The Dominative p -Laplacian Karl K. Brustad Aalto University KTH - - PowerPoint PPT Presentation

The Dominative p-Laplacian

Karl K. Brustad

Aalto University

KTH 26.-28. August 2019

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The p-Laplace Equation

∆pu := div

• |∇u|p−2∇u
• = 0,

2 ≤ p < ∞. Variational integral: ✂ |∇u|p dx. ∆u = div (∇u) = 0, p = 2. Infinity-Laplacian: ∆∞u := ∇uHu∇uT. Nonlinear when p > 2: ∆p[u + v] = ∆pu + ∆pv.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The fundamental solution

wn,p(x) :=        − p−1

p−n|x|

p−n p−1 ,

p = n, − ln |x|, p = n, −|x|, p = ∞ or n = 1. Satisfies ∆pwn,p(x) = 0 for x = 0 in Rn.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Ellipticity & Viscosity

Definition

A function u : Ω → (−∞, ∞] is a (viscosity) supersolution to the equation Dw = 0 in Ω if

• u is lower semicontinuous.
• u is finite on a dense subset of Ω.
• whenever φ ∈ C 2 touches u from below at some point

x0 ∈ Ω, then Dφ(x0) ≤ 0.

Definition

An operator Du = F(x, u, ∇u, Hu) is (degenerate) elliptic if X ≤ Y ⇒ F(x, s, q, X) ≤ F(x, s, q, Y ) for all (x, s, q) ∈ Ω × R × Rn.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Superposition principle for the fundamental solutions

V (x) :=

N

• i=1

ciwn,p(x − yi), ci ≥ 0, yi ∈ Rn.

Theorem (Crandall/Zhang (2003))

V is p-superharmonic in Rn.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Superposition principle for the fundamental solutions

V (x) :=

N

• i=1

ciwn,p(x − yi), ci ≥ 0, yi ∈ Rn.

Theorem (Crandall/Zhang (2003))

∆pV ≤ 0.

Theorem (Lindqvist/Manfredi (2008))

∆p ✂ wn,p(x − y)ρ(y) dy

• ≤ 0,

ρ ≥ 0. ∆pV (x) = − (p−2)(n+p−2)

p−1

|∇V (x)|p−2

N

• i=1

ci sin2 θi(x) |x − yi|

n+p−2 p−1

, θi(x) angle between x − yi and ∇V (x).

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Preliminaries.

The Hessian matrix: Hu := ∂2u ∂xi∂xj n

i,j=1

. n real eigenvalues: λ1 ≤ · · · ≤ λn, λi = λi(Hu). Bounds on the Rayleigh quotient: λ1 ≤ ξTHuξ |ξ|2 ≤ λn, 0 = ξ ∈ Rn.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The Dominative p-Laplacian

∆pu = div(|∇u|p−2∇u) = |∇u|p−2

• (p − 2)∇uHu∇uT

|∇u|2 + ∆u

• .

Dpu := (p − 2)λn(Hu) + ∆u, 2 ≤ p < ∞. Domination: Dpu ≤ 0 ⇒ ∆pu ≤ |∇u|p−2Dpu ≤ 0.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The Dominative p-Laplacian

Sublinearity: Dp[u + v] ≤ Dpu + Dpv, Dp[cu] = cDpu, for c ≥ 0. Ap : Rn → S(n), Ap(ξ) := (p − 2)ξξT + I, Dpu = max

|ξ|=1 tr(Ap(ξ)Hu),

Dpu ≤ 0, Dpv ≤ 0 ⇒ Dp[u + v] ≤ 0.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The Dominative p-Laplacian

Equivalence: The gradient ∇wn,p is an eigenvector to the Hessian Hwn,p corresponding to the largest eigenvalue λn(Hwn,p). ∇wn,pHwn,p∇wT

n,p

|∇wn,p|2 = λn(Hwn,p). Dpwn,p = 0.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

An explanation of the superposition principle

∆pwn,p = 0 and Dpwn,p = 0, (Equivalence) Dpu ≤ 0, Dpv ≤ 0 ⇒ Dp[u + v] ≤ 0, (Sublinearity) Dpw ≤ 0 ⇒ ∆pw ≤ 0. (Domination) We immediately get ∆p N

• i=1

ciwn,p(x − yi)

• ≤ 0.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Taking it further

New questions. When is a generic sum N

i=1 ui p-superharmonic?

Sufficient conditions: Dpui ≤ 0 ⇒ ∆p N

• i=1

ui

• ≤ 0.

Necessary conditions?: u ∈ C 2 is addable if

1 ∆p[u + l] ≤ 0 for every linear function l(x) = aTx. 2 ∆p[u + cwn,p ◦ T] ≤ 0 for every c ≥ 0 and every

translation T(x) = x − x0.

3 ∆p[u + u ◦ T] ≤ 0 for every isometry T.

1-3 is equivalent. 1-3 is equivalent to u being dominative p-superharmonic.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Cylindrical functions

Next question: What are the addable p-harmonic functions?

• Addable = dominative p-superharmonic.
• 0 = ∆pu ≤ |∇u|p−2Dpu ≤ 0.

The double equation: Dpu = 0 = ∆pu. A function f is radial if there exists a one-variable function F so that f (x) = F(|x|).

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Cylindrical functions

Definition

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. QTQ = Ik, so that f (x) = F

• |QT(x − x0)|
• for some x0 ∈ Rn. We say that a function w in Rn is a

cylindrical fundamental solution (to the p-Laplace Equation) if w is in the form w(x) = C1wk,p

• QT(x − x0)
• + C2,

C1 ≥ 0, for some k, Q and x0 as above. Dpw = 0 = ∆pw.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Cylindrical functions

Theorem

Let 2 < p < ∞ and let u ∈ C 2(Ω). If ∆pu = 0 = Dpu in Ω, then u is locally a cylindrical fundamental solution. The addable p-harmonic functions are exactly the cylindrical fundamental solutions.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Isoparametric functions

A nonconstant smooth function u : M → R on a Riemannian manifold M is called isoparametric if there exists functions f and g so that |∇u| = f (u) and ∆u = g(u). A regular level-set of an isoparametric function is called an isoparametric hypersurface.

Theorem (Segre 1938)

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

1 a hyperplane Rn−1, 2 a sphere Sn−1, 3 a generalized cylinder Sk−1 × Rn−k, k = 2, . . . , n − 1.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Isoparametric functions

0 = |∇u|p−2

• (p − 2)∇uHu∇uT

|∇u|2 + ∆u

• ,

0 = (p − 2)λn + ∆u. ∇uHu∇uT |∇u|2 = λn, Hu∇uT = λn∇uT. c curve in a level set of u: d dt 1 2|∇u(c(t))|2 = ∇uHuc′ = λn∇uc′ = 0. f (u) = 1 2|∇u|2, f ′(u)∇u = ∇uHu = λn∇u ∆u = −(p − 2)λn = −(p − 2)f ′(u) =: g(u).

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Corollary

Let 2 < p < ∞. The largest family, F, of p-harmonic functions containing the affine functions, in which every sum of the members, x →

N

• i=1

ui(x), ui ∈ F, N ∈ N, is p-superharmonic, is the set of cylindrical fundamental solutions. That is, the family of functions on the form x → C1Wk,p

• |QT(x − x0)|
• + C2,

C1 ≥ 0, where k ∈ {1, . . . , n}, Q ∈ Rn×k with QTQ = Ik, and where Wk,p(r) :=

• − p−1

p−k r

p−k p−1 ,

p = k, − ln r, p = k.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

Fundamental properties of the dominative p-Laplacian

1 Domination: A dominative p-superharmonic function is

p-superharmonic.

2 Cylindrical equivalence: The cylindrical dominative- and

• rdinary p-superharmonic functions are the same.

3 Sublinearity: If u and v are dominative p-superharmonic,

then so is u + v.

4 Nesting property:

• If u is dominative p-superharmonic, then it is dominative

q-superharmonic for every q ∈ [2, p].

• If u is dominative p-subharmonic, then it is dominative

q-subharmonic for every q ∈ [p, ∞].

5 Rotational invariance:

(Dpu) ◦ T = Dp[u ◦ T] for all isometries T : Rn → Rn.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The doubly nonlinear diffusion

• perator

Lp,qu := div

• uq−1|∇u|p−2∇u
• = uq−1|∇u|p−2
• ∆h

pu + (q − 1)|∇u|2

u

• q = 1 : p-Laplace,

p = 2 : Porous Medium. Superposition principles for the fundamental solution in the cases 2 ≤ p < n, q ≥ 1 and p > n, 0 < q ≤ 1 (J.Tyson 2016).

Definition (The dominative (p, q)-Laplacian)

Dp,qu := Dpu + (q − 1)|∇u|2 u , 2 ≤ p < ∞, q ≥ 1, u > 0. Lp,qu ≤ uq−1|∇u|p−2Dp,qu.

The Dominative p-Laplacian Karl K. Brustad Introduction The discovery by Crandall- Zhang An explanation of the superposition principle Superposition

• f p-harmonic

functions Further applications of sublinear

• perators:

Superposition in the doubly nonlinear diffusion equation

The dominative (p, q)-Laplacian

Dp,qu = Dpu + (q − 1) |∇u|2

u

is sublinear for positive smooth functions since |∇u|2 u = max

c≤− 1

4 |b|2 (∇ub + cu) .

2 ≤ p < n, q ≥ 1.

• Fundamental solution wn,p,q(x) := |x|

p−n p−2+q > 0.

• λmax(Hwn,p,n) = ∆h

∞wn,p,q which implies

Dp,qwn,p,q = 0 = Lp,qwn,p,q. V (x) :=

N

• i=1

ciwn,p,q(x − xi), ci > 0 is (p, q)-superharmonic in Rn since Lp,qV ≤ V q−1|∇V |p−2Dp,qV ≤ 0,

• ptw. x = xi.

No test functions from below at x = xi.