March 30, 2017

Seminar at Universidade de Aveiro

,

On maximum principles for higher-order fractional Laplacians

Alberto Saldaña De Fuentes

joint work with N. Abatangelo (Brussels) and S. Jarohs (Frankfurt)

Maximumprinciples!

Existence anduniqueness

Sub-SuperSolutionMethod

QualitativeProperties

Symmetryresults

Positivity

A prioriestimates

Non-existence

Other toolsHopf

Lemma

Harnackinequalities

Regularity

Alberto Saldana

A bit of history on maximum principles

L =∑Ni,j=1 aij∂iju+

∑Ni=1 bi∂iu+ cu,

∑Ni,j=1 aijξiξj ≥ λ|ξ|2

1839 C.F. Gauss and S. Earnshaw (separately),(sub)harmonic functions.

1892 A. Paraf, L with c < 0, N = 2.

1894 T. Moutard, M.P. for L with c < 0, N ≥ 1.

1905 E. Picard, M.P. for L with c ≤ 0, N = 2.

1927 E. Hopf, M.P. for L with L∞ coefficients.

1952 E. Hopf and O.A. Oleinik (separately), M.P. withboundary estimates.

Gauss

E. Hopf

Alberto Saldana

The simplest case of a maximum principle

Let u ∈ C2(0, 1) such that −u′′ > 0, then u minimizes at the boundary 0, 1,i.e.,

min(0,1)

u = min∂(0,1)

u

Alberto Saldana

Maximum principles for the Laplacian

Let Ω be a smooth bounded domain (open and connected) and u ∈ C2(Ω).

−∆u ≥ 0 =⇒ minΩu = min

∂Ωu (weak maximum principle),

−∆u ≥ 0

minΩu = u(x0)

x0 ∈ Ω

=⇒ u ≡ u(x0) in Ω (strong maximum principle).

Alberto Saldana

Positivity preserving properties

Let Ω be a smooth bounded domain and u ∈ C2(Ω).

−∆u ≥ 0 in Ω

u ≥ 0 on ∂Ω=⇒ u ≥ 0 in Ω

The solution inherits the sign of the data. Many physical interpretations ofthis are possible: heat diffusion equilibrium.

Alberto Saldana

Positivity preserving properties

Let Ω be a smooth bounded domain and u ∈ C2(Ω).

−∆u ≥ 0 in Ω

u ≥ 0 on ∂Ω=⇒ u ≥ 0 in Ω

The solution inherits the sign of the data. Many physical interpretations ofthis are possible: population diffusion equilibrium.

Alberto Saldana

Positivity preserving properties

Let Ω be a smooth bounded domain and u ∈ C2(Ω).

−∆u ≥ 0 in Ω

u ≥ 0 on ∂Ω=⇒ u ≥ 0 in Ω

The solution inherits the sign of the data. Many physical interpretations ofthis are possible: stress distribution/ elastic membrane.

Alberto Saldana

However, not all operators have maximum principles...

Let ∆2 denote the bilaplacian, that is, ∆2u = −∆(−∆u).

A counterexample for the bilaplacian [Shapiro & Tegmark, ’94]

Let E := (x, y) ∈ R2 : Q(x, y) := x2 + 25y2 − 1 < 0 and

u(x, y) := Q(x, y)2((1− x)2(4− 3x)− ε)

with ε > 0 small. Then ∆2u ≥ 0 in E and u = ∂νu = 0 on ∂E but u changessign .

Alberto Saldana

But... why??

To gain some insight, let us have a look at models for suspension bridges!

Some simplified model is the following: Let u denote the displacement of thebridge caused by a load f , then u satisfies

uxxxx = −ku+ +W + f(x) in (0, L),

where W is the weight of the bridge itself and ku+ is the effect of thesuspension cables. We complement this equation with

u(0) = u(L) = uxx(0) = uxx(L) = 0 (Navier boundary conditions)

u(0) = u(L) = ux(0) = ux(L) = 0 (Dirichlet boundary conditions)

This equation follows from minimizing an energy functional of the type

I(u) =

∫ L

0

|u′′|2 + . . . ,

where the term |u′′|2 is an approximation of the curvature under theassumption of small deformations.

Alberto Saldana

I(u) =

∫ L

0

|u′′|2 + . . . ,

But minimizing curvature-type terms |u′′| (at the expense of u′) can becounterintuitive sometimes...

Minimizing u′ Minimizing u′′

Alberto Saldana

On the other hand, not everything is lost

Positivity preserving properties such as

(−∆)mu ≥ 0 in Ω

Dαu|∂Ω = 0 for |α| ≤ m− 1=⇒ u ≥ 0 in Ω,

still hold for some domains!

Example : Ω = B1(0) ⊂ RN or a small smooth perturbation of a ball.

Why? : The Green function is explicitly known (Lauricella in 1896 for m = 2and N = 2; and Boggio in 1905 for N ≥ 1) and it is positive!

GΩ,(−∆)m(x, y) = C|x− y|2m−N∫ ρ(x,y)

0

tm−1

(t+ 1)N2

dt

where

ρ(x, y) :=(1− |x|2)+(1− |y|2)+

|x− y|2.

Alberto Saldana

“Some”domains?

The deep reason why some domains have positive Greenfunction GΩ,(−∆)m is not known.

It has also been observed numerically that positivitypreserving almost occurs: solutions inherit the sign of the dataexcept for a small contribution.

Alberto Saldana

Our goal

Let us have a look at what happens between −∆ and ∆2.

Question:

At which point does the maximum principle stops being valid? That is, forwhich s ∈ (1, 2) is a fractional Laplacian (−∆)s positivity preserving?

We need:

F A good notion of (−∆)s for s ∈ (1, 2).

F A proof or a counterexample of the Maximum Principle.

Alberto Saldana

The fractional setting for s ∈ (0, 1)

Let s ∈ (0, 1), Ω ⊂ RN a smooth bounded domain and consider the Hilbertspace

Hs0(Ω) := u ∈ Hs(RN ) : u ≡ 0 on RN\Ω

equipped with the norm ‖u‖Hs0(Ω) = (‖u‖L2(Ω) + Es(u, u))

12 where

Es(ϕ,ψ) = CN,s

∫RN

∫RN

(ϕ(x)− ϕ(y))(ψ(x)− ψ(y))

|x− y|N+2sdxdy

In fact, Es(ϕ,ψ) =∫RN |ξ|2sϕ(ξ)ψ(ξ) dξ.

Here:

Hs(RN ) = u ∈ L2(RN ) : (1 + |ξ|2)s2 u ∈ L2(RN ), s ∈ R,

u(ξ) =

∫ ∞−∞

u(x)e−2πix·ξ dx is the Fourier transform of u.

Alberto Saldana

The fractional setting for s ∈ (0, 1)

Let s ∈ (0, 1), Ω ⊂ RN a smooth bounded domain, and f ∈ L2(Ω). A functionu ∈ Hs0(Ω) is a weak solution of (−∆)su = f in Ω if

Es(u, v) =

∫Ω

f v dx for all v ∈ Hs0(Ω).

Regularity: If f ∈ Cα(Ω) then actually u ∈ Cs(RN ) ∩ C2s+ε(Ω) and solves

(−∆)su(x) := cN,s limε→0

∫|x−y|>ε

u(x)− u(y)

|x− y|N+2sdy = f(x), x ∈ Ω

with u ≡ 0 on RN\Ω.Limiting: (−∆)su→ u as s→ 0, (−∆)su→ −∆u as s→ 1, if u ∈ C∞c (RN ).

Alberto Saldana

And if s > 1?

These notions cannot be plainly extended to s > 1, in fact, if∫Ω

∫Ω

(u(x)− u(y))2

|x− y|N+2sdxdy <∞ for s > 1 (Ω open and connected)

then u must be a constant in Ω.

Alberto Saldana

What people usually did...

One may consider the functional I : Hs0(Ω)→ R given by

I(u) :=

∫RN

|(−∆)s2u|2

2− f u dx,

and the minimizer is the weak solution of

(−∆)su = f in Ω, u ≡ 0 on RN\Ω.

z Pohozaev identity and integration by parts (Ros-Oton & Serra (2015)).

z Regularity (Grubb (2015)).

z Spectral properties (Musina & Nazarov (2015)).

z Nonlinear problems (Fazly & Wei (2016), Lopes & Maris (2008),Maalaoui & Martinazzi & Schikorra (2015), Palatucci & Pisante (2015))

Alberto Saldana

Our variational framework for s > 1

Let m ∈ N0, σ ∈ (0, 1), and s = m+ σ. For u, v ∈ Hs0(Ω) let

Es(u, v) :=

Eσ(∆

m2 u,∆

m2 v), if m is even,

N∑k=1

Eσ(∂k∆m−1

2 u, ∂k∆m−1

2 v), if m is odd,(1)

where, as before,

Eσ(ϕ,ψ) = CN,s

∫RN

∫RN

(ϕ(x)− ϕ(y))(ψ(x)− ψ(y))

|x− y|N+2σdxdy.

So, for instance,

E1+σ(u, u) = CN,s

∫RN

∫RN

|∇u(x)−∇u(y)|2

|x− y|N+2σdxdy.

Alberto Saldana

Maximum principles for weak solutions, s = 1

Let Ω be a bounded domain, u ∈ H10 (Ω), and E(u, v) :=

∫Ω∇u∇v dx.

E(u, v) ≥ 0 ∀v ∈ H10 (Ω), v ≥ 0 =⇒ u ≥ 0 in Ω

Proof.

Since u− := −minu, 0, u+ := maxu, 0 ∈ H10 (Ω), we have that

0 ≤ E(u, u−) = E(u+, u−)− E(u−, u−) = −∫

Ω

|∇u−|2 dx ≤ 0 =⇒ u− ≡ 0.

Note that E(u+, u−) = 0.

Alberto Saldana

Maximum principles for weak solutions, s ∈ (0, 1)

Let Ω be a bounded domain and u ∈ Hs0(Ω).

Es(u, v) ≥ 0 ∀v ∈ Hs0(Ω), v ≥ 0 =⇒ u ≥ 0 in Ω

Proof.

Since u− := −minu, 0, u+ := maxu, 0 ∈ Hs0(Ω), we have that

0 ≤ Es(u, u−) = Es(u+ − u−, u−) = Es(u

+, u−)− Es(u−, u−) ≤ 0

and therefore u− ≡ 0. Note that

Es(u+, u−) = −2CN,s

∫RN

∫RN

u+(x)u−(y)

|x− y|N+2sdxdy ≤ 0.

Alberto Saldana

What about s > 1 ?

Let Ω be a bounded smooth domain and u ∈ Hs0(Ω), does it hold that

Es(u, v) ≥ 0 ∀v ∈ Hs0(Ω), v ≥ 0 =⇒ u ≥ 0 in Ω ?

We actually have that

Bourdaud & Meyer (1991)

u− := −minu, 0, u+ := maxu, 0 ∈ Hs0(Ω) for s ∈ (0,3

2).

Could it be that the maximum principle holds for s ∈ (0, 32 )?

Well...

Alberto Saldana

Let’s try...

Let Ω be a bounded smooth domain and u ∈ Hs0(Ω) with s ∈ (1, 32 ). Then

Es(u, u−) = Es(u

+ − u−, u−) = Es(u+, u−)− Es(u−, u−),

where

Es(u−, u−) = CN,s

∫RN

∫RN

|∇u−(x)−∇u−(y)|2

|x− y|N+2σdxdy ≥ 0

Es(u+, u−) = −2CN,s

∫RN

∫RN

∇u+(x) · ∇u−(y)

|x− y|N+2σdxdy

Naively, two integration by parts would yield that

−∫RN

∫RN

∇u+(x) · ∇u−(y)

|x− y|N+2σdxdy = (N + 2σ)s

∫RN

∫RN

u+(x)u−(y)

|x− y|N+2sdxdy ≥ 0.

This calculation cannot be done that easily, but it has recently been showedthat Es(u

+, u−) ≥ 0 (Musina & Nazarov, 2017).

Alberto Saldana

Counterexample for s ∈ (1, 2) [Abatangelo, Jarohs, A.S.]

X Let s = 1 + σ, B = B1(0), and U = B1(3e1).

X Ψ(x) := (1− |x|2)s+ ∈ Hs0(B) =⇒ (−∆)sΨ = c in B for some c > 0 and

Es(Ψ, ϕ) = c

∫B

ϕ dx for all ϕ ∈ Hs0(B).

X g ∈ C∞c (U) nonnegative with ‖g‖L∞(U) ≤ 1 and g ≡ 0 in RN\U . Then,

Es(g, ϕ) =∫U

(−∆)sg ϕ dx for ϕ ∈ Hs0(U) and (−∆)sg ∈ L∞(RN ).

X u ∈ Hs0(B ∪ U) be given by u := aΨ−g , where a > 0 is a large constant.

Alberto Saldana

Counterexample for s ∈ (1, 2)

We now show that Es(u, ϕ) ≥ 0 for all nonnegative ϕ ∈ Hs0(B ∪ U).Let ϕ ∈ Hs0(B ∪ U) nonnegative and

ϕB := ϕχB ∈ Hs0(B), ϕU := ϕχU ∈ Hs0(U).

Since ϕ = ϕB + ϕU , it suffices to show that Es(u, ϕB) ≥ 0 and Es(u, ϕU ) ≥ 0.

Alberto Saldana

Counterexample for s ∈ (1, 2)

Indeed,

Es(u, ϕB) = aEs(Ψ, ϕB)− Es(g, ϕB)

= ac

∫B

ϕB dx− C∫B

∫U

ϕB(x)g(y)

|x− y|N+2sdydx

≥ ac∫B

ϕB dx− C|U |∫B

ϕB(x)dx ≥ 0 for a large enough,

since |x− y|N+2s ≥ 1 for x ∈ B and y ∈ U and ‖g‖L∞(U) ≤ 1.

Alberto Saldana

Counterexample for s ∈ (1, 2)

And...

Es(u, ϕU ) = aEs(Ψ, ϕU )− Es(g, ϕU )

= aC

∫B

∫U

Ψ(x)ϕU (y)

|x− y|N+2sdydx−

∫U

(−∆)sg ϕU dx

≥ aC ′∫B

Ψ(x)dx

∫U

ϕU (y) dy − ‖(−∆)sg‖L∞(RN )

∫U

ϕU dx

≥ 0 for a large enough,

since |x− y| ≤ 5 for x ∈ B and y ∈ U , Es(g, ϕ) =∫U

(−∆)sg ϕU dx, and(−∆)sg ∈ L∞(RN ). Therefore Es(u, ϕ) = Es(u, ϕB) + Es(u, ϕU ) ≥ 0 and theproof is finished.

Alberto Saldana

Some remarks

∅ The two disjoint balls can be connected with a thin tube, to make thedomain connected.

∅ The counterexample can be extended to any s ∈ (k, k + 1) for k ∈ N odd.

∅ A counterexample to maximum principles for s ∈ (k, k + 1) for k ∈ N evenremains an open question, but we do not expect the M.P. to hold for anys > 1.

∅ Surprisingly, the only counterexample to maximum principles for oddpowers of the Laplacian was obtained last year (2016), by Guido Sweers(a polynomial in an ellipse for s = 3).

Alberto Saldana

A positive case

Maximum principles for (−∆)s do hold in balls for any s > 0, since the Greenfunction is positive and it is given by Boggio’s formula:

GB,(−∆)s(x, y) = CN,s|x− y|2s−N∫ ρ(x,y)

0

ts−1

(t+ 1)N2

dt,

where

ρ(x, y) :=(1− |x|2)+(1− |y|2)+

|x− y|2.

♠ s = 2 and N = 2 Lauricella, 1896.

♠ s ∈ N Boggio, 1905.

♠ s ∈ (0, 1) Riesz, 1938 and Blumenthal-Getoor-Ray, 1961.

♠ s > 1 Dipierro-Grunau, 2016, and Abatangelo-Jarohs-S., 2016.

Alberto Saldana

Another positive case

Maximum principles for (−∆)s do hold in two disjoint balls Ω = B ∪B1(te1)for s ∈ (m,m+ 1) with m even and t ≥ 3, since the Green function is positive.

Theorem (N. Abatangelo, S. Jarohs, and A.S., 2017)

Let s = m+ σ > 0, t ≥ 3, Bt := B1(te1), and

ΓB(x, y) = (−1)mγN,σ(1− |x|2)s

(|y|2 − 1)s|x− y|N.

The unique (Dirichlet) Green function GΩ of (−∆)s in Ω = B ∪Bt satisfies

GΩ(x, y) = GB(x, y) +

∫Bt

ΓB(x, z)GΩ(y, z) dz for x, y ∈ B, x 6= y,

GΩ(x, y) =

∫Bt

ΓB(x, z)GΩ(z, y) dz for x ∈ B, y ∈ Bt,

Alberto Saldana

Another positive case

Theorem (N. Abatangelo, S. Jarohs, and A.S., 2017)

Let N ∈ N, m ∈ N0, σ ∈ (0, 1), s = m+ σ. Then,

GΩ > 0 in (x, y) ∈ (B ×B) ∪ (Bt ×Bt) : x 6= y

and

GΩ > 0 in (B ×Bt) ∪ (Bt ×B) if m is even,

GΩ < 0 in (B ×Bt) ∪ (Bt ×B) if m is odd.

Alberto Saldana

The take away

General maximum principles only hold for s ∈ [0, 1].

It is not the belonging of u+ and u− to the right space what makes themaximum principle valid.

There is still much to be studied in terms of what makes a domain have aPPP or not and what kinds of weaker PPP can be used.

Alberto Saldana

Some interesting references:

Introductions to higher-order physical and mathematical models:

♣ Polyharmonic Boundary Value Problems (2010), Gazzola, Grunau,Sweers.

♣ Mathematical Models for Suspension Bridges (2015), Gazzola.

♣ Spatial Patterns, Higher Order Models in Physics andMechanics (2001), Peletier, Troy.

♣ On Sign Preservation for Clotheslines, Curtain Rods, ElasticMembranes and Thin Plates (2016), Sweers.

Our preprint available in arXiv.org:

♣ On the maximum principle for higher-order fractionalLaplacians (2016), Abatangelo, Jarohs, S.

Alberto Saldana