Superposition in the $p$-Laplace Equation

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by  Karl K. Brustad

SUPERPOSITION IN THE P-LAPLACE EQUATION.
KARL K. BRUSTAD
Abstract. That a superposition of fundamental solutions to the
p-Laplace Equation is p-superharmonic – even in the non-linear
6 cases p > 2 – has been known since M. Crandall and J. Zhang
1
published their paper Another Way to Say Harmonic in2003. We
0
give a simple proof and extend the result by means of an explicit
2
formula for the p-Laplacian of the superposition.
n
a
J

8
1 1. Introduction

] Our object is a superposition of fundamental solutions for the p-
P
Laplace Equation
A
. (1.1) ∆ u := div |∇u|p−2∇u = 0.
h p
at Although the equation is non-lin(cid:0)ear, the fun(cid:1)ction
m
ρ(y)
[ V(x) = dy, ρ ≥ 0, 2 ≤ p < n
1 ZRn |x−y|np−−1p
v is a supersolution in Rn, i.e. ∆ V ≤ 0 in the sense of distributions. It
p
2
is a so-called p-superharmonic function – see Definition 2 on page 6 –
9
4 according to which it has to obey the comparison principle. The case
4 p = 2 reduces to the Laplace Equation ∆u = 0 with the Newtonian
0
potential
.
1
ρ(y)
0 V(x) = dy,
6 |x−y|n−2
Rn
1 Z
which is a superharmonic function.
:
v M. Crandall and J. Zhang discovered in [CZ03] that the sum
i
X
N
a
r i
, a > 0
a n−p i
i=1 |x−yi|p−1
X
of fundamental solutions is a p-superharmonic function. Their proof
was written in terms of viscosity supersolutions. A different proof was
given in [LM08]. The purpose of our note is a simple proof of the
following theorem:
Date: January 18, 2016.
1
2 KARL K.BRUSTAD
Theorem 1. Let 2 ≤ p < n. For an arbitrary concave function K,
∞
a
(1.2) W(x) := i +K(x), y ∈ Rn, a ≥ 0,
n−p i i
i=1 |x−yi|p−1
X
is p-superharmonic in Rn, provided the series converges at some point.
Through Riemann sums one can also include potentials like
ρ(y)
dy +K(x), ρ ≥ 0.
n−p
ZRn |x−yi|p−1
Similar results are given for the cases p = n and p > n and, so far
as we know, the extra concave term K(x) is a new feature. The key
aspect of the proof is the explicit formula (3.2) for the p-Laplacian of
the superposition. Although the formula is easily obtained, it seems to
have escaped attention up until now.
Finally, we mention that in [GT10] the superposition of fundamental
solutions has been extended to the p-Laplace Equation in the Heisen-
berg group. (Here one of the variables is discriminated.) In passing, we
show in Section 6 that similar results are not valid for the evolutionary
equations
∂ ∂
u = ∆ u and (|u|p−2u) = ∆ u
p p
∂t ∂t
where u = u(x,t). We are able to bypass a lenghty calculation in our
counter examples.
2. The fundamental solution
Consider a radial function, say
f(x) = v(|x|)
where we assume that v ∈ C2(0,∞). By differentiation
v′
(2.1) ∇f = xT, |∇f| = |v′|,
|x|
xxT v′ xxT v′
Hf = v′′ + I − , ∆f = v′′ +(n−1) ,
|x|2 |x| |x|2 |x|
(cid:18) (cid:19)
when x 6= 0.
The Rayleigh quotient formed by the Hessian matrix Hf = ∂2f
∂xi∂xj
above will play a central role. Notice that for any non-zero z ∈hRn, wie
have that
zT xxT z
= cos2θ
|z| |x|2 |z|
3
where θ is the angle between the two vectors x and z. This yields the
expedient formula
zT(Hf)z v′
(2.2) = v′′cos2θ+ sin2θ, x,z 6= 0.
|z|2 |x|
Since the gradient of a radial function is parallel to x, the Rayleigh
quotient in the identity
∇f(Hf)∇fT
(2.3) div |∇f|p−2∇f = |∇f|p−2 (p−2) +∆f
|∇f|2
(cid:18) (cid:19)
(cid:0) (cid:1)
reduces to v′′. The vanishing of the whole expression is then equivalent
to
v′
(2.4) (p−1)v′′ +(n−1) = 0
|x|
which, integrated once, implies that a radially decreasing solution w is
on the form
(2.5) w(x) = v(|x|) where v′(|x|) = −c|x|1p−−n1.
The constant c = c > 0 can now be chosen so that
n,p
∆ w +δ = 0
p
in the sense of distributions. Thus
(2.6) w(x) = −cn,ppp−−n1|x|pp−−n1, when p 6= n,
(−cn,nln|x|, when p = n
is the fundamental solution to the p-Laplace Equation (1.1).
3. Superposition of fundamental solutions
We now form a superposition of translates of the fundamental solu-
tion and compute its p-Laplacian. To avoid convergence issues all sums
are, for the moment, assumed finite.
Lemma 1. Let w be the fundamental solution to the p-Laplace equa-
tion. Define the function V as
N
(3.1) V(x) := a w(x−y ), a > 0, y ∈ Rn.
i i i i
i=1
X
Then, in any dimension and for any p 6= 1 1, ∆ V is of the same sign
p
wherever it is defined in Rn. Furthermore, the dependence of the sign
on p and n is as indicated in figure 1.
1When p = 1 there are no non-constant radial solutions of (1.1). Instead we
get the zero mean curvature equation in which a solution’s level sets are minimal
surfaces.
4 KARL K.BRUSTAD
n
4
3
2
1
−2 −1 0 1 2 3 4 p
Figure 1. ∆ V ≤ 0, ∆ V = 0, ∆ V ≥ 0
p p p
Proof. We simplify the notation by letting w and v denote that the
i i
functions w and v are to be evaluated at x−y and|x−y |, respectively.
i i
First, the linearity of the Hessian and the Laplacian enable us to
write
∇V(HV)∇TV
∆ V = |∇V|p−2 (p−2) +∆V
p
|∇V|2
(cid:18) (cid:19)
N ∇V(Hw )∇TV
= |∇V|p−2 a (p−2) i +∆w .
i |∇V|2 i
i=1 (cid:18) (cid:19)
X
Secondly, by (2.1) and (2.2) this is
N v′
= |∇V|p−2 a (p−2) v′′cos2θ + i sin2θ
i i i |x−y | i
Xi=1 (cid:18) (cid:16) i (cid:17)
v′
+ v′′ +(n−1) i
i |x−y |
i (cid:19)
N v′
= |∇V|p−2 a (p−2) i −v′′ sin2θ
i |x−y | i i
Xi=1 (cid:18) (cid:16) i (cid:17)
v′
+ (p−1)v′′ +(n−1) i
i |x−y |
i (cid:19)
where θ is the angle between x−y and ∇V(x). And finally, as w is
i i
a fundamental solution, the last two terms disappear by (2.4). We get
N v′
∆ V = (p−2)|∇V|p−2 a i −v′′ sin2θ .
p i |x−y | i i
i=1 (cid:18) i (cid:19)
X
5
It only remains to use the formula (2.5) for v′ to compute that
i
|x−vi′y | −vi′′ = −cn,pp+p−n−1 2|x−yi|2−p−n−1p
i
and the sign of ∆ V can easily be read off the final identity
p
N sin2θ
(3.2) ∆ V(x) = −c (p−2)(p+n−2)|∇V|p−2 a i .
p n,p p−1 i p+n−2
i=1 |x−yi| p−1
X
(cid:3)
Remark 1. Thethreegreenlinesinfigure1deservesomeattention. The
line p = 2 is obvious since the equation becomes linear. So is the line
n = 1 asthe“angle”between two numbers is0 orπ. The littlesurprise,
perhaps, is the case p + n = 2. Then the terms in V will be on the
form a |x−y |2 and it all reduces to the rather unexciting explanation
i i
that a linear combination of quadratics is again a quadratic.
4. Adding more terms
We will now examine what will happen to the sign of the p-Laplace
operator when an extra term, K(x), is added to the linear combination
(3.1). We will from now on only consider p > 2. Restricted to this
case, the factor C := c (p−2)(p+n−2) in (3.2) stays positive.
n,p n,p p−1
Let V be as in Lemma 1 and let K ∈ C2. For efficient notation,
write ξ = ξ(x) := ∇V(x)+∇K(x). Then
ξH(V +K)ξT
∆ (V +K) = |ξ|p−2 (p−2) +∆(V +K)
p
|ξ|2
(cid:18) (cid:19)
ξ(HV)ξT
= |ξ|p−2 (p−2) +∆V
|ξ|2
(cid:18) (cid:19)
ξ(HK)ξT
+|ξ|p−2 (p−2) +∆K .
|ξ|2
(cid:18) (cid:19)
Now, the second to last term equals
−Cn,p|ξ|p−2 ai|x−yi|2−p−n−1p sin2αi ≤ 0
i
X
where α is the angle between x − y and ∇V(x) + ∇K(x). Thus it
i i
suffices to ensure that the last term also is non-positive in order for the
p-Laplace to hold its sign. Lemma 2 presents a sufficient condition.
Lemma 2. Let p > 2 and define V as in (3.1). Then
(4.1) ∆ (V(x)+K(x)) ≤ 0
p
for all concave functions K ∈ C2(Rn) wherever the left-hand side is
defined.
6 KARL K.BRUSTAD
Proof. zT(HK)z ≤ 0 for all z ∈ Rn since the Hessian matrix of a con-
cave function K is negative semi-definite. Also K is superharmonic
since the eigenvalues of HK are all non-positive, i.e. ∆K ≤ 0. There-
fore,
ξ(HK)ξT
∆ (V(x)+K(x)) ≤ |ξ|p−2 (p−2) +∆K ≤ 0.
p |ξ|2
(cid:18) (cid:19)
(cid:3)
Remark 2. Though K ∈ C2 being concave is sufficient, it is not neces-
sary. A counter example is provided by the quadratic form
1
K(x) = xTAx, where A = diag(1−m,1,...,1), m = p+n−2.
2
ThenK isnotconcave, butacalculationwillconfirmthat(p−2)ξ(HK)ξT+
|ξ|2
∆K ≤ 0 and hence ∆ (V + K) ≤ 0. In fact, a stronger result than
p
Lemma 2 is possible: Let f be C2 at x for i = 1,...,N and let
i
λi ≤ λi ≤ ··· ≤ λi
1 2 n
be the eigenvalues of the Hessian matrix Hf (x). If
i
λi +···+λi +(p−1)λi ≤ 0 ∀i,
1 n−1 n
then ∆ ( f ) ≤ 0 at x.
p i i
P 5. p-superharmonicity
We now prove that
∞
W(x) := a w(x−y )+K(x), a ≥ 0, y ∈ Rn, K concave
i i i i
i=1
X
is a p-superharmonic function in Rn. The three cases 2 < p < n, p = n
and p > n are different and an additional assumption, (5.3), seems
to be needed when p ≥ n. In the first case, only convergence at one
point is assumed. We start with the relevant definitions and a useful
Dini-type lemma.
Definition 1. Let Ω be a domain in Rn. A continuous function h ∈
W1,p(Ω) is p-harmonic if
loc
(5.1) |∇h|p−2∇h∇φT dx = 0
Z
for each φ ∈ C∞(Ω).
0
Definition 2. A function u: Ω → (−∞,∞] is p-superharmonic in
Ω if
i) u 6≡ ∞.
ii) u is lower semi-continuous in Ω.
iii) If D ⊂⊂ Ω and h ∈ C(D) is p-harmonic in D with h ≤ u ,
∂D ∂D
then h ≤ u in D.
(cid:12) (cid:12)
(cid:12) (cid:12)
7
Furthermore, if u ∈ C2(Ω), it is a standard result that u is p-
harmonic if and only if ∆ u = 0 and u is p-superharmonic if and
p
only if ∆ u ≤ 0.
p
Also, a function u in C(Rn)∩W1,p(Rn) is p-superharmonic if
loc
(5.2) |∇u|p−2∇u∇φT dx ≥ 0
Rn
Z
for all 0 ≤ φ ∈ C∞(Rn). See [Lin86].
0
Lemma 3. Let(f ) be anincreasingsequenceof lowersemi-continuous
N
(l.s.c.) functions defined on a compact set C converging point-wise to
a function f ≥ 0. Then, given any ǫ > 0 there is an N ∈ N such that
ǫ
f (x) > −ǫ
N
for all x ∈ C and all N ≥ N .
ǫ
The standard proof is omitted.
In the following, K is any concave function in Rn. We let K , δ > 0
δ
denote the smooth convolution φ ∗K with some mollifier φ . One can
δ δ
show that K is concave and
δ
K → K
δ
locally uniformly on Rn as δ → 0+.
5.1. The case 2<p<n. Let δ > 0. If y ∈ Rn and a > 0, the function
i i
N
a
Wδ(x) := i +K (x)
N n−p δ
i=1 |x−yi|p−1
X
is p-superharmonic except possibly at the poles y (Lemma 2). Defin-
i
ing Wδ(y ) := ∞, we claim that Wδ is p-superharmonic in the
N i N
whole Rn.
We have to verify Def. 2. Clearly, i) and ii) are valid. For the
comparison principle in iii) we select D ⊂⊂ Rn (i.e. D is bounded)
and let h ∈ C(D) be p-harmonic in D with h ≤ Wδ . If any,
∂D N ∂D
isolate the points y in D with ǫ-balls B := B(y ,ǫ) where ǫ > 0 is so
i i (cid:12)i (cid:12)
small so that Wδ ≥ max h. This is possible(cid:12)because h(cid:12)is bounded
N Bi D
and because lim Wδ(x) = ∞. Then Wδ is C2 on D \ ∪B so, by
x→(cid:12)yi N N i
Lemma 2, ∆ W (cid:12)≤ 0 on this set. Also, h ≤ Wδ by
p N ∂(D\∪Bi) N ∂(D\∪Bi)
the construction of the ǫ-balls, so h ≤ Wδ on this set since Wδ is
(cid:12)N (cid:12) N
p-superharmonic there. Naturally, h ≤ Wδ(cid:12) on ∪B , so the(cid:12) inequality
N i
will hold in the whole domain D. This proves the claim.
Now N → ∞. Assume that the limit function
∞
a
Wδ(x) := i +K (x)
n−p δ
i=1 |x−yi|p−1
X
8 KARL K.BRUSTAD
is finite at least at one point in Rn. We claim that Wδ is p-
superharmonic.
By assumption Wδ 6≡ ∞ and it is a standard result that the limit of
an increasing sequence of l.s.c functions is l.s.c.
Part iii). Suppose that D ⊂⊂ Rn and h ∈ C(D) is p-harmonic in D
with h ≤ Wδ . Then (Wδ −h) is an increasing sequence of l.s.c.
∂D ∂D N
functionsonthecompactset ∂D withpoint-wiselimit(Wδ−h) ≥ 0.
(cid:12) (cid:12) ∂D
If ǫ > 0(cid:12), then (W(cid:12)δ −h) > −ǫ for a sufficiently big N by Lemma 3.
N ∂D (cid:12)
That is (cid:12)
(cid:12)
(cid:12)(h−ǫ) < Wδ
∂D N ∂D
so(h−ǫ) ≤ Wδ sinceh−ǫis(cid:12)p-harmoni(cid:12)candWδ isp-superharmonic.
D N D (cid:12) (cid:12) N
Finally, since Wδ ≤ Wδ we get
(cid:12) N(cid:12)
(cid:12) (cid:12)
(h−ǫ) ≤ Wδ
D D
and as ǫ was arbitrary, the require(cid:12)d inequa(cid:12)lity h ≤ Wδ in D is obtained
(cid:12) (cid:12)
and the claim is proved.
Let δ → 0 and set
∞
a
i
W(x) := +K(x).
n−p
i=1 |x−yi|p−1
X
We claim that W is p-superharmonic.
Part i) and ii) are immediate. For part iii), assume D ⊂⊂ Rn and
h ∈ C(D) is p-harmonic in D with h ≤ W . Let ǫ > 0. Then
∂D ∂D
there is a δ > 0 such that
(cid:12) (cid:12)
(cid:12) (cid:12)
|K(x)−K (x)| < ǫ
δ
at every x ∈ D. We have
Wδ = W +K −K > W −ǫ ≥ h−ǫ
δ
on∂D. Andagain,sinceh−ǫisp-harmonicandWδ isp-superharmonic,
we get Wδ ≥ h−ǫ in D. Thus
W ≥ Wδ −ǫ ≥ h −2ǫ.
D D D
This proves the claim,(cid:12)settles t(cid:12)he case 2 <(cid:12) p < n and completes the
(cid:12) (cid:12) (cid:12)
proof of Theorem 1.
We now turn to the situation p ≥ n and introduce the assumption
∞
(5.3) A := a < ∞.
i
i=1
X
9
5.2. The case p=n. Let δ > 0. The partial sums
N
Wδ(x) := − a ln|x−y |+K (x)
N i i δ
i=1
X
are p-superharmonic in Rn by the same argument as in the case 2 <
p < n.
Let N → ∞. We claim that
∞
Wδ(x) := − a ln|x−y |+K (x)
i i δ
i=1
X
is p-superharmonic in Rn provided the sum converges absolutely2
at least at one point.
Assume for the moment that, given a radius R > 0, it is possible to
find numbers C so that
i
ln|x−y | ≤ C for all x ∈ B := B(0,R),and
i i R
∞
(5.4)
the series a C =: S converges.
i i R
i=1
X
Define the sequence (f ) in B by
N R
N
f (x) := −a ln|x−y |+a C +K (x), f(x) := lim f (x).
N i i i i δ N
N→∞
i=1
X(cid:0) (cid:1)
Then (f ) is an increasing sequence of l.s.c functions implying that f
N
is l.s.c. in B and that
R
Wδ = f −S
R
is as well. Since R can be arbitrarily big, we conclude that Wδ does
not take the value −∞ and is l.s.c. in Rn.
For part iii) we show that f obeys the comparison principle. Assume
D ⊂⊂ B and h ∈ C(D) is p-harmonic in D with h ≤ f . Then
R ∂D ∂D
(f − h) is an increasing sequence of l.s.c. functions on the compact
N (cid:12) (cid:12)
set ∂D with point-wise limit (cid:12) (cid:12)
(f −h) ≥ 0.
∂D
If ǫ > 0, then (f −h) > −ǫ fo(cid:12)r a sufficiently big N by Lemma 3.
N ∂D (cid:12)
That is
(cid:12)
(cid:12) (h−ǫ) < f
∂D N ∂D
so(h−ǫ) D ≤ fN D sinceh−ǫisp(cid:12)(cid:12)-harmoni(cid:12)(cid:12)candfN isp-superharmonic.
Finally, since f ≤ f we get
N
(cid:12) (cid:12)
(cid:12) (cid:12)
(h−ǫ) ≤ f
D D
(cid:12) (cid:12)
2Conditional convergence is not suffi(cid:12)cient. A(cid:12) counter example is ai = 1/i2,
i δ
|yi|= exp((−1) i), yielding W (x)=−∞ for all yi 6=x6=0.
10 KARL K.BRUSTAD
and as ǫ was arbitrary, the required inequality h ≤ f in D is obtained.
Hence Wδ(x) = f(x) − S is a p-superharmonic function in any ball
R
B .
R
The claim is now proved if we can establish the existence of the
numbers C satisfying (5.4). By a change of variables we may assume
i
that the convergence is at the origin. That is
∞
L := a |ln|y || < ∞.
i i
i=1
X
We have
ln|x−y | ≤ ln(|x|+|y |)
i i
≤ ln(2max{|x|,|y |})
i
= max{ln|x|,ln|y |}+ln2,
i
so
C := max{lnR,ln|y |}+ln2
i i
will do since (for R > 1/2) the sequence of partial sums N a C is
i=1 i i
increasing and bounded by Aln2R+L.
P
The final limit δ → 0 causes no extra problems.
∞
W(x) := − a ln|x−y |+K(x)
i i
i=1
X
is p-superharmonic in Rn.
This settles the case p = n.
5.3. The case p>n. Let δ > 0. Consider again the partial sums
N
p−n
WNδ(x) := − ai|x−yi|p−1 +Kδ(x).
i=1
X
As before Wδ is p-superharmonic in Rn, but now a different
N
approach is required for the proof. For ease of notation, write
N
p−n
u(x) := − a |x−y |α +K(x), 0 < α := < 1,
i i
p−1
i=1
X
where K ∈ C∞(Rn) is concave. We will show that u satisfies the
integral inequality (5.2).
Clearly, u is continuous and |u|pdx < ∞ on any bounded domain
Ω
Ω. Also,
R xT p 1
|∇(|x|α)|p = α ∝
|x|2−α |x|(1−α)p
(cid:12) (cid:12)
where one can show that (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(1−α)p < n.
Thus |∇u|pdx < ∞ locally so u ∈ C(Rn)∩W1,p(Rn).
loc
R