.
THE NEWTONIAN POTENTIAL
39
we find that u(x) is C" in the neighborhood U ;since r'-" is harmonic, we find that Au = 0 in any such neighborhood. That the function is lower semicontinuous follows immediately from the theorem of Fatou: if & converges to x,, , then u(xo) I lim inf u(xk).If we compute the integral of u(x) over a sphere of radius R we have
the interchange of integrations being permitted by the fact that the function Ix - yI2-" is a Borel function in the 2n-dimensional space and positive there. The inner integral is the potential computed with a measure v consisting of Lebesgue measure confined to the sphere 1x1 c R; since r2-" is Lebesgue' integrable this potential is a bounded function; it is surely lower semicontinuous and it obviously takes its maximum at the origin, where it is finite. Thus we have to compute the integral of a bounded Borel function relative to the measure p which has finite total mass. The integral is therefore finite. Hence u(x) is locally integrable, and therefore is infinite only on a set of Lebesgue measure 0. We suppose finally that u(x) 5 M on F. Let x be outside F, and x' the nearest point of F to x. (If there is more than one such point, x' is any convenient choice of the nearest point.) Now, for all y in F, Ix' - yI
5 21x - yl ,
whence Ix - yI2-" 5 2n-21x' - y l z - " .
From this it follows directly that u(x) 5 2"-' u(x') S 2"-'M. We use these facts to compute two interesting potentials. Let
U ( x ) = Jdw(Y)/lx - A"-'; here w is the uniform distribution of unit mass over the sphere Jyl = 1. The function is clearly harmonic outside F = [lyl = 13 and is a function only of radius, owing to the symmetry. Thus, inside the sphere, since U(0) = 1, the functionisconstant, U ( x ) = 1, and outside thesphere wehave U ( x ) = BIxJ'-", since U ( x ) must vanish as 1x1 grows large. We will show presently that U ( x )is continuous, hence B = 1. Let x vary on some line, say the positive x,-coordinate axis, and let z be the intersection of that line with the unit sphere. By the lower semicontinuity of U ( x ) , or what is the same thing, Fatou's theorem, U(z) 5 1 and hence l/(z- yl"-' is w-integrable. As x varies on the positive x,-axis, z is the nearest point to x belonging to the support of w, hence, as we have seen, Ix - yl'-" 5 2"-'1z - y12-" for all y such that Iyl = 1; from the Lebesgue convergence theorem, then, U ( x )approaches U(z) and U ( x )is continuous.
40
I. INTRODUCTION
Another interesting potential is
the measure being Lebesgue measure restricted to the unit ball IyI 5 1. As we have already observed, this function is finite at all points, since r 2 - , is locally integrable in R". We write it in spherical coordinates to obtain dw(2)
n-2
r d r , where y = rz with r = Iyl and lzl = 1 .
Thus, if U ( x ) denotes the potential which we have just studied, then 1
V ( x ) = J- rLI(z) d r w, o r
If we compute AV, which is well defined except at the surface of the unit sphere, we find AV = o,(2 - n) inside the sphere, and A V = 0 outside it. We consider a form of Green's formula: If V is an open set with a suitably regular boundary B, and u(x) and v(x) are two C2-fun$ions defined in the closure of V, then
AUV- AVUd x =
au
a0
-V - - u d S , I,an an
where dS is the element of surface area of B and du/dn is the exterior normal derivative of u(x). The formula is easily verified when V is a rectangular parallelepiped, however, we shall need it when V is a sphere. Let V be the volume between two concentric spheres in R" centered about the origin, that is, the set E c r c R; let u(x) be a Cz-function which vanishes for 1x1 >= Rand let v(x) = r2-",a function which is smooth and harmonic in V. We apply Green's formula, and note that the integrand vanishes on the outer boundary of V, whence
dx =
-J .aur ( E , 0) dw(8) W , E + ( 2 - n)w, / u ( c , 0) d w ( 0 )
8.
THE NEWTONIAN POTENTIAL
41
Since IxI2-" is integrable (we have n 2 3), the left-hand side tends with decreasing E to A U ( X ) / ~ Xd~x"; the - ~ first term on the right-hand approaches 0, since the gradient of the C2-function u(x) is uniformly bounded; the last term evidently approaches (2 n)w,u(O).Since the origin can be chosen anywhere, we have finally
s
where the integral may be taken over the whole space. Another interesting consequence of the Green's formula is obtained by applying it when v(x) = 1 and V is the sphere of radius R.We have then jvAu(x) d x
=s 7
au ( R , 0)d o ( @ R " - ' W ~ . r
We shall write F ( r ) = ju(r, 0) dw(0) and note that F(0) = u(0). Since F ( r ) is absolutely continuous and F'(r) = j(au/dr)(r, 0) do(@ we have
j
R 1 F'(R) = - lsRI
Au d x ,
SR
where SR denotes the ball 1x1 < R and IS,l its measure. Thus, finally, F(R) - u(0) =
IRnr -1 -
0
IS.1 s,
Au d x d r .
Thus, if we suppose AM2 0 inside a sphere 1x1 < R , then for all r < R we have u(0) S / u ( r , 0) d w ( 0 ) .
It follows that if u ( x ) is C2 in an open set G where AMis nonnegative, then for any x in G and any r smaller than the distance of x to the boundary, we have u(x) S
Iu ( x + r z )
dw(z)
I n this case we say that u is subharmonic in G . Similarly, if AM5 0 in G we have the reverse inequality; u(x) is superharmonic in G. Finally, for A M = 0 in G, we have equality, and we say that u(x) is harmonic in G. In the next section we introduce a seemingly more general definition of harmonic functions, while in Section 27 the subharmonic functions are considered in some detail.