New Tools for Nonlocal Elliptic Problemsstolo/conf/hiver14/MiniCourseNew.pdf · !Physics & Maths 2...

New Tools for Nonlocal Elliptic ProblemsMini-Course @ Winter School 2014, St. Etienne de Tinee

Enno Lenzmann1

Department of MathematicsUniversity of Basel

February 4, 2014

1Joint work with Rupert Frank (CalTech) and Luis Silvestre (Chicago)

E. Lenzmann Mini-Course Nonlocal Elliptic Problems

Overview

Class of Problems

Lu + f (x , u) = 0 in Rn

L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).

f is nonlinearity

Contents of Mini-Course:

1 Motivation & History

2 New Results

3 Tools & Methods

4 Applications & Outlook

Overview

Class of Problems

Lu + f (x , u) = 0 in Rn

f is nonlinearity

1 Motivation & History→ Physics & Maths

2 Results→ Symmetry, Uniqueness, Oscillations

3 Tools & Methods→ s-Harmonic Extension, Topological Bounds, Hamiltonian Estimates

4 Applications & Outlook→ Stability and Blowup for Nonlinear Dispersive PDE

Overview

Class of Problems

Lu + f (x , u) = 0 in Rn

f is nonlinearity

Overview

Class of Problems

Lu + f (x , u) = 0 in Rn

f is nonlinearity

Overview

Class of Problems

Lu + f (x , u) = 0 in Rn

f is nonlinearity

Nonlocality of Type I

Question: Where do nonlocal problems arise?

Example: Let Ω ⊂ Rn and L local (elliptic) differential operator, e. g.,Laplacian L = −4

Classical Boundary-Value ProblemLu = 0 in Ωu = g on ∂Ω

Nonlocality of Type I

Question: Where do nonlocal phenomena arise?

Example: Let Ω ⊂ Rn and L local (elliptic) differential operator, e. g.,Laplacian L = −4

Classical Boundary-Value ProblemLu = 0 in Ωu = g on ∂Ω

rpNonlocality due to change of g

g = g+“Pertubation around p ∈ ∂Ω”

=⇒ u(x) 6= u(x) ∀x ∈ Ω

Boundary-Value Problem from Skandinavia...

Elliptic & Parabolic vs. Hyperbolic

Deeper reason for nonlocal phenomena lies in the class of PDE.

Elliptic and Parabolic PDE such as

4u = 0 und ∂tu −4u = 0

exhibit Nonlocality via Boundary-/Initial conditions.

Hyperbolic PDE such as wave equation

c2∂ttu −4u = 0

preserve Locality (with c > 0 speed of propagation)

Nonlocality of Type II

Nonlocality in equations involving Pseudo-Differentialoperators

EquationLu = f in Ω ⊂ Rn

with Pseudo-Differentialoperator L.

E. g. fractional Laplacian L = (−4)s with s > 0 in Rn defined via Fouriertransform as

((−4)s f )(ξ) := |ξ|2s f (ξ)

For s ∈ (0, 1) we have integral formula by Aronszajn-Smith:

((−4)s f )(x) = cn,s

f (x)− f (y)

|x − y |n+2sdy

Harmonic Extension

Problem

Let f : Rn → R be bounded. Find F : Rn+1+ → R bounded such that

4F = 0 in Rn+1+

F = f on ∂Rn+1+

4F = 0

Derivative on boundary ∂Rn+1+

∂tF (·, t)∣∣t=0

= −(−4)1/2f

Harmonic Extension

Problem

Let f : Rn → R be bounded. Find F : Rn+1+ → R bounded such that

4F = 0 in Rn+1+

F = f on ∂Rn+1+

4F = 0

Derivative on boundary ∂Rn+1+

∂tF (·, t)∣∣t=0

= −(−4)1/2f

s-Harmonic Extension: (−4)s as Dirichlet-Neumann map

Observation by Caffarelli-Silvestre ’06 (see also Graham-Zworski ’04,Ostrovskii-Molchanov ’69)

Let s ∈ (0, 1) und f : Rn → R bounded. Let F = F (x , t) be solution to∇ · (t1−2s∇F ) = 0 in Rn+1

F = f on ∂Rn+1+

∇ · (t1−2s∇F ) = 0

(Weighted) Boundary Derivative on ∂Rn+1+

t1−2s∂tF (·, t)∣∣t=0

= −cs(−4)s f

The Five-Fold Way to (−4)s

Fractional Laplacian (−4)s in Rn with s ∈ (0, 1) can be represented as:

1 Multiplication by |ξ|2s in Fourier space.

2 Singular integral operator∫ f (x)−f (y)

|x−y|n+2s dy .

3 Generator of the (heat) semigroup e−t(−4)st≥0.

4 By spectral calculus,

(−4)s =sin(πs)

∫ ∞0

µs−1 −4−4+ µ

5 Dirchlet–Neumann operator for Ls = ∇ · (t1−2s∇·) on Rn+1+ .

Remarks:

Hence many different situations, where (−4)s arises.

Large toolbox due to diverse defintions.

Extension Principle in Retrospective

For many interesting pseudo-differentialoperators L it holds:

Extension Principle

Lu = f in Rn ⇐⇒

LlocalU = 0 in Rn+1+

∂U∂n

= f auf ∂Rn+1+

Examples: L = (−4)s , (−4+ 1)s with s ∈ (0, 1), L = LILW (water waves)etc.

Manifolds and domains: Hn, Tn, and Ω ⊂ Rn etc.

Regularity, Existence/Nonexistence, Harnack inequalities, uniquecontinuation for L etc.

Oscillation bounds for H = (−4)s + V .

Montonicity formulae (Hamiltonian estimates) for radial solutions.

Why study Nonlocal Equations?

Concrete Examples from Physics & Maths

Stable Levy Processes & Anomalous Diffusion:

pt = e−t(−4)s , ∂tu = (−4)su + f (u)

Song-Wu ’99, Bogdan et al. ’99, Banuelos et al. ’04, Caffarelli & Vazquez ’11, ...

Relativistic Schrodinger Operators:

H = (−4)s + V

Herbst ’77, Carmona-Master-Simon ’90, ...

Relativsitic Quantum Mechanics, Gravitatonal Collapse, Long-RangeSystems:

i∂tu = (−4)su + f (u) (fNLS)

Fefferman-de la Llave ’86, Lieb-Yau ’87, Elgart-Schlein ’05, Frohlich-Lenzmann

’07, ...

Fluid Dynamics:

∂tu + ∂x(−4)su − |u|p−1∂xu = 0 (gBO)

Weinstein ’87, Amick-Toland ’91, Kenig-Martel-Robbiano ’11, ...

Concrete Examples from Physics & Maths

Stable Levy Processes & Anomalous Diffusion:

pt = e−t(−4)s , ∂tu = (−4)su + f (u)

Song-Wu ’99, Bogdan et al. ’99, Banuelos et al. ’04, Caffarelli & Vazquez ’11, ...

Relativistic Schrodinger Operators:

H = (−4)s + V

Herbst ’77, Carmona-Master-Simon ’90, ...

Relativsitic Quantum Mechanics, Gravitatonal Collapse, Long-RangeSystems:

i∂tu = (−4)su + f (u) (fNLS)

Fefferman-de la Llave ’86, Lieb-Yau ’87, Elgart-Schlein ’05, Frohlich-Lenzmann

’07, ...

Fluid Dynamics:

∂tu + ∂x(−4)su − |u|p−1∂xu = 0 (gBO)

Weinstein ’87, Amick-Toland ’91, Kenig-Martel-Robbiano ’11, ...

More Examples

Fractional Obstacle and Free Boundary Problems:

min((−4)su, u − ϕ) = 0

Silvestre ’05, Caffarelli-Salsa-Silvestre ’08, ...

Layer Solutions (Peirls-Nabarro) Phase Transitions:

(−4)su = f (u) in Rn with uxn > 0

Toland ’97, Cabre-Sola-Morales ’05, Cabre-Sire ’12, ...

Conformal Geometry: Fractional Yamabe Problem, Paneitz operators,etc.Graham-Zworski ’03, Chang-Mar ’11, Mar-Mazzeo-Sire ’12, ...

Nonlocal ‘Minimal’ Surfaces:

L(A,B) =

∫ ∫χA(x)χB(y)

|x − y |n+2sdx dy .

Caffarelli-Roquejoffre-Savin ’10, Davila-del Pino-Wei ’13, ...

n/2-Fractional Harmonic Maps:

(−4)n/2u ∧ u = 0 for u : Rn → Sn−1

da Lio-Riviere ’11, Schikorra ’12, ...

More Examples

Fractional Obstacle and Free Boundary Problems:

min((−4)su, u − ϕ) = 0

Silvestre ’05, Caffarelli-Salsa-Silvestre ’08, ...

Layer Solutions (Peirls-Nabarro) Phase Transitions:

(−4)su = f (u) in Rn with uxn > 0

Toland ’97, Cabre-Sola-Morales ’05, Cabre-Sire ’12, ...

Conformal Geometry: Fractional Yamabe Problem, Paneitz operators,etc.Graham-Zworski ’03, Chang-Mar ’11, Mar-Mazzeo-Sire ’12, ...

Nonlocal ‘Minimal’ Surfaces:

L(A,B) =

∫ ∫χA(x)χB(y)

|x − y |n+2sdx dy .

Caffarelli-Roquejoffre-Savin ’10, Davila-del Pino-Wei ’13, ...

n/2-Fractional Harmonic Maps:

(−4)n/2u ∧ u = 0 for u : Rn → Sn−1

da Lio-Riviere ’11, Schikorra ’12, ...

Scope of this Mini-Course

We will study nonlocal elliptic problems with the fractional Laplacian (−4)s

given by((−4)su)(ξ) = |ξ|2s u(ξ)

Type of Problem

(−4)su + f (x , u) = 0 in Rn

We’ll focus on s ∈ (0, 1) with f (x , u) either linear or nonlinear.

Symmetry, uniqueness and non-degeneracy of ground states u(x) > 0.

Applications to time-dependent problems.

Remarks on bounded domains Ω ⊂ Rn with exterior Dirichlet conditions

(−4)su + f (x , u) = 0 on Ω and u ≡ 0 on Rn \ Ω

First: Brief recap for “classical” local case when s = 1

Scope of this Mini-Course

We will study nonlocal elliptic problems with the fractional Laplacian (−4)s

given by((−4)su)(ξ) = |ξ|2s u(ξ)

Type of Problem

(−4)su + f (x , u) = 0 in Rn

We’ll focus on s ∈ (0, 1) with f (x , u) either linear or nonlinear.

Symmetry, uniqueness and non-degeneracy of ground states u(x) > 0.

Applications to time-dependent problems.

Remarks on bounded domains Ω ⊂ Rn with exterior Dirichlet conditions

(−4)su + f (x , u) = 0 on Ω and u ≡ 0 on Rn \ Ω

First: Brief recap for “classical” local case when s = 1

Recap: Classical Local Theory

Elliptic Problem

−∆u + f (x , u) = 0 in Ω ⊂ Rn

In many cases (P) stems from variational problem given by

E(u) =

∫|∇u|2 +

∫F (u, x)

Linear Setting: Theory of Schrodinger Operators H = −∆ + V (x) with

Existence and Regularity of Eigenfunctions Hun = λnun

Bounds on N(V ) (number of eigenvalues)

Estimates on λn (e. g. spectral gaps)

Nodal Properties of un.

Unique continuation, absence of embedded eigenvalues...

Elliptic Problem

−∆u + f (x , u) = 0 in Ω ⊂ Rn

E(u) =

∫|∇u|2 +

∫F (x , u)

Nonlinear Setting:

Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0

Regularity

Symmetry (Gidas-Ni-Nirenberg ’79): If Ω = BR(0) or Ω = Rn then

u(x) > 0 ⇒ u = u(|x − x0|) > 0

for large class of local nonlinearities f = f (u) (e. g. locally Lipschitz).

Uniqueness and Nondegeneracy of Ground States u(x) > 0 (by reductionto ODE).

Elliptic Problem

−∆u + f (x , u) = 0 in Ω ⊂ Rn

E(u) =

∫|∇u|2 +

∫F (x , u)

Nonlinear Setting:

Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0

Regularity

Symmetry (Gidas-Ni-Nirenberg ’79): If Ω = BR(0) or Ω = Rn then

u(x) > 0 ⇒ u = u(|x − x0|) > 0

for large class of local nonlinearities f = f (u) (e. g. locally Lipschitz).

Uniqueness and Nondegeneracy of Ground States u(x) > 0 (by reductionto ODE).

Ground States: ODE Analysis

By moving planes argument, can restrict to radial functions and hence ODE.

Initial-Value Problem

−ϕ′′ − n−1rϕ′ + f (ϕ) = 0 for r > 0

ϕ(r) > 0 and ϕ(r)→ 0 as r →∞

For n = 1 uniqueness of ϕ is very simple.For n ≥ 2 analysis substantially harder.

Coffman ’73: Cubic case f (ϕ) = ϕ− ϕ3 and n = 3

Lieb ’77: Choquard-Pekar f (ϕ) = ϕ− (|x |−1 ∗ |ϕ|2)ϕ and n = 3

McLeod & Serrin ’81: f (ϕ) = ϕ− ϕp for some p and n ≥ 2.

Kwong ’89: General power-case f (ϕ) = ϕ− ϕp with 1 < p < n+2n−2

andn ≥ 2.

Many further results...

All proofs depend on ODE techniques!

Ground States: ODE Analysis

By moving planes argument, can restrict to radial functions and hence ODE.

Initial-Value Problem

−ϕ′′ − n−1rϕ′ + f (ϕ) = 0 for r > 0

ϕ(r) > 0 and ϕ(r)→ 0 as r →∞

For n = 1 uniqueness of ϕ is very simple.For n ≥ 2 analysis substantially harder.

Coffman ’73: Cubic case f (ϕ) = ϕ− ϕ3 and n = 3

Lieb ’77: Choquard-Pekar f (ϕ) = ϕ− (|x |−1 ∗ |ϕ|2)ϕ and n = 3

McLeod & Serrin ’81: f (ϕ) = ϕ− ϕp for some p and n ≥ 2.

Kwong ’89: General power-case f (ϕ) = ϕ− ϕp with 1 < p < n+2n−2

andn ≥ 2.

Many further results...

All proofs depend on ODE techniques!

Warm-Up: One-Dimensional Case

Consider ODE for the ground state problem with power-type nonlinearity.

ODE Problem

−ϕ′′ + ϕ− |ϕ|p−1ϕ = 0 on R+

ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞

Define Energy by

E(r) =1

2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1

2|ϕ|2 + 1

p+1|ϕ|p+1

Simple calculation shows dE/dr = 0 and hence E(r) ≡ const. Combinedwith limr→∞ E(r) = 0 (by decay of ϕ) this gives

E(r) =1

2|ϕ′(r)|2 + V (ϕ(r)) ≡ 0

Since ϕ′(0) = 0, must have V (ϕ(0)) = 0 and hence ϕ(0) = ( p+12

p−1 .

By-Product: No sign-changing ϕ solution with ϕ(∞) = 0 exists!

Warm-Up: One-Dimensional Case

Consider ODE for the ground state problem with power-type nonlinearity.

ODE Problem

−ϕ′′ + ϕ− |ϕ|p−1ϕ = 0 on R+

ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞

Define Energy by

E(r) =1

2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1

2|ϕ|2 + 1

p+1|ϕ|p+1

Simple calculation shows dE/dr = 0 and hence E(r) ≡ const. Combinedwith limr→∞ E(r) = 0 (by decay of ϕ) this gives

E(r) =1

2|ϕ′(r)|2 + V (ϕ(r)) ≡ 0

Since ϕ′(0) = 0, must have V (ϕ(0)) = 0 and hence ϕ(0) = ( p+12

p−1 .

By-Product: No sign-changing ϕ solution with ϕ(∞) = 0 exists!

Next Step: Higher-Dimensional Case

ODE Problem for (n ≥2)

−ϕ′′ − n−1rϕ′ + ϕ− |ϕ|p−1ϕ = 0 on R+

ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞

Again, consider energy given by

E(r) =1

2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1

2|ϕ|2 + 1

p+1|ϕ|p+1

Simple calculation shows monotonicity

dr= −n − 1

r|ϕ′(r)|2 ≤ 0

Intuitive Picture (McLeod, Tao): Motion of particle with position ϕ(r) andvelocity ϕ′(r) at ‘time’ r in potential V (ϕ) subject to friction force n−1

r2 ϕ′.

Fractional Case

Fractional Elliptic Problem

(−∆)su + f (x , u) = 0 in Ω ⊂ Rn

Linear Setting: Fractional Schrodinger Operators H = (−∆)s + V (x) with

Existence and Regularity of Eigenfunctions Hun = λnun.Simon 79 via e−t(−4)s on Rn; Serra/Ros-Oton & Grubb 13 on domains

Bounds on N(V ) (number of eigenvalues).Daubechies ’85 Lieb-Thirring estimates for (−4)s

Nodal Properties of un.Frank-Lenzmann-Silvestre ’12 and ’13

Unique continuation, absence of embedded eigenvalues...Fall & Felli ’13, Seo ’13, Ruland ’13

Fractional Case

(−∆)su + f (x , u) = 0 in Ω ⊂ Rn

E(u) =

∫|(−4)s/2u|2 +

∫F (u, x)

Nonlinear Setting:

Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0Weinstein ’84, Albert-Bona-Saut ’97, ...

Symmetry Every positive solution u(x) > 0 solving

(−4)su − un+2sn−2s = 0 in Rn

is radial (mod translation). Proof by moving planes for integral equation.Y.Y. Li ’04, Chen-Li-Ou ’06, Ma-Zhao ’10, ...

Uniqueness and Nondegeneracy of Ground States u(x) > 0. ?? See nextslide

Fractional Case

(−∆)su + f (x , u) = 0 in Ω ⊂ Rn

E(u) =

∫|(−4)s/2u|2 +

∫F (u, x)

Nonlinear Setting:

Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0Weinstein ’84, Albert-Bona-Saut ’97, ...

Symmetry Every positive solution u(x) > 0 solving

(−4)su − un+2sn−2s = 0 in Rn

is radial (mod translation). Proof by moving planes for integral equation.Y.Y. Li ’04, Chen-Li-Ou ’06, Ma-Zhao ’10, ...

Uniqueness and Nondegeneracy of Ground States u(x) > 0. ?? See nextslide

Uniqueness and Nondegenarcy for (−4)s

Amick-Toland ’91: Uniqueness (mod translations) of Q(x) > 0 solving

(−4)1/2Q + Q − Q2 = 0 in R

Proof by complex analysis. “Magic identities”. Very rigid.

Y.Y. Li ’04, Chen-Li-Ou ’06: Uniqueness (mod translations/scalings) ofQ(x) > 0 solving

(−4)sQ − Qn+2sn−2s = 0 in Rn

Use of conformal symmetry. Very rigid.

Frank-Lenzmann ’11, Frank-Lenzmann-Silvestre ’13: Uniqueness andNondegeneracy of ground states Q(x) > 0 solving

(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1

with 0 < α < α∗(s, n).Proof by estimates. New (robust) methods. Substitutes of ODE results.

(−4)1/2Q + Q − Q2 = 0 in R

(−4)sQ − Qn+2sn−2s = 0 in Rn

(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1

(−4)1/2Q + Q − Q2 = 0 in R

(−4)sQ − Qn+2sn−2s = 0 in Rn

(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1

The Three Methods

Nonlocal Problem

(−4)su + f (u, x) = 0 in Rn

1 Topological Bounds: For s-harmonic Extension U(x , t) show that

# Nodal Domains of U ≤ C

Yields bounds on # sign changes of U = u on ∂Rn+1+ .

2 “Continuity Argument”: Take s ∈ (0, 1) as parameter and consider

(−4)sus + fs(us , x) = 0

and try to take limit s → 1.

3 Monotonicity Formula: For radial u(r), obtain monotone quantity

H(r) = cs

∫ ∞0

(∂rU(r , t))2 − (∂tU(r , t))2t1−2sdt − F (u(r), r)

Use the fact that H′(r) ≤ 0. Hamiltonian Estimate. Cabre-Sola-Morales

’05, Cabre-Sire ’12, Frank-Lenzmann-Silvestre ’13

The Three Methods

Nonlocal Problem

(−4)su + f (u, x) = 0 in Rn

(−4)sus + fs(us , x) = 0

H(r) = cs

∫ ∞0

The Three Methods

Nonlocal Problem

(−4)su + f (u, x) = 0 in Rn

(−4)sus + fs(us , x) = 0

H(r) = cs

∫ ∞0

New Results in Linear Case

Linear Problem

(−4)su + Vu = 0 in Rn

New Results:

Cauchy-Lipschitz Theorem: For suitable V and radial u(r), we have

u(0) = 0 ⇒ u ≡ 0

Simplicity of all radial eigenvalues of H = (−4)s + V .

Sturm Oscillation: Estimates number of zeros for radial eigenfunctions ukof

(−4)suk + Vuk = λkuk .

Key to nondegeneracy proof for nonlinear ground states.

New Tools for Nonlocal Elliptic Problemsstolo/conf/hiver14/MiniCourseNew.pdf · !Physics & Maths 2...

Documents

Transcript of New Tools for Nonlocal Elliptic Problemsstolo/conf/hiver14/MiniCourseNew.pdf · !Physics & Maths 2...