Superposition of p-superharmonic functions

Karl K. Brustad

doi:10.1515/acv-2017-0030

Open Access Published by De Gruyter December 7, 2017

Superposition of p-superharmonic functions

Karl K. Brustad

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2017-0030

Abstract

The dominative p-Laplace operator is introduced. This operator is a relative to the p-Laplacian, but with the distinguishing property of being sublinear. It explains the superposition principle in the p-Laplace equation.

Keywords: Dominative; superposition

MSC 2010: 35J15; 35J60; 35J70

1 Introduction

The p-Laplace equation

(1.1)Δp⁢u:=div⁡(|∇⁡u|p-2⁢∇⁡u)=0

is the Euler–Lagrange equation for the variational integral ∫Ω|∇⁡u|p⁢dx. When p=2, we have Dirichlet’s integral and Laplace’s equation Δ⁢u=0. For p=∞ we define

Δ∞⁢u:=∑i,j=1n∂2⁡u∂⁡xi⁢∂⁡xj⁢∂⁡u∂⁡xi⁢∂⁡u∂⁡xj=0.

The object of our work is a superposition principle, originally discovered by Crandall and Zhang in [5].

Although the p-Laplace equation is nonlinear when p≠2, a principle of superposition for the fundamental solutions (note the sign change of the scaling constant as p crosses n)

(1.2)wn,p⁢(x):={-p-1p-n⁢|x|p-np-1,p≠n,-ln⁡|x|,p=n,-|x|,p=∞⁢ or ⁢n=1,

is valid. That is, if p≥2 and ci≥0, then the superposition

(1.3)V⁢(x):=∑i=1Nci⁢wn,p⁢(x-yi)

satisfies

Δp⁢V≤0

away from the poles y1,…,yN. Moreover, the sum is p-superharmonic in the whole of ℝn according to Definition 3 in Section 4.^[1] In [1] an explicit formula for Δp⁢V⁢(x) was derived. There it was shown that an arbitrary concave function also may be added to the sum (1.3). The result can be extended to infinite sums, and via Riemann sums one obtains that the potentials

V⁢(x)=∫ℝnρ⁢(y)⁢wn,p⁢(x-y)⁢dy,ρ≥0,

are p-superharmonic functions, provided that V⁢(x)≢∞. See [14] and [6].

It has been a little mystery why the sum (1.3) is p-superharmonic. It has not been clear what the underlying reasons are, or how far the superposition could be extended. It turns out that a class of functions called dominative p-superharmonic functions plays a central rôle in these questions. We introduce them through the sublinear operator

(1.4)𝒟p⁢u:=λ1+⋯+λn-1+(p-1)⁢λn,p≥2,

where λ1≤⋯≤λn are the eigenvalues of the Hessian matrix

ℋ⁢u:=(∂2⁡u∂⁡xi⁢∂⁡xj)i,j=1n.

The fundamental solutions are members of this class and for C2 functions we have:

Proposition.

Let u∈C2⁢(Ω). Then

𝒟p⁢u≤0⟹Δp⁢u≤0 in Ω.

In general, the inequality 𝒟p⁢u≤0 must be interpreted in the viscosity sense, see Section 4. As we shall see, the superposition principle is governed by the equation 𝒟p⁢u≤0.

Needless to say, the eigenvalues of ℋ⁢u have been much studied. In [16] the equation λ1=0 is found. The related equation λn=0 is produced by the dominative operator in the limit p→∞:

1p𝒟pu→λn=:𝒟∞u.

The supersolutions 𝒟∞⁢u≤0 are, in fact, the concave functions. We also mention the papers [11] and [18] where symmetric functions of the eigenvalues are investigated. So far as we know, the dominative p-Laplace equation is new for p≠∞.

Our main results are Theorem 1 and Theorem 2 below. Theorem 1 gives sufficient and necessary conditions for a sum to be p-superharmonic. In short: a generic sum is p-superharmonic if and only if its terms are dominative p-superharmonic functions. Furthermore, it will be demonstrated that one cannot expect a sum of p-harmonic functions (like (1.3)) to be p-superharmonic unless the functions involved have a high degree of symmetry (Definition 1).

In Theorem 2 we extend the superposition principle for the fundamental solutions to arbitrary radial p-superharmonic functions. Its proof is obtained by showing that important properties of the dominative p-Laplace operator in the smooth case, also hold in the viscosity sense.

Throughout the paper, we restrict ourselves to the case 2≤p≤∞ and n≥2. An open subset of ℝn is denoted by Ω.

Theorem 1.

Let 2≤p≤∞. The following conditions hold pointwise in Rn.

Let u1,…,uN be dominative p-superharmonic C2 functions. Then
Δp⁢[∑i=1Nui]≤0.
Let u be C2. Then the following claims are equivalent.
1. For every linear function l⁢(x)=aT⁢x,a∈ℝn,
  Δp⁢[u+l]≤0.
2. For all constants c≥0 and every translation T⁢(x)=x-x0,
  Δp⁢[u+c⁢wn,p∘T]≤0.
3. For every isometry T:ℝn→ℝn,
  Δp⁢[u+u∘T]≤0.
4. u is a dominative p-superharmonic function.
If u, in addition, is p-harmonic and 2<p<∞, then
1. u is locally a cylindrical fundamental solution (see Definition 1).

Theorem 2.

Let 2≤p≤∞ and let u1,…,uN be radial p-superharmonic functions in Rn. Then the sum

∑i=1Nui⁢(x-yi)+K⁢(x),yi∈ℝn,

is p-superharmonic in Rn for any concave function K.

To keep things simple, we have not extended Theorem 2 to cover cylindricalp-superharmonic functions.

A function f in ℝn is radial if there exists a one-variable function F so that f⁢(x)=F⁢(|x|). As usual,

|x|:=x12+⋯+xn2

denotes the Euclidean norm of x. An equivalent definition is the symmetry condition f⁢(Q⁢x)=f⁢(x) for every n×n orthogonal matrix Q. Radial functions have concentric spherical level-sets, and so do the translated ones f⁢(x-x0). We generalize this notion to functions having concentric cylindrical level-sets:

Definition 1.

A function f in ℝn is cylindrical (or k-cylindrical) if there exists a one-variable function F, an integer 1≤k≤n, and an n×k matrix Q with orthonormal columns, i.e. QT⁢Q=Ik, so that

f⁢(x)=F⁢(|QT⁢(x-x0)|)

for some x0∈ℝn.

We say that a function u in ℝn is a cylindrical fundamental solution (to the p-Laplace equation) if u is in the form

u⁢(x)=C1⁢wk,p⁢(QT⁢(x-x0))+C2,C1≥0,

for some k, Q and x0 as above, and where wk,p is given by (1.2).

Notice that a 1-cylindrical fundamental solution

u⁢(x)=-C1⁢|qT⁢(x-x0)|+C2=aT⁢x+b

is affine in the regions where it is differentiable, while an n-cylindrical fundamental solution

u⁢(x)=C1⁢wn,p⁢(QT⁢(x-x0))+C2=C1⁢wn,p⁢(x-x0)+C2

is a translated radial function.

A calculation shows (Proposition 1, part b) that the cylindrical fundamental solutions solve the p-Laplace equation, except on the (n-k)-dimensional affine space {x:QT⁢(x-x0)=0}, where the functions become singular. When p<∞, a Dirac delta is produced. For example, setting Q=(𝐞1,…,𝐞k), x0=0 and splitting x=(y,z)∈ℝn in y∈ℝk, z∈ℝn-k yields

∫ℝnΔp⁢wk,p⁢(QT⁢x)⁢ϕ⁢(x)⁢dx=∫ℝn-k∫ℝkΔp⁢wk,p⁢(y)⁢ϕ⁢(y,z)⁢dy⁢dz

=-C⁢∫ℝn-kϕ⁢(0,z)⁢dz,C=C⁢(k,p)>0,

for every ϕ∈C0∞⁢(ℝn). Moreover, we shall see that these solutions have an essential property that is not shared by any other p-harmonic function:

Property.

The gradient of a cylindrical fundamental solution is an eigenvector corresponding to the largest eigenvalue of the Hessian matrix.

2 The dominative p-Laplace operator

2.1 Preliminary basics and notation

For a function u∈C2⁢(Ω), we denote by ℋ⁢u=ℋ⁢u⁢(x) the Hessian matrix of u at x. We list some elementary and useful facts about this matrix.

ℋ⁢u is a symmetric n×n matrix: ℋ⁢uT=ℋ⁢u.
ℋ⁢u has n real eigenvalues, which we label in increasing order:
λ1≤…≤λn.
The largest eigenvalue, λn, has special importance and is denoted by λu to indicate its origin. Sometimes we are inconsistent with the notation and write λX for the largest eigenvalue of a symmetric matrix X.
The eigenvectors, ξ1,…,ξn, of ℋ⁢u can be chosen to be orthonormal: ξiT⁢ξj=δi⁢j. A unit eigenvector corresponding to the largest eigenvalue λu is labeled ξu.
tr⁡ℋ⁢u=ux1⁢x1+⋯+uxn⁢xn=λ1+⋯+λn=Δ⁢u. In general
Δ⁢u=∑i=1nziT⁢ℋ⁢u⁢zi
for every orthonormal set {z1,…,zn}⊆ℝn.
We have for any vector z∈ℝn,
λ1⁢|z|2≤zT⁢ℋ⁢u⁢z≤λu⁢|z|2,
and
λu=max|z|=1⁡zT⁢ℋ⁢u⁢z.
Conversely, if z∈ℝn, |z|=1, satisfies
zT⁢ℋ⁢u⁢z=λu,
then
ℋ⁢u⁢z=λu⁢z.

We adapt the convention that vectors/points in space are column vectors, except gradients which are to be read as row vectors.

2.2 Definition and fundamental properties

Definition 2.

We define the Dominative^[2]p-Laplace operator, 𝒟p, as

𝒟p⁢u:={(p-2)⁢λu+Δ⁢u,when ⁢2≤p<∞,λu,when ⁢p=∞.

A C2 function u is dominative p-superharmonic if 𝒟p⁢u≤0 at each point in its domain.

The expression (1.4) is, of course, an alternative representation when p<∞. Observe that 𝒟2=Δ2=Δ.

Remark 1.

The operator can be written as

𝒟p⁢u=max|ξ|=1⁡tr⁡Ap⁢(ξ)⁢ℋ⁢u,Ap⁢(ξ):=(p-2)⁢ξ⁢ξT+I,

and is, in this form, sublinear and degenerate elliptic all the way down to p=1. For p≤2 it reads

𝒟p⁢u=(p-2)⁢λ1+Δ⁢u.

Nevertheless, to avoid complicating the exposition (e.g., Proposition 1 (3) will not hold if p<2), we shall let p≥2 throughout the paper.

In low dimensions it is possible to express 𝒟p⁢u in terms of the second-order partial derivatives uxi⁢xj. In ℝ2 it can be calculated to be

𝒟p⁢u=p2⁢(ux⁢x+uy⁢y)+p-22⁢(ux⁢x-uy⁢y)2+4⁢ux⁢y2,

𝒟∞⁢u=12⁢(ux⁢x+uy⁢y)+12⁢(ux⁢x-uy⁢y)2+4⁢ux⁢y2.

We clearly see the nonlinearity introduced when p>2.

The motivation behind Definition 2 came from the following observations: By carrying out the differentiation in (1.1), we arrive at the identity

(2.1)Δp⁢u=|∇⁡u|p-2⁢((p-2)⁢∇⁡u⁢ℋ⁢u⁢∇⁡uT|∇⁡u|2+Δ⁢u).

The normalized variant of the ∞-Laplacian appearing in (2.1) satisfies

(2.2)Δ∞N⁢u:=Δ∞⁢u|∇⁡u|2=∇⁡u⁢ℋ⁢u⁢∇⁡uT|∇⁡u|2≤λu=𝒟∞⁢u.

Thus the normalized p-Laplacian also satisfies

(2.3)ΔpN⁢u:=Δp⁢u|∇⁡u|p-2=(p-2)⁢Δ∞N⁢u+Δ⁢u≤(p-2)⁢λu+Δ⁢u=𝒟p⁢u

when 0≤p-2<∞. When u is a fundamental solution, we have equality in (2.2) and (2.3). Since 𝒟p is sublinear and invariant under translations, a very simple proof of the superposition principle for the fundamental solutions (1.3) is produced. However, the above calculations are, for the moment, not valid at the poles or at critical points.

Proposition 1 (Fundamental properties of Dp. Smooth case).

Let u,v∈C2⁢(Ω). Then the following holds pointwise for 2≤p≤∞.

Domination: We have
Δp⁢u≤{|∇⁡u|p-2⁢𝒟p⁢u,2≤p<∞,|∇⁡u|2⁢𝒟∞⁢u,p=∞.
Sublinearity: We have
1. 𝒟p⁢[u+v]≤𝒟p⁢u+𝒟p⁢v, and
2. 𝒟p⁢[α⁢u]=α⁢𝒟p⁢u, α≥0.
Cylindrical Equivalence: Let 1≤k≤n and let u be k-cylindrical.
1. Assume that the corresponding one-variable function U=U⁢(r) satisfies U′r≤U′′. If k<n, we also require that U′′≥0. Then
  (2.4)Δp⁢u={|∇⁡u|p-2⁢𝒟p⁢u,2≤p<∞,|∇⁡u|2⁢𝒟∞⁢u,p=∞.
2. If u is a k-cylindrical fundamental solution, then
  𝒟p⁢u=0=Δp⁢u.
3. If 2<p<∞, k≥2, and 𝒟p⁢u=0=Δp⁢u, then u is a k-cylindrical fundamental solution.
Nesting Property: The following hold:
1. If 𝒟p⁢u≤0, then 𝒟q⁢u≤0 for every 2≤q≤p.
2. If 𝒟p⁢u≥0, then 𝒟q⁢u≥0 for every p≤q≤∞.
Invariance: We have
𝒟p⁢[u∘T]=(𝒟p⁢u)∘T
in T-1⁢(Ω) for all isometries T:ℝn→ℝn.

Proof.

We prove (1)–(5) separately.

(1) Domination: Since

Δ∞⁢u=∇⁡u⁢ℋ⁢u⁢∇⁡uT≤|∇⁡u|2⁢λu=|∇⁡u|2⁢𝒟∞⁢u,

we also get, when ∇⁡u≠0,

Δp⁢u=|∇⁡u|p-2⁢((p-2)⁢Δ∞N⁢u+Δ⁢u)

≤|∇⁡u|p-2⁢((p-2)⁢λu+Δ⁢u)

=|∇⁡u|p-2⁢𝒟p⁢u

for 0≤p-2<∞. If ∇⁡u=0 or p=2, the claim is trivial: The p-Laplacian is zero at critical points when p>2.

(2) Sublinearity: Since

𝒟∞⁢[u+v]=λu+v

=ξu+vT⁢ℋ⁢[u+v]⁢ξu+v

=ξu+vT⁢(ℋ⁢u+ℋ⁢v)⁢ξu+v

=ξu+vT⁢ℋ⁢u⁢ξu+v+ξu+vT⁢ℋ⁢v⁢ξu+v

≤λu+λv

=𝒟∞⁢u+𝒟∞⁢v,

we also get

𝒟p⁢[u+v]=(p-2)⁢λu+v+Δ⁢[u+v]

≤(p-2)⁢(λu+λv)+Δ⁢u+Δ⁢v

=𝒟p⁢u+𝒟p⁢v

for 0≤p-2<∞. Also, if α≥0 and λu is the largest eigenvalue of ℋ⁢u, then α⁢λu is the largest eigenvalue of ℋ⁢[α⁢u]=α⁢ℋ⁢u. Thus λα⁢u=α⁢λu. This means that

𝒟∞⁢[α⁢u]=λα⁢u=α⁢λu=α⁢𝒟∞⁢u

and

𝒟p⁢[α⁢u]=(p-2)⁢λα⁢u+Δ⁢[α⁢u]=α⁢𝒟p⁢u

for p<∞.

(3) Cylindrical Equivalence: (a) Let 1≤k≤n, x0∈ℝn, let Q be an n×k matrix with QT⁢Q=Ik, and let

u⁢(x)=U⁢(|QT⁢(x-x0)|).

Write y:Ω⊆ℝn→ℝk,

y⁢(x):=QT⁢(x-x0).

Assume first that u is C2 at a point x1∈Ω, where y⁢(x1)=0. Then, for a number h and vector q, we have y⁢(x1+h⁢q)=h⁢QT⁢q and

±∇⁡u⁢(x1)⁢q=limh→0⁡u⁢(x1±h⁢q)-u⁢(x1)h

=limh→0⁡U⁢(|h⁢QT⁢q|)-U⁢(0)h,

which is independent of the sign. Therefore, ∇⁡u⁢(x1)=0 and the equality in (2.4) is trivial.

The Jacobian matrix of y is D⁢y=QT, and

∇⁡|y|=yT|y|⁢D⁢y=y^T⁢QT,y^:=y|y|.

Moreover,

∇⁡1|y|=-yT|y|3⁢D⁢y

D⁢y^=D⁢y|y|+y⁢∇⁡1|y|=1|y|⁢(D⁢y-y⁢yT|y|2⁢D⁢y)=1|y|⁢(QT-y^⁢y^T⁢QT).

Now, u⁢(x)=U⁢(|y⁢(x)|) and

∇⁡u=U′⁢∇⁡|y|=U′⁢y^T⁢QT.

The Hessian matrix is

ℋ⁢u=D⁢[∇⁡uT]

=D⁢[U′⁢Q⁢y^]

=Q⁢y^⁢U′′⁢∇⁡|y|+U′⁢Q⁢D⁢y^

=U′′⁢Q⁢y^⁢y^T⁢QT+U′|y|⁢Q⁢(QT-y^⁢y^T⁢QT)

=Q⁢{U′′⁢y^⁢y^T+U′|y|⁢(Ik-y^⁢y^T)}⁢QT

with trace

Δ⁢u=U′′+(k-1)⁢U′|y|.

There are n-k perpendicular constant eigenvectors in the null-space of QT with zero eigenvalues. The (transposed) gradient is in the column-space of Q and is an eigenvector:

ℋ⁢u⁢∇⁡uT=Q⁢{U′′⁢y^⁢y^T+U′|y|⁢(Ik-y^⁢y^T)}⁢QT⁢(U′⁢Q⁢y^)

=U′⁢Q⁢{U′′⁢y^⁢y^T+U′|y|⁢(Ik-y^⁢y^T)}⁢y^

=U′⁢Q⁢{U′′⁢y^+0}

=U′′⁢∇⁡uT.

Finally, there are k-1 eigenvectors ξ=ξ⁢(x)∈ℝn in the column-space of Q that are perpendicular to ∇⁡u, i.e. y^T⁢QT⁢ξ=0 and ξ=Q⁢ξ~ for some ξ~∈ℝk:

ℋ⁢u⁢ξ=Q⁢{U′′⁢y^⁢y^T+U′|y|⁢(Ik-y^⁢y^T)}⁢QT⁢ξ

=Q⁢{0+U′|y|⁢(QT⁢ξ-0)}

=U′|y|⁢Q⁢QT⁢ξ

=U′|y|⁢Q⁢QT⁢Q⁢ξ~=U′|y|⁢Q⁢ξ~=U′|y|⁢ξ.

Thus the n eigenvalues of ℋ⁢u are U′|y| with multiplicity k-1, 0 with multiplicity n-k and U′′ with multiplicity 1.

By the assumption U′r≤U′′ and U′′≥0, if k<n, it is clear that the largest eigenvalue is λu=U′′ and it follows that

Δ∞⁢u=∇⁡u⁢ℋ⁢u⁢∇⁡u=|∇⁡u|2⁢U′′=|∇⁡u|2⁢𝒟∞⁢u

and

Δp⁢u=|∇⁡u|p-2⁢((p-2)⁢∇⁡u⁢ℋ⁢u⁢∇⁡uT|∇⁡u|2+Δ⁢u),∇⁡u≠0,

=|∇⁡u|p-2⁢((p-2)⁢λu+Δ⁢u)

=|∇⁡u|p-2⁢𝒟p⁢u.

Again, the equality is trivial if ∇⁡u=0.

(b) Now assume that u⁢(x)=U⁢(|y|) is a C2k-cylindrical fundamental solution:

U⁢(r)={-C1⁢p-1p-k⁢rp-kp-1+C2,p≠k,-C1⁢ln⁡r+C2,p=k,-C1⁢r+C2,p=∞⁢ or ⁢k=1,

with C1≥0, C2∈ℝ. Then

U′⁢(r)={-C1⁢r1-kp-1,2≤p<∞,-C1,p=∞,

and

U′′⁢(r)={-C1⁢1-kp-1⁢r2-p-kp-1,2≤p<∞,0,p=∞.

We see that U′′≥0≥U′r in every case, so 𝒟∞⁢u=λu=U′′=0 if p=∞, and

Δ∞⁢u=|∇⁡u|2⁢𝒟∞⁢u=0.

Also,

𝒟p⁢u=(p-2)⁢U′′+U′′+(k-1)⁢U′|y|

=(p-1)⁢U′′+(k-1)⁢U′|y|

=-C1⁢((1-k)⁢|y|2-p-kp-1+(k-1)⁢|y|1-kp-1-1)=0

and

Δp⁢u=|∇⁡u|p-2⁢𝒟p⁢u=0

when p<∞.

(2.5)0=ΔpN⁢u=(p-1)⁢U′′+(k-1)⁢U′r

with general solution U satisfying

U′⁢(r)=C⁢r-k-1p-1,C∈ℝ.

By the definition of the cylindrical fundamental solutions, we only need to show that C≤0. The equation 𝒟p⁢u=0 gives

(2.6)0=(p-2)⁢max⁡{U′′,U′r}+U′′+(k-1)⁢U′r,k=n,

(2.7)0=(p-2)⁢max⁡{U′′,U′r,0}+U′′+(k-1)⁢U′r,2≤k<n.

Subtract (2.5) from (2.6) or (2.7) and divide by p-2 to obtain the condition

0=max⁡{U′′,U′r}-U′′.

That is,

-C⁢k-1p-1⁢r-k+p-2p-1=U′′≥U′r=C⁢r-k+p-2p-1

which is true only if C≤0.

(4) Nesting Property: Let 2≤q≤p≤∞ and assume 𝒟p⁢u≤0. For each x∈Ω we consider two cases. If 𝒟∞⁢u=λu<0, then every eigenvalue is negative and

𝒟q⁢u=λ1+⋯+λn-1+(q-1)⁢λu<0.

If 𝒟∞⁢u=λu≥0, then also

𝒟q⁢u=(q-2)⁢λu+Δ⁢u≤(p-2)⁢λu+Δ⁢u=𝒟p⁢u≤0.

Now let 2≤p≤q≤∞ and assume

𝒟p⁢u=(p-1)⁢λu+λ1+⋯+λn-1≥0.

Then, obviously, λu≥0 and

𝒟q⁢u=(q-2)⁢λu+Δ⁢u≥(p-2)⁢λu+Δ⁢u=𝒟p⁢u≥0.

(5) Invariance: An isometry in ℝn is on the form T⁢(x)=QT⁢(x-x0) for some x0∈ℝn and some constant orthogonal n×n matrix Q: QT⁢Q=I. Define v⁢(x):=u⁢(T⁢(x)) on T-1⁢(Ω), i.e. T⁢(x)∈Ω. Write y:=T⁢(x). Then ∇⁡v⁢(x)=∇⁡u⁢(y)⁢QT and

ℋ⁢v⁢(x)=Q⁢ℋ⁢u⁢(y)⁢QT.

So λv⁢(x)=λu⁢(y), proving the case p=∞, since

λv⁢(x)=max|z|=1⁡zT⁢ℋ⁢v⁢(x)⁢z=max|z|=1⁡zT⁢Q⁢ℋ⁢u⁢(y)⁢QT⁢z=max|z|=1⁡zT⁢ℋ⁢u⁢(y)⁢z=λu⁢(y).

Also

Δ⁢v⁢(x)=tr⁡ℋ⁢v⁢(x)=tr⁡Q⁢ℋ⁢u⁢(y)⁢QT=tr⁡ℋ⁢u⁢(y)=Δ⁢u⁢(y).

Therefore

(𝒟p⁢[u∘T])⁢(x)=(𝒟p⁢v)⁢(x)

=(p-2)⁢λv⁢(x)+Δ⁢v⁢(x)

=(p-2)⁢λu⁢(y)+Δ⁢u⁢(y)

=(𝒟p⁢u)⁢(T⁢(x)).∎

3 Proof of Theorem 1

That dominative p-superharmonicity of the terms is a sufficient condition for the sum to be p-superharmonic is now an immediate consequence of Proposition 1:

Proof of Theorem 1 (i).

Let 2≤p≤∞ and let u1,…,uN be dominative p-superharmonic C2 functions. That is, 𝒟p⁢ui≤0. Write

V⁢(x):=∑i=1Nui⁢(x)

to denote the sum. Then V is C2 and

Δp⁢V≤|∇⁡V|p-2⁢𝒟p⁢[∑i=1Nui](by Proposition 1 (1))

≤|∇⁡V|p-2⁢∑i=1N𝒟p⁢ui(by Proposition 1 (2))

≤0.

The calculations are the same when p=∞. ∎

Notice that a sum of fundamental solutions, V⁢(x)=∑i=1Nci⁢wn,p⁢(x-yi) in a domain not containing the singularities, is just a special case by (3) and (5) of Proposition 1.

We restate and prove the first part of Theorem 1 (ii).

Proposition 2.

Let 2≤p≤∞ and let u∈C2⁢(Ω). Then the following properties are equivalent.

u⁢(x)+aT⁢x is p-superharmonic for every linear function aT⁢x.
u⁢(x)+c⁢wn,p⁢(x-y) is p-superharmonic in Ω∖{y} for every c≥0 and every y∈ℝn.
u+u∘T is p-superharmonic in Ω∩T-1⁢(Ω) for every isometry T:Ω→ℝn.
u is a dominative p-superharmonic function.

Proof.

We show

The upward implications are immediate from the fundamental properties of 𝒟p. As for the downward implications, assume that u is not dominative p-superharmonic. Then there is a point x0∈Ω so that 𝒟p⁢u⁢(x0)>0.

(a) ⇒ (d) By the contrapositive assumption 𝒟p⁢(x0)>0, the implication is proved if we can find a constant a∈ℝn so that Δp⁢[u+aT⁢x]>0 at x0. To this end, let ξu=ξu⁢(x0) be a unit eigenvector of ℋ⁢u⁢(x0) corresponding to the largest eigenvalue λu and let v⁢(x):=aT⁢x be the linear function with

a:=ξu-∇⁡uT⁢(x0).

Then, at x0,

∇⁡u+∇⁡v=∇⁡u+aT=ξuT,

so |∇⁡(u+v)|=1 and

Δ∞⁢[u+v]=Δ∞N⁢[u+v]

=(∇⁡u+∇⁡v)⁢(ℋ⁢u+ℋ⁢v)⁢(∇⁡u+∇⁡v)T

=ξuT⁢(ℋ⁢u+0)⁢ξu

=λu

=𝒟∞⁢u⁢(x0)>0

if p=∞. Also, if p<∞,

Δp⁢[u+v]=ΔpN⁢[u+v]

=(p-2)⁢Δ∞N⁢[u+v]+Δ⁢[u+v]

=(p-2)⁢λu+Δ⁢u+0

=𝒟p⁢u⁢(x0)>0.

(b) ⇒ (d) The implication is proved if we can find a y∈ℝn and a c≥0 so that

Δp⁢[u⁢(x)+c⁢wn,p⁢(x-y)]>0

at x=x0. To this end, let ξu=ξu⁢(x0) be a unit eigenvector of ℋ⁢u⁢(x0) corresponding to the largest eigenvalue λu, and denote by q:=∇⁡uT⁢(x0) the gradient of u at x0. The idea is to consider a fundamental solution with centre far away from x0 in the proper direction, and then scale it in order to achieve a convenient cancellation in the sum of the gradients.

Introduce a (large) parameter s and let the centre of the scaled fundamental solution

fs⁢(x):=cs⁢wn,p⁢(x-ys)

be at ys:=x0-q+s⁢ξu. Let zs be the point

zs:=x0-ys=q-s⁢ξu.

Then

∇⁡fs⁢(x0)=cs⁢∇⁡wn,p⁢(zs)=cs⁢Wn,p′⁢(|zs|)⁢zsT|zs|,Wn,p⁢(|x|):=wn,p⁢(x),

which equals -zsT=-(q-s⁢ξu)T if we choose the scale cs to be

cs:=-|zs|Wn,p′⁢(|zs|)={|q-s⁢ξu|n+p-2p-1,2≤p<∞,|q-s⁢ξu|,p=∞.

We may read the fraction n+p-2p-1 as 1 if p=∞.

We now get, at x0, ∇⁡u+∇⁡fs=qT-(q-s⁢ξu)T=s⁢ξuT and

ΔpN⁢[u+fs]=(p-2)⁢(∇⁡u+∇⁡fs)⁢(ℋ⁢u+ℋ⁢fs)⁢(∇⁡u+∇⁡fs)T|∇⁡u+∇⁡fs|2+Δ⁢u+Δ⁢fs

=(p-2)⁢ξuT⁢(ℋ⁢u+ℋ⁢fs)⁢ξu+Δ⁢u+Δ⁢fs

=𝒟p⁢u+(p-2)⁢ξuT⁢ℋ⁢fs⁢ξu+Δ⁢fs

if p<∞ and

Δ∞N⁢[u+fs]=𝒟∞⁢u+ξuT⁢ℋ⁢fs⁢ξu

if p=∞. Since 𝒟p⁢u⁢(x0)>0 and does not depend on s, we finish the proof by making the remaining term(s) arbitrarily close to zero.

The Hessian matrix of the fundamental solution is

ℋ⁢wn,p⁢(x)=-Wn,p′⁢(|x|)|x|⁢(n+p-2p-1⁢x⁢xT|x|2-I),n+∞-2∞-1:=1,

so at zs we get

ℋ⁢fs⁢(x0)=cs⁢ℋ⁢wn,p⁢(zs)=-cs⁢Wn,p′⁢(|zs|)|zs|⁢(n+p-2p-1⁢zs⁢zsT|zs|2-I)=n+p-2p-1⁢zs⁢zsT|zs|2-I,

Δ⁢fs⁢(x0)=n+p-2p-1-n.

Thus, when p<∞,

(p-2)⁢ξuT⁢ℋ⁢fs⁢ξu+Δ⁢fs=(p-2)⁢(n+p-2p-1⁢(zsT⁢ξu)2|zs|2-1)+n+p-2p-1-n

=(p-2)⁢n+p-2p-1⁢((zsT⁢ξu)2|zs|2-1)→0

as s→∞ since

lims→∞⁡(zsT⁢ξu)2|zs|2=lims→∞⁡(qT⁢ξu-s)2|q-s⁢ξu|2=1.

Likewise, when p=∞,

ξuT⁢ℋ⁢fs⁢ξu=(zsT⁢ξu)2|zs|2-1→0.

To summarize: If 𝒟p⁢u⁢(x0)>0 and if s is large enough, then for

zs:=∇⁡uT⁢(x0)-s⁢ξu⁢(x0),fs⁢(x):=-|zs|Wn,p′⁢(|zs|)⁢wn,p⁢(x-x0+zs)

we have^[3]

Δp⁢[u+fs]⁢(x0)=sαp⁢ΔpN⁢[u+fs]⁢(x0)>0

and the sum u+fs is not p-superharmonic.

(c) ⇒ (d) Without loss of generality we may assume x0 to be the origin. i.e. 0∈Ω and 𝒟p⁢u⁢(0)>0. We shall prove the implication by finding an isometry T and a point y0∈Ω, equal or close to 0, so that T⁢(y0)∈Ω and Δp⁢[u+u∘T]>0 at y0. To this end, let y∈ℝn, |y|=1 be a fixed direction in space defining the line

ℓ:={α⁢y:α∈ℝ}.

The projection onto ℓ is given by the 1-rank matrix

P:=y⁢yT.

We have, as for every projection,

P⁢x∈ℓ and P⁢P=P.

The reflection about ℓ is now given by R⁢x:=P⁢x-(x-P⁢x). That is,

R=2⁢P-I.

A reflection satisfies

R|ℓ=id and R⁢R=I.

After carefully choosing y, T⁢(x):=R⁢x will be our isometry.

Define the superposition

V⁢(x):=u⁢(x)+u⁢(R⁢x)2.

The main idea of the proof is that, on ℓ, ∇⁡V will be pointing in the y-direction: The chain rule gives ∇⁡V⁢(x)=12⁢(∇⁡u⁢(x)+∇⁡u⁢(R⁢x)⁢R) and ℋ⁢V⁢(x)=12⁢(ℋ⁢u⁢(x)+R⁢ℋ⁢u⁢(R⁢x)⁢R) and Δ⁢V⁢(x)=12⁢(Δ⁢u⁢(x)+Δ⁢u⁢(R⁢x)). For x∈ℓ we have R⁢x=x, and

∇⁡V=∇⁡u⁢I+R2=∇⁡u⁢P∈ℓ,

ℋ⁢V=ℋ⁢u+R⁢ℋ⁢u⁢R2,

Δ⁢V=Δ⁢u.

This gives, when ∇⁡V=∇⁡u⁢P≠0,

ΔpN⁢V|ℓ=(p-2)⁢∇⁡V⁢ℋ⁢V⁢∇⁡VT|∇⁡V|2+Δ⁢V

=(p-2)⁢1|∇⁡u⁢P|2⁢∇⁡u⁢P⁢ℋ⁢u+R⁢ℋ⁢u⁢R2⁢P⁢∇⁡uT+Δ⁢u

=(p-2)⁢∇⁡u⁢P⁢ℋ⁢u⁢P⁢∇⁡uT|∇⁡u⁢P|2+Δ⁢u

=(p-2)⁢yT⁢ℋ⁢u⁢y+Δ⁢u

since R⁢P=P and 0≠P⁢∇⁡uT is parallel to y. Similarly

Δ∞N⁢V|ℓ=yT⁢ℋ⁢u⁢y.

Now choose y:=ξu, where ξu=ξu⁢(0) is a unit eigenvector of ℋ⁢u⁢(0) corresponding to the largest eigenvalue λu. The isometry T is then

T⁢(x):=R⁢x=(2⁢P-I)⁢x=(2⁢ξu⁢ξuT-I)⁢x.

Since 0∈ℓ, and unless ∇⁡u⁢(0)⁢ξu=0, it follows that

12⁢ΔpN⁢[u+u∘T]|x=0=ΔpN⁢V⁢(0)=𝒟p⁢u⁢(0)>0

and Δp⁢[u+u∘T]=2αp+1⁢|∇⁡u⁢P|αp⁢ΔpN⁢[u+u∘T]>0 at x=0 since ∇⁡u⁢(0)⁢P=∇⁡u⁢(0)⁢ξu⁢ξuT≠0.

If ∇⁡u⁢(0)⁢ξu=0, we complete the proof with a continuity argument: Since λu⁢(0)>0,^[4]𝒟p⁢u⁢(0)>0 and Ω is open, and since the Hessian is continuous, there must be a common ϵ>0 so that

ξuT⁢ℋ⁢u⁢(t⁢ξu)⁢ξu>0,
(p-2)⁢ξuT⁢ℋ⁢u⁢(t⁢ξu)⁢ξu+Δ⁢u⁢(t⁢ξu)>0 if p<∞,
T⁢(t⁢ξu)=t⁢ξu∈Ω for all t∈[0,ϵ].

A Taylor expansion of the gradient about 0 in the ξu-direction then gives

∇⁡u⁢(ϵ⁢ξu)=∇⁡u⁢(0)+ϵ⁢ξuT⁢ℋ⁢u⁢(t0⁢ξu)

for some t0∈[0,ϵ]. So

∇⁡u⁢(ϵ⁢ξu)⁢ξu=0+ϵ⁢ξuT⁢ℋ⁢u⁢(t0⁢ξu)⁢ξu>0,

and again, since ϵ⁢ξu∈ℓ,

12⁢Δp⁢[u+u∘T]|x=ϵ⁢ξu=|2⁢∇⁡u⁢(ϵ⁢ξu)⁢P|p-2⁢((p-2)⁢yT⁢ℋ⁢u⁢(ϵ⁢ξu)⁢y+Δ⁢u⁢(ϵ⁢ξu))

=(2⁢∇⁡u⁢(ϵ⁢ξu)⁢ξu)p-2⁢((p-2)⁢ξuT⁢ℋ⁢u⁢(ϵ⁢ξu)⁢ξu+Δ⁢u⁢(ϵ⁢ξu))>0

if p<∞ and

12⁢Δ∞⁢[u+u∘T]|x=ϵ⁢ξu=(2⁢∇⁡u⁢(ϵ⁢ξu)⁢ξu)2⁢ξuT⁢ℋ⁢u⁢(ϵ⁢ξu)⁢ξu>0

if p=∞. Thus the sum u+u∘T is not p-superharmonic in Ω∩T-1⁢(Ω). ∎

We finish the proof of Theorem 1 by showing the equivalence of (d) and (e). The nontrivial implication is (d) ⇒ (e). Namely that if 2<p<∞ and u∈C2⁢(Ω) is both p-harmonic and dominative p-superharmonic, then u is locally a cylindrical fundamental solution. Since the hypothesis and the domination implies

0=Δp⁢u≤|∇⁡u|p-2⁢𝒟p⁢u≤0,

the claim follows from Proposition 3 below. It is partially the converse of the Cylindrical Equivalence (part (3) of Proposition 1).

Proposition 3.

Let 2<p<∞ and let u∈C2⁢(Ω).^[5] If

(3.1)Δp⁢u=0=𝒟p⁢u in Ω,

then u is locally a cylindrical fundamental solution.

The proof relies on a rather deep result in differential geometry. We refer to [3], [19] and [17] for the details of the following exposition.

A nonconstant smooth function u:M→ℝ on a Riemannian manifold M is called isoparametric if there exist functions f and g so that

12⁢|∇⁡u|2=f⁢(u) and Δ⁢u=g⁢(u).

A regular level-set of an isoparametric function is called an isoparametric hypersurface. The investigation of isoparametric functions and hypersurfaces was originally motivated by the study of wave propagation.

The isoparametric hypersurfaces in the Euclidean case M⊆ℝn have been classified completely. Apparently, this was first done by Segre (see [17]) in 1938:

Theorem 3 (Segre).

A connected isoparametric hypersurface in Rn is, upon scaling and an Euclidean motion, an open part of one of the following hypersurfaces:

a hyperplane ℝn-1,
a sphere Sn-1,
a generalized cylinder Sk-1×ℝn-k, k=2,…,n-1.

Moreover, the family of cylinders u-1⁢(c) is concentric. Thus u is a function of the distance to the common “axis” of the cylinders, the axis being an (n-k)-dimensional affine subspace, k=1,…,n, in ℝn. Call this subset 𝒜k.

The axis 𝒜k is isomorphic to ℝn-k,

𝒜k≅ℝn-k≅{(000In-k)x:x∈ℝn}=:𝒜k~,

where, obviously,

dist⁡(x,𝒜k~)=x12+⋯+xk2=|(Ik⁢ 0)⁢x|,0∈ℝk×n-k.

Translating and rotating back to 𝒜k via an isometry QnT⁢(x-x0), Qnn×n orthogonal, we find that

u⁢(x)=U⁢(dist⁡(x,𝒜k))=U⁢(|(Ik⁢ 0)⁢QnT⁢(x-x0)|)=U⁢(|QkT⁢(x-x0)|),

where Qk is the n×k matrix consisting of the first k columns of Qn.

Thus an isoparametric function is cylindrical.

Proof of Proposition 3.

If u is affine, it can locally be written as a 1-cylindrical fundamental solution. If u is not an affine function, let x0∈Ω be a point with a neighbourhood Ω′⊆Ω where u has connected level-sets and where ∇⁡u≠0, ℋ⁢u≠0. By the discussion above and part (c) of Proposition 1, it is sufficient to show that u is a smooth isoparametric function.

As ∇⁡u≠0 and p>2, our equation (3.1)

|∇⁡u|p-2⁢((p-2)⁢∇⁡u⁢ℋ⁢u⁢∇⁡uT|∇⁡u|2+Δ⁢u)=0=(p-2)⁢λu+Δ⁢u

implies ∇⁡u⁢ℋ⁢u⁢∇⁡uT=λu⁢|∇⁡u|2 and the gradient is therefore an eigenvector of the Hessian:

ℋ⁢u⁢∇⁡uT=λu⁢∇⁡uT.

Let 𝐜 be a differentiable curve in a level-set of u. Then 0=dd⁢t⁢u⁢(𝐜⁢(t))=∇⁡u⁢d⁢𝐜d⁢t and

dd⁢t⁢12⁢|∇⁡u⁢(𝐜⁢(t))|2=∇⁡u⁢ℋ⁢u⁢d⁢𝐜d⁢t=λu⁢∇⁡u⁢d⁢𝐜d⁢t=0.

Thus the length of the gradient is constant on the level-sets and can be written as a function only of u. Say,

(3.2)12⁢|∇⁡u⁢(x)|2=f⁢(u⁢(x))>0.

We need to show that f is differentiable, because if so, differentiation of (3.2) yields

(3.3)∇⁡u⁢ℋ⁢u=f′⁢(u)⁢∇⁡u

and λu=f′⁢(u). Using (3.1) once more, we then find that also Δ⁢u is a function of u:

(3.4)Δu=-(p-2)λu=-(p-2)f′(u)=:g(u).

Fix x∈Ω′ and let 𝐱⁢(t) now be an integral curve of the gradient field starting from x:

d⁢𝐱d⁢t⁢(t)=∇⁡uT⁢(𝐱⁢(t)),𝐱⁢(0)=x.

Then define the function h as h⁢(t):=u⁢(𝐱⁢(t)). We see that h is C2 and

h′⁢(t)=∇⁡u⁢(𝐱⁢(t))⁢∇⁡uT⁢(𝐱⁢(t))=2⁢f⁢(u⁢(𝐱⁢(t)))=2⁢f⁢(h⁢(t))>0.

Thus h is strictly monotone and

12⁢h′′⁢(0)h′⁢(0)=12⁢limt→0⁡h′⁢(t)-h′⁢(0)h⁢(t)-h⁢(0)=limt→0⁡f⁢(h⁢(t))-f⁢(h⁢(0))h⁢(t)-h⁢(0)=f′⁢(h⁢(0))=f′⁢(u⁢(x)).

This is enough to conclude that (3.3), and thus (3.4), is valid.

As for the regularity of u, observe that if F is an anti-derivative of (2⁢f)p-22 and ϕ⁢(x):=F⁢(u⁢(x)), then ϕ is C2 and

∇⁡ϕ=F′⁢(u)⁢∇⁡u=|∇⁡u|p-2⁢∇⁡u.

That is, ϕ is harmonic by (3.1). It follows that ϕ is real-analytic, and so is u since ∇⁡u=|∇⁡ϕ|-p-2p-1⁢∇⁡ϕ. ∎

This concludes the proof of Theorem 1.

It is worth noting that Proposition 3 is not true for p=2 and p=∞. The case p=2 is obvious since 𝒟2≡Δ and every harmonic function satisfies (3.1). When p=∞, a counter-example is provided by a function

u⁢(x)=dist⁡(x,∂⁡ℰ) in Ω,

where ℰ is a non-spherical solid ellipse in, say, ℝ2 and where Ω⊆ℰ is a small ball sufficiently close to the boundary of the ellipse. Then u is neither affine nor a circular cone (i.e. not a cylindrical ∞-fundamental solution). But u is smooth with |∇⁡u|=1, so ∇⁡u⁢ℋ⁢u=0 and λu=0 is the larger eigenvalue as u is concave (see [7, Lemma 1 and Lemma 2, Appendix]). It follows that

𝒟∞⁢u=λu=0=∇⁡u⁢ℋ⁢u⁢∇⁡uT=Δ∞⁢u in Ω.

4 Viscosity solutions

The equation 𝒟p⁢u=0 needs to be interpreted in the viscosity sense (v.s.). We refer to [4], [10] and [9] for the general theory of viscosity solutions. For our purpose, only the basic notions of the concept are needed.

A PDE in the form F⁢(∇⁡u,ℋ⁢u)=0 is said to be degenerate elliptic if for any two symmetric matrices X and Y such that Y-X is positive semi-definite, i.e. X≤Y, we have

F⁢(q,X)≤F⁢(q,Y)

for all q∈ℝn.^[6]

In our case

0=𝒟p⁢u=Fp⁢(ℋ⁢u),

where

Fp⁢(X):={(p-2)⁢λX+tr⁡X,p<∞,λX,p=∞.

So if zT⁢X⁢z≤zT⁢Y⁢z for all z∈ℝn, then

λX=ξXT⁢X⁢ξX≤ξXT⁢Y⁢ξX≤λY

and F∞⁢(X)≤F∞⁢(Y). Also, for any orthonormal set {z1,…,zn},

tr⁡X=∑i=1nziT⁢X⁢zi≤∑i=1nziT⁢Y⁢zi=tr⁡Y,

so Fp⁢(X)≤Fp⁢(Y) when 0≤p-2<∞. Thus the dominative p-Laplace equation 𝒟p⁢u=0 is degenerate elliptic.

It is known that the p-Laplace equation is degenerate elliptic for 2≤p≤∞.

4.1 Definitions and fundamental properties

Consider a degenerate elliptic equation

(4.1)F⁢(∇⁡u,ℋ⁢u)=0.

Definition 3.

We say that u:Ω→(-∞,∞] is a viscosity supersolution of the PDE (4.1) if the following hold:

u is lower semi-continuous (l.s.c.).
u<∞ in a dense subset of Ω.
If x0∈Ω and ϕ∈C2 touches u from below at x0, i.e.
ϕ⁢(x0)=u⁢(x0),ϕ⁢(x)≤u⁢(x) for x near x0,
we require that
F⁢(∇⁡ϕ⁢(x0),ℋ⁢ϕ⁢(x0))≤0.

The viscosity subsolutions v:Ω→[-∞,∞) are defined in a similar way: they are upper semi-continuous and the test functions touch from above. Finally, a function w:Ω→ℝ is a viscosity solution if it is both a viscosity supersolution and a viscosity subsolution. Necessarily, w∈C⁢(Ω).

We say that u is dominative p-superharmonic if u is a viscosity supersolution to the equation 𝒟p⁢w=0. We write this as 𝒟p⁢u≤0 v.s. (viscosity sense). The p-superharmonic functions were traditionally defined by the comparison principle and weak integral formulations – and not by viscosity. According to [8], however, the two concepts are equivalent and we may therefore define the p-superharmonic functions in terms of viscosity as well. The comparison principle then becomes a theorem:

Theorem 4 (Comparison Principle).

Let 2≤p≤∞. Assume that v is p-subharmonic and that u is p-superharmonic in Ω. Let D⊂⊂Ω. Then

v|∂⁡D≤u|∂⁡D⟹v≤u in D.

Before we extend the fundamental properties of the dominative operator (Proposition 1) to the setting of viscosity, we establish that a dominative ∞-superharmonic function is the same as a concave function:

Proposition 4.

Let Ω⊆Rn be open and convex. Then

𝒟∞⁢u≤0⁢ v.s. in Ω⇔u is concave in Ω.

This is [15, Proposition 4.1] or, alternatively, [16, Theorem 2.2] in disguise. Note that continuity is automatically given by either direction: It is well known that concave functions are continuous in open domains. Also, if 𝒟∞⁢u≤0 v.s., then u is ∞-superharmonic by Proposition 5 and is therefore continuous by [13, Lemma 6.7].

Proposition 5 (Fundamental properties of Dp. Viscosity sense).

The following hold for 2≤p≤∞.

Domination: If u is dominative p-superharmonic, then u is p-superharmonic.
Sublinearity: If 𝒟p⁢u≤0 and 𝒟p⁢v≤0 v.s. and u is radial, then 𝒟p⁢[u+v]≤0 v.s.
Radial Equivalence: If u is a radial p-superharmonic function, then 𝒟p⁢u≤0 v.s.
Nesting Property: The following hold:
1. If 𝒟p⁢u≤0 v.s., then 𝒟q⁢u≤0 v.s. for every 2≤q≤p. In particular, if u is locally concave, then 𝒟q⁢u≤0 v.s. for all 2≤q≤∞.
2. If 𝒟p⁢u≥0 v.s., then 𝒟q⁢u≥0 v.s. for every p≤q≤∞. In particular, if u is subharmonic, then 𝒟q⁢u≥0 v.s. for all 2≤q≤∞.
Invariance: If 𝒟p⁢u≤0 v.s. in ℝn, then 𝒟p⁢[u∘T]≤0 v.s. in ℝn for all isometries T:ℝn→ℝn.

As shown below, claims (1), (4) and (5) follow easily from the corresponding properties in the smooth case (Proposition 1). The proofs of (2) and (3) are more difficult and are postponed until the survey of Radial p-superharmonic functions in Section 5.

Proof.

As said, we prove (1), (4) and (5).

(1) Domination: Assume 𝒟p⁢u≤0 v.s. Let x0 be a point in the domain of u, and assume that ϕ is a test function of u from below at x0. Then 𝒟p⁢ϕ⁢(x0)≤0 and

Δp⁢ϕ⁢(x0)≤{|∇⁡ϕ⁢(x0)|p-2⁢𝒟p⁢ϕ⁢(x0)≤0,2≤p<∞|∇⁡ϕ⁢(x0)|2⁢𝒟∞⁢ϕ⁢(x0)≤0,p=∞,

by the smooth case Domination. Hence Δp⁢u≤0 v.s. and u is p-superharmonic.

(4) Nesting Property: Let 2≤q≤p and suppose 𝒟p⁢u≤0 v.s. Let x0 be a point in the domain of u, and assume that ϕ is a test function of u from below at x0. Then 𝒟p⁢ϕ⁢(x0)≤0 and

𝒟q⁢ϕ⁢(x0)≤0

by the smooth case Nesting Property and 𝒟q⁢u≤0 v.s. The additional claim follows from Proposition 4.

Let p≤q≤∞ and suppose 𝒟p⁢u≥0 v.s. Let x0 be a point in the domain of u, and assume that ϕ is a test function of u from above at x0. Then 𝒟p⁢ϕ⁢(x0)≥0 and

𝒟q⁢ϕ⁢(x0)≥0

by the smooth case Nesting Property and 𝒟q⁢u≥0 v.s.

(5) Invariance: Assume 𝒟p⁢u≤0 v.s. in ℝn and let T:ℝn→ℝn be an isometry. Let x0∈ℝn, and assume that ϕ is a test function for u∘T from below at x0. Then the function ϕ^:=ϕ∘T-1 is a test function for u from below at y0:=T⁢(x0), since

ϕ^⁢(y0)=ϕ∘T-1⁢(T⁢(x0))=ϕ⁢(x0)=u⁢(y0)

and, for y near y0, T-1⁢(y) is near x0 and

ϕ^⁢(y)=ϕ⁢(T-1⁢(y))≤(u∘T)⁢(T-1⁢(y))=u⁢(y).

Thus 𝒟p⁢ϕ^⁢(y0)≤0 and

𝒟p⁢ϕ⁢(x0)=(𝒟p⁢[ϕ^∘T])⁢(x0)

=(𝒟p⁢ϕ^)⁢(T⁢(x0)) (by the smooth case 𝐼𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒)

=𝒟p⁢ϕ^⁢(y0)

≤0

and 𝒟p⁢[u∘T]≤0 v.s. in ℝn. ∎

4.2 The superposition principle for radial p-superharmonic functions

Proposition 5 contains everything we need in order to show that radial p-superharmonic functions can be added and perturbed by a concave function:

Proof of Theorem 2.

Let 2≤p≤∞ and let u1,…,uN be radial p-superharmonic functions in ℝn. We must show that the sum

∑i=1Nui⁢(x-yi)+K⁢(x),yi∈ℝn,

is p-superharmonic in ℝn for any concave function K.

Observe first that if 𝒟p⁢u≤0, 𝒟p⁢v≤0 v.s., where u is radial, then 𝒟p⁢[v∘T-1]≤0 v.s. for every isometry T by the Invariance (5). By the Sublinearity (2) it then follows that 𝒟p⁢[u+v∘T-1]≤0 v.s. and

𝒟p⁢[u∘T+v]=𝒟p⁢[(u+v∘T-1)∘T]≤0⁢ v.s.,

again by (5).

From the Radial Equivalence (3), 𝒟p⁢ui≤0 v.s. for each i=1,…,N and also 𝒟p⁢K≤0 v.s. by the Nesting Property (4). Denoting the translations by Ti⁢(x):=x-yi, and adding the functions ui∘Ti one by one, starting with K, we obtain that

𝒟p⁢[∑i=1Nui∘Ti+K]≤0 v.s.

We now use the Domination (1) to conclude that the sum

∑i=1Nui∘Ti+K

is p-superharmonic in ℝn. ∎

5 Radial p-superharmonic functions

A radial p-superharmonic function u:ℝn→(-∞,∞] is of the form u⁢(x)=U⁢(|x|) for some l.s.c. one-variable function U:[0,∞)→(-∞,∞]. By comparing with constant functions (which are p-harmonic), it is clear that U must be decreasing (= non-increasing). Also, since the set {x:u⁢(x)=∞} has measure zero [12], the origin is the only possible pole of u. Therefore, u is bounded in every annulus

Aab:=B⁢(0,b)∖B⁢(0,a)¯,0<a<b<∞.

Equivalently, U is bounded on the interval (a,b).

Let

wn,p⁢(x)=Wn,p⁢(|x|),2≤p≤∞,n≥2,

denote the fundamental solution to the p-Laplace equation in ℝn, see (1.2) in the Introduction. In this section, the important properties of the fundamental solution is that any scaled version, C1⁢wn,p+C2 (C1,C2∈ℝ), is still a radial C∞p-harmonic function in ℝn∖{0}. And when C1≥0, it is also dominative p-harmonic by part (b) of Proposition 1.

Furthermore, we shall frequently use the fact that the function Wn,p:[0,∞)→(-∞,∞] is strictly decreasing. A simple calculation shows that a scaled fundamental solution is uniquely determined by its values at two different positive radii.

Given a radial p-superharmonic function u⁢(x)=U⁢(|x|) in ℝn and two numbers 0<a<b, we define ha⁢b on ℝn∖{0} as the scaled fundamental solution ha⁢b⁢(x)=Ha⁢b⁢(|x|), where

(5.1)Ha⁢b⁢(r):=Ca⁢b⁢[Wn,p⁢(r)-Wn,p⁢(b)]+U⁢(b),

(5.2)Ca⁢b:=U⁢(a)-U⁢(b)Wn,p⁢(a)-Wn,p⁢(b).

$Figure 1 We have Ha⁢b{H_{ab}}≤U{\leq U} on [a,b]{[a,b]}, while Ha⁢b{H_{ab}}≥U{\geq U} outside the interval.$

Figure 1

We have Ha⁢b≤U on [a,b], while Ha⁢b≥U outside the interval.

The point of this is that ha⁢b is p-harmonic, smooth and it satisfies

Ha⁢b⁢(a)=U⁢(a) and Ha⁢b⁢(b)=U⁢(b).

(See Figure 1.) Thus ha⁢b=u on the boundary of the annulus Aab and by the comparison principle we must have

ha⁢b≤u in ⁢Aab.

Equivalently,

Ha⁢b≤U in ⁢(a,b).

We now deduce other immediate properties of Ha⁢b and the scaling constant Ca⁢b.

Lemma 1.

Let 2≤p≤∞ and let u⁢(x)=U⁢(|x|) be a given radial p-superharmonic function in Rn. For numbers a and b with 0<a<b define the scaled fundamental solution ha⁢b⁢(x)=Ha⁢b⁢(|x|) with scaling constant Ca⁢b as in (5.1) and (5.2).

We have the opposite inequality outside the annulus Aab:
Ha⁢b≥U in ⁢(0,a]∪[b,∞).
0≤Ca⁢b≤Cb⁢c<∞ whenever 0<a<b<c.
The mappings a↦Ca⁢b and c↦Cb⁢c are increasing.
The one-sided limits
Cb-:=lima→b-⁡Ca⁢b,Cb+:=limc→b+⁡Cb⁢c
exist and
0≤Cb-≤Cb+<∞.

Observe that the existence of the limits (4) implies that U⁢(r) has one-sided derivatives at every r≠0. For example,

U⁢(a)-U⁢(b)a-b=U⁢(a)-U⁢(b)Wn,p⁢(a)-Wn,p⁢(b)⁢Wn,p⁢(a)-Wn,p⁢(b)a-b

which goes to Cb-⁢Wn,p′⁢(b) as a→b-.

Proof.

(1) Suppose for the sake of contradiction that there is a number d>b so that Ha⁢b⁢(d)<U⁢(d). See Figure 2. The function ha⁢d⁢(x)=Ha⁢d⁢(|x|) satisfies

Ha⁢d⁢(a)=U⁢(a)=Ha⁢b⁢(a)(by definition),

Ha⁢d⁢(d)=U⁢(d)>Ha⁢b⁢(d)(by assumption),

Ha⁢d⁢(b)≤U⁢(b)=Ha⁢b⁢(b)(by the comparison principle).

By the Intermediate Value Theorem, there is an c∈[b,d) so that Ha⁢d⁢(c)=Ha⁢b⁢(c). Since also Ha⁢d⁢(a)=Ha⁢b⁢(a), the two functions are identical. This is the contradiction to the assumption Ha⁢d⁢(d)>Ha⁢b⁢(d). The proof for 0<r<a is symmetric.

$Figure 2 Impossible situation. If there is a d with d>b{d>b} such that Ha⁢b⁢(d){H_{ab}(d)}<U⁢(d){<U(d)}, then there exists a c∈[b,d){c\in[b,d)} so that Ha⁢d⁢(c)=Ha⁢b⁢(c){H_{ad}(c)=H_{ab}(c)}. Thus Ha⁢d≡Ha⁢b{H_{ad}}\equiv{H_{ab}}.$

Figure 2

Impossible situation. If there is a d with d>b such that Ha⁢b⁢(d)<U⁢(d), then there exists a c∈[b,d) so that Ha⁢d⁢(c)=Ha⁢b⁢(c). Thus Ha⁢d≡Ha⁢b.

(2) We have

0≤Ca⁢b:=U⁢(a)-U⁢(b)Wn,p⁢(a)-Wn,p⁢(b)<∞

for all 0<a<b since U is decreasing and Wn,p is strictly decreasing. Moreover,

Hb⁢c⁢(b+ϵ)≤U⁢(b+ϵ)≤Ha⁢b⁢(b+ϵ)

whenever b+ϵ is between b and c, see Figure 3. The first inequality follows from the comparison principle, and the second inequality follows from (1). Therefore

Ca⁢b⁢Wn,p′⁢(b)=Ha⁢b′⁢(b)

=limϵ→0+⁡Ha⁢b⁢(b+ϵ)-Ha⁢b⁢(b)ϵ

≥limϵ→0+⁡Hb⁢c⁢(b+ϵ)-Hb⁢c⁢(b)ϵ

=Hb⁢c′⁢(b)

=Cb⁢c⁢Wn,p′⁢(b)

and Ca⁢b≤Cb⁢c since Wn,p′⁢(b)<0.

$Figure 3 Proof of (2): We have Hb⁢c≤Ha⁢b{H_{bc}\leq H_{ab}} on (b,c){(b,c)} soHb⁢c′⁢(b)≤Ha⁢b′⁢(b){H_{bc}^{\prime}(b)\leq H_{ab}^{\prime}(b)}.$

Figure 3

Proof of (2): We have Hb⁢c≤Ha⁢b on (b,c) soHb⁢c′⁢(b)≤Ha⁢b′⁢(b).

$Figure 4 Proof of (3): We have Ha′⁢b≥Ha⁢b{H_{a^{\prime}b}\geq H_{ab}} on (0,b){(0,b)} soHa′⁢b′⁢(b)≤Ha⁢b′⁢(b){H_{a^{\prime}b}^{\prime}(b)\leq H_{ab}^{\prime}(b)}.$

Figure 4

Proof of (3): We have Ha′⁢b≥Ha⁢b on (0,b) soHa′⁢b′⁢(b)≤Ha⁢b′⁢(b).

(3) Let 0<a<a′<b. By the comparison principle,

Ha′⁢b⁢(a′)=U⁢(a′)≥Ha⁢b⁢(a′).

Since Ha′⁢b⁢(b)=Ha⁢b⁢(b), the two functions are either identical or Ha′⁢b>Ha⁢b on (0,b). It follows that

Ca⁢b⁢Wn,p′⁢(b)=Ha⁢b′⁢(b)

=limϵ→0+⁡Ha⁢b⁢(b-ϵ)-Ha⁢b⁢(b)-ϵ

≥limϵ→0+⁡Ha′⁢b⁢(b-ϵ)-Ha′⁢b⁢(b)-ϵ

=Ha′⁢b′⁢(b)

=Ca′⁢b⁢Wn,p′⁢(b)

and Ca⁢b≤Ca′⁢b since Wn,p′⁢(b)<0. The proof for b<c′<c is symmetric. (See Figure 4.)

(4) The claims in (4) are immediate consequences of (2) and (3). ∎

We are now able to reveal a crucial fact about radial p-superharmonic functions: They have a smooth p-harmonic test function touching from above at every finite value.^[7]

Lemma 2.

Let 2≤p≤∞. Moreover, let u be a radial p-superharmonic function and let x0∈Rn. If u⁢(x0)<∞, then there exists a fundamental solution

h⁢(x):=C1⁢wn,p⁢(x)+C2,C1≥0,

touching u from above at x0:

h⁢(x0)=u⁢(x0) 𝑎𝑛𝑑 h⁢(x)≥u⁢(x) near x0.

Proof.

If x0=0 and u is bounded at the origin, the constant function h⁢(x)≡u⁢(0) will do.

Write u⁢(x)=U⁢(|x|) and b:=|x0|>0. Let Cb∈[Cb-,Cb+], where the end-points of the, possibly singleton but non-empty, interval are defined in part (4) of Lemma 1. We claim that the scaled fundamental solution h⁢(x)=H⁢(|x|) given by

H⁢(r):=Cb⁢[Wn,p⁢(r)-Wn,p⁢(b)]+U⁢(b)

touches u from above at x0.

Obviously, H⁢(b)=U⁢(b) and if 0<a<b, then Ca⁢b≤Cb by parts (3) and (4) of Lemma 1 and

H⁢(a)=Cb⁢[Wn,p⁢(a)-Wn,p⁢(b)]+U⁢(b)

≥Ca⁢b[Wn,p(a)-Wn,p(b)]+U(b) (Wn,p(a)-Wn,p(b)>0)

=Ha⁢b⁢(a)=U⁢(a).

If b<c, then similarly Cb≤Cb⁢c and

H⁢(c)=Cb⁢[Wn,p⁢(c)-Wn,p⁢(b)]+U⁢(b)

≥Cb⁢c[Wn,p(c)-Wn,p(b)]+U(b) (Wn,p(c)-Wn,p(b)<0)

=U⁢(c).

The lemma is proved. ∎

Observe that H is uniquely determined if and only if U is differentiable at r=b. That is, if and only if Cb+=Cb-.

With these new tools, we restate and prove the Radial Equivalence (3) and the Sublinearity (2) of Proposition 5.

Proposition 6 (Radial Equivalence).

Let 2≤p≤∞. If u is a radial p-superharmonic function in Rn, then

𝒟p⁢u≤0 in ⁢ℝn

in the viscosity sense.

Proof.

Let x0∈ℝn and assume that u has a test function ϕ from below at x0. We need to show 𝒟p⁢ϕ⁢(x0)≤0. Obviously, u⁢(x0)<∞. By Lemma 2 there is a scaled fundamental solution h touching u from above at x0. Thus,

ϕ⁢(x0)=u⁢(x0)=h⁢(x0)

and, for x close to x0,

ϕ⁢(x)≤u⁢(x)≤h⁢(x),

which implies the Hessian matrix inequality ℋ⁢ϕ⁢(x0)≤ℋ⁢h⁢(x0). It follows that

𝒟p⁢ϕ⁢(x0)≤𝒟p⁢h⁢(x0)=0

from the fact that 𝒟p is degenerate elliptic and from the smooth case Cylindrical Equivalence. ∎

Proposition 7 (Sublinearity).

Let 2≤p≤∞. Assume Dp⁢u≤0 and Dp⁢v≤0 in the viscosity sense in Rn, where u is radial. Then

𝒟p⁢[u+v]≤0

in the viscosity sense in Rn.

Proof.

Let x0∈ℝn and assume that ϕ is a test function to u+v from below at x0. Clearly, u⁢(x0)<∞ since otherwise u+v=∞ at x0 and there would be no test function there. Also, u is p-superharmonic by the Domination (1) of Proposition 5. Hence, by Lemma 2, there exists a scaled fundamental solution h touching u from above at x0:

h⁢(x0)=u⁢(x0),u⁢(x)≤h⁢(x) near x0.

Again, 𝒟p⁢h=0 by the smooth case Cylindrical Equivalence.

Define ψ⁢(x):=ϕ⁢(x)-h⁢(x). Then ψ is C2 and

ψ⁢(x0)=u⁢(x0)+v⁢(x0)-u⁢(x0)=v⁢(x0)

and

ψ⁢(x)≤u⁢(x)+v⁢(x)-u⁢(x)=v⁢(x) near x0,

so ψ is a test function for v from below at x0. This means that 𝒟p⁢ψ⁢(x0)≤0, and it follows that

𝒟p⁢ϕ=𝒟p⁢[ψ+h]≤𝒟p⁢ψ+𝒟p⁢h≤0

at x0 by the smooth case Sublinearity. Hence,

𝒟p⁢[u+v]≤0

in the viscosity sense in ℝn. ∎

The proof of Proposition 5, and hence Theorem 2, is now completed.

Communicated by Juan Manfredi

Acknowledgements

I thank Peter Lindqvist for useful discussions and for naming the operator. I thank Fredrik Arbo Høeg for checking calculations. Also, I thank Juan Manfredi for pointing out the work [16].

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Received: 2017-05-22

Revised: 2017-11-13

Accepted: 2017-11-16

Published Online: 2017-12-07

Published in Print: 2020-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Superposition of p-superharmonic functions

Abstract

1 Introduction

Proposition.

Theorem 1.

Theorem 2.

Definition 1.

Property.

2 The dominative p-Laplace operator

2.1 Preliminary basics and notation

2.2 Definition and fundamental properties

Definition 2.

Remark 1.

Proposition 1 (Fundamental properties of Dp. Smooth case).

Proof.

3 Proof of Theorem 1

Proof of Theorem 1 (i).

Proposition 2.

Proof.

Proposition 3.

Theorem 3 (Segre).

Proof of Proposition 3.

4 Viscosity solutions

4.1 Definitions and fundamental properties

Definition 3.

Theorem 4 (Comparison Principle).

Proposition 4.

Proposition 5 (Fundamental properties of Dp. Viscosity sense).

Proof.

4.2 The superposition principle for radial p-superharmonic functions

Proof of Theorem 2.

5 Radial p-superharmonic functions

Lemma 1.

Proof.

Lemma 2.

Proof.

Proposition 6 (Radial Equivalence).

Proof.

Proposition 7 (Sublinearity).

Proof.

Acknowledgements

References

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