Game Theoretical Methods in PDEs seminars... · 2020-03-11 · Linear PDEs (D) and...

Game Theoretical Methods in PDEsTutorial

Marta Lewicka

University of Pittsburgh

1

Linear PDEs (∆) and probability

• Initial position of token: x0 ∈Ω⊂ R2.

• Moves: random increments of length ε.

• uε(x0): probability of hitting Oε(Γ1),

the first time Oε(∂Ω) is hit.

THM uε ⇒ v∈ C (Ω) as ε→ 0, and v is a viscosity solution to ∆v = 0:

∆φ(x0)≤ 0 ∆φ(x0)≥ 0

In fact, v is the classical solution to:∆v = 0 in Ω

v = 1 on Γ1 and v = 0 on ∂Ω\Γ1

2

Proof that v is a viscosity solution

uε(x) = 14(uε(x+ εe1)+uε(x− εe1)+uε(x+ εe2)+uε(x− εe2))

⇔ 0 = (uε(x+ εe1)−uε(x))+(uε(x− εe1)−uε(x))

+(uε(x+ εe2)−uε(x))+(uε(x− εe2)−uε(x))

• uε ⇒ v in Ω as ε→ 0 (can be proven rigorously).

Take φ smooth, (v−φ)(x0) = 0≤ (v−φ)(x) ∀x.

Find xε→ x0: (uε−φ)(xε)≤ (uε−φ)(x)+ε3 ∀x

⇔ uε(x)−uε(xε)≥ φ(x)−φ(xε)− ε3

• 0≥ (φ(xε+ εe1)−φ(xε))+(φ(xε− εe1)−φ(xε))

+(φ(xε+ εe2)−φ(xε))+(φ(xε− εe2)−φ(xε))−4ε3

(Taylor) = ε2∂11φ(xε)+ ε2∂22φ(xε)+o(ε2) = ε2∆φ(xε)+o(ε2)

• Conclude: 0≥ ∆φ(xε)+o(1). Hence: ∆φ(x0)≤ 0.3

Non-linear PDEs (∆p)

∆pu = div(|∇u|p−2∇u) = |∇u|p−2∆u+⟨

∇(|∇u|p−2) : ∇u⟩

= |∇u|p−2(

∆u+(p−2)〈∇2u : ∇u|∇u|⊗

∇u|∇u|〉

)= |∇u|p−2(∆u+(p−2) ∆∞u)

• Taylor expand: u(x) = u(x0)+ 〈∇u(x0),x− x0〉+ 1

2〈∇2u(x0) : (x− x0)⊗ (x− x0)〉+o(ε2)

Average:ffl

Bε(x0)u = u(x0)+

12〈∇

2u(x0) :ffl(x− x0)⊗ (x− x0)〉

= u(x0)+12〈∇

2u(x0) : ε2

4 Id〉= u(x0)+ε2

8 ∆u+o(ε2)[• Thm (Manfredi-Parviainen-Rossi 2012)

Let u ∈ C (Ω) satisfy: ∀x0 ∈Ω u(x0) =ffl

Bε(x0)u+o(ε2). Then ∆u = 0 in Ω.

]• 1

2(supBε(x0)u+ infBε(x0)

u)≈ 12

(u(x0+ ε

∇u(x0)|∇u(x0)|

)+u(x0− ε∇u(x0)|∇u(x0)|

))

= u(x0)+ε2

2

⟨∇2u(x0) : ∇u(x0)

|∇u(x0)|⊗ ∇u(x0)|∇u(x0)|

⟩+o(ε2)

= u(x0)+ε2

2 ∆∞u(x0)+o(ε2)4

Non-linear PDEs (∆p)

∆pu = div(|∇u|p−2∇u) = |∇u|p−2∆u+⟨

∇(|∇u|p−2) : ∇u⟩

= |∇u|p−2(

∆u+(p−2)〈∇2u : ∇u|∇u|⊗

∇u|∇u|〉

)= |∇u|p−2(∆u+(p−2) ∆∞u)

• We have:ffl

Bε(x0)u = u(x0)+

ε2

8 ∆u+o(ε2)

• 12(supBε(x0)

u+ infBε(x0)u) = u(x0)+

ε2

2 ∆∞u(x0)+o(ε2)

Hence: 4p+2

fflBε(x0)

u+ p−2p+2

12(supBε(x0)

u+ infBε(x0)u)

= u(x0)+ε2

2(p+2)(∆u+(p−2)∆∞u)+o(ε2)

Concluding: If ∆pu = 0 then:

u(x0) =α2 supBε(x0)

u+ α2 infBε(x0)

u+βffl

Bε(x0)u+o(ε2)

(Ω⊂ R2, 2≤ p < ∞) (α = p−2p+2, α+β = 1)

5

Tug-of-War game for ∆p

X = Ω∪Γε. Rules of the game: • Token initially at x0 ∈Ω.

• probability α2 → player I moves by at most ε

• probability α2 → player II moves by at most ε

• probability β→ computer

moves randomly by at most ε

Game stops, when token position xn ∈ Γε

Then player II pays F(xn) to player I. Boundary data: F : Γε→ R.

Strategies: σI = σkI : Xk+1→ X∞k=1, σII = σk

II : Xk+1→ X∞k=1Borel functions, σk

I,II(x0, . . .xk) ∈ Bε(xk),

and σkI,II(x0, . . .xk) = xk if xk ∈ Γε.

.6


X = Ω∪Γε. Strategies: σI,II = σkI,II : Xk+1→ X∞k=1,

Probability measures:

γk[x0, . . .xk] =α2δ

σkI (x0,...xk)

+ α2δ

σkII(x0,...xk)

+β

LbBε(xk)|Bε| if xk ∈Ω

δxk if xk ∈ Γε

• Probabilities Px0,kσI,σII :

Px0,kσI,σII(A1× . . .Ak) =

´A1

. . .´

Ak1 dγk−1[x0, . . .xk−1] . . . dγ0[x0].

• They generate (Kolmogoroff extension) probability Px0σI,σII

on sets of histories (x0,x1, . . .) ∈ X∞.

• Define the expectation: Ex0σI,σII[Fτ] =

´X∞ F(xτ) dPx0

σI,σII

where τ = mink; xk ∈ Γε and F(xτ) =

F(xk) if τ = k+∞ if τ =+∞

7


minimum gain of player I: uI(x0) = supσIinfσII E

x0σI,σII[Fτ]

maximum loss of player II: uII(x0) = infσII supσIEx0

σI,σII[Fτ]

THM 1 (Peres-Schramm-Sheffield-Wilson 2008 (p = ∞))(Peres-Sheffield 2008 (1 < p < ∞))(Manfredi-Parviainen-Rossi 2012 (2≤ p < ∞))

uI = uII = uε := the unique p-harmonious function: u(x) =α

2sup

Bε(x)u+

α

2inf

Bε(x)u+β

Bε(x)

u in Ω

u = F in Γε

THM 2 When ε→ 0 then uε ⇒ v the unique viscosity solution to:∆pv = 0 in Ω

u = F on ∂Ω

8

Plan for the lecture

• complete proofs of Theorem 1 and Theorem 2

• proof of the Harnack inequality for ∆pv = 0,

using tug-of-war game interpretation

• limiting cases p→ ∞ and p→ 1

• p = ∞: infinity Laplacian ∆∞v = 0

and absolutely minimizing Lipschitz extensions

• p = 1: motion by mean curvature vt−|∇v|div( ∇v|∇v|) = 0

and “geometric” equations, Kohn-Serfaty game

9

Basic facts on ∆pv = div(|∇v|p−2∇v), 1 < p < ∞

• p = 2: Laplacian, p > 2: degenerate elliptic, 1 < p < 2: singular

• Euler-Lagrange eqn for: Ip(v) = 1p´

Ω|∇v|p on v ∈W 1,p(Ω)

⇒ existence and uniqueness of a weak solution to:

∆pv = 0v|∂Ω = g

• weak solutions are C 1,αloc , in general not C 2 when ∇v(x) = 0

(Uraltseva: p > 2, Lewis, DiBenedetto: 1 < p < 2),

• a continuous v is called a viscosity solution if:

∇φ(x0) 6= 0⇒ ∆pφ(x0)≤ 0

∇φ(x0) 6= 0⇒ ∆pφ(x0)≥ 0

• weak solutions = viscosity solutions (Juutinen-Lindqvist-Manfredi 2001)

• Open problems: If v|Bρ= 0 v≡ 0 in Ω

?⇒(true for n = 2 by complex analysis methods: Manfredi 1988)

10

Proof of Thm 1 (β 6= 0): existence of p-harmonious u(after Luiro, Parviainen, Saksman 2013)

Let u : X → R bounded, Borel. Look for fixed points of the operator:

(Tu)(x) =

α2 supBε(x)u+ α

2 infBε(x)u+βffl

Bε(x)u in Ω

u(x) in Γε

• Tu is also bounded, Borel. • Let u0 =

infΓε

F in Ω

F in Γε

• Then u1 = Tu0 ≥ u0 and u2 = Tu1 ≥ Tu0 = u1 and ... u j+1 ≥ u j

Hence u j u. In fact u j ⇒ u in Ω.

• So: u ⇔ u j+1 = Tu j ⇒ Tu, so Tu = u and u = F on Γε.

.11



(Tu)(x) =

α2 supBε(x)u+ α

2 infBε(x)u+βffl

Bε(x)u in Ω

u(x) in Γε


infΓε

F in Ω

F in Γε




• Uniqueness: If u 6= v are two solutions, call M = sup(u− v)> 0.

∀x u(x)− v(x)= α2(supBε(x)u−supBε(x)v)+α

2(infBε(x)u−infBε(x)v)

+βffl

Bε(x)(u−v)≤αsupBε(x)(u−v)+βffl

Bε(x)(u−v)

≤ αM+βffl

Bε(x)(u− v)

.12



(Tu)(x) =

α2 supBε(x)u+ α

2 infBε(x)u+βffl

Bε(x)u in Ω

u(x) in Γε


infΓε

F in Ω

F in Γε




• Uniqueness: If u 6= v are two solutions, call M = sup(u− v)> 0.

∀x u(x)− v(x)≤ αM+βffl

Bε(x)(u− v).

If u(xn)−v(xn)→M and xn→ x0∈X then: M ≤ αM+βffl

Bε(x0)(u− v).

But β = 1−α, so: M ≤ffl

Bε(x0)(u− v). Hence u− v≡M in Bε(x0).

By extension: u− v≡M in X , contradicting u = v on Γε.

13

Proof of Thm 1: u is the game value

Lemma Let u : Ω→ R be bounded and Borel. Fix ε,δ > 0.

There exists suboptimal selections Ssup,Sin f : Ω→Ω, such that:

• Ssup,Sin f are Borel functions, and ∀x ∈Ω Ssup(x),Sin f (x)∈ Bε(x)

• ∀x ∈Ω u(Ssup(x))≥ supBε(x)u−δ, u(Sin f (x))≤ infBε(x)u+δ

Example Ssup may not exist with closed balls Bε(x) !

Let A⊂R3 be bounded, Borel, and such that: A+ B1(0) is not Borel.

Let u = χA. Assume that: ∀x u(Ssup(x))≥ supB1(x)u−12.

Then: (Ssup)−1(A) = x ∈ R3; supB1(x)u = 1= A+ B1(0)

Therefore Ssup cannot be a Borel function.

14


Recall: uI(x0) = supσIinfσII E

x0σI,σII[Fτ],

uII(x0) = infσII supσIEx0

σI,σII[Fτ].Clearly: uI ≤ uII.

• We show that uII ≤ u. Symetrically u≤ uI. Then uI = uII = u.

• Fix η > 0. Choose σII suboptimal selection, so that:

u(σkII(x0 . . .xk))≤ infBε(xk)

u+ η

2k+1

• Let σI be any strategy for player I.

uII(x0)≤ supσIEx0

σI,σII[Fτ]≤ supσI

Ex0σI,σII

[u(xτ)+

η

2τ

]≤ supσI

Ex0σI,σII

[u(x0)+

η

20

]= u(x0)+η

⇑

because

u(xk)+η

2k, (Borel σ-algebra)k

is a supermartingale.

15


Lemma

u(xk)+η

2k, F x0k


Proof Ex0σI,σII

u(xk)+

η

2k | Fx0

k−1

(x0 . . .xk−1) =

=´

X(u+η

2k) dγk−1[x0 . . .xk−1]

= α2 u(σI(x0 . . .xk−1))+

α2 u(σII(x0 . . .xk−1))+β

fflBε(xk−1)

u dy+ η

2k

≤ α2 supBε(xk−1)

u+ α2

(infBε(xk−1)

u+ η

2k

)+β

fflBε(xk−1)

u dy+ η

2k

= u(xk−1)+(α2 +1) η

2k ≤ u(xk−1)+η

2k−1

Theorem (Doob’s optional stopping)

If fk,Fk is a supermartingale and τ1,τ2 are bounded stopping times,

then: τ1 ≤ τ2 ⇒ E[ fτ2]≤ E[ fτ1].

16

p-harmonious u and the martingale calculation

Similarly, one can prove:

Lemma Let σI, σII be any two fixed strategies. Then:

infσII Ex0σI,σII

[u(xτ∗)]≤ u(x0)≤ supσIEx0

σI,σII[u(xτ∗)]

for every stopping time τ∗ ≤ τ.

Proof Use the suboptimal selection strategy σII:

infσII Ex0σI,σII

[u(xτ∗)]≤ Ex0σI,σII

[u(xτ∗)]≤ Ex0σI,σII

[u(xτ∗)+

η

2τ∗]

≤ Ex0σI,σII

[u(x0)+

η

20

]= u(x0)+η

because

u(xk)+η

2k, F x0k


17

Proof of Thm 2: uε ⇒ v continuous(after Manfredi, Parviainen, Rossi 2012)

Lemma (∼ Ascoli-Arzela theorem for discontinuous functions)

Let uε : Ω→ R be a sequence of functions such that:

(i) ∃C ∀x ∈ Ω ∀ε |uε(x)| ≤C

(ii) ∀η > 0 ∃r0 ∃ε0 ∀x,y ∈ Ω|x− y|< r0

∀ε < ε0 |uε(x)−uε(y)|< η

Then uε ⇒ v in Ω and v is a continuous function.

• Take uε : Ω∪Γε→ R to be the unique p-harmonious functions

with parameter ε and boundary data F in Γε

• uε is equibounded. Need to verify condition (ii).

• Case 1: x ∈Ω, y ∈ ∂Ω ⇒ Case 2: x,y ∈Ω.

18

Proof of Thm 2: verification of (ii) for x0 ∈Ω, y ∈ ∂Ω

(after Manfredi, Parviainen, Rossi 2012)

|uε(x0)−uε(y)|= |uε(x0)−F(y)| ≤ |uε(x0)−F(z)|+Lδ

= |supσI

infσII

Ex0σI,σII[F(xτ)−F(z)]|+Lδ≤ Lsup

σIinfσII

Ex0σI,σII[|xτ− z|]+Lδ

Take any σI and σII = σzII “pulling towards z”

σzII(xk) =

xk+ ε

z−xk|z−xk|

if xk 6∈ Bδ(z)xk if xk ∈ Bδ(z)

Compute: Ex0σI,σ

zII

|xk− z|−Cε2k | F x0

k−1

(x0 . . .xk−1)

≤ α2(|xk−1− z|+ε)+ α

2(|xk−1− z|−ε)+βffl

Bε(xk−1)|s− z| ds−Cε2k

= α|xk−1− z|+β

(|xk−1− z|+ ε2

8 ∆(|s− z|)+o(ε2))−Cε2k

≤ |xk−1− z|−Cε2(k−1) for C >> 1.

Hence:|xk− z|−Cε2k,F x0

k


19



By Doob’s Thm: Ex0σI,σ

zII[|xτ− z|−Cε2τ]≤ Ex0

σI,σzII[|x0− z|] = |x0− z|

Hence: Ex0σI,σ

zII[|xτ− z|]≤ |x0− z|+Cε2Ex0

σI,σzII[τ]

Consider:∆g =−2(n+2) in BR(z)\ Bδ(z)g = 0 on ∂Bδ(z)∂g∂~n = 0 on ∂BR(z)

i.e: g(s) = g(|s− z|) and g(r) =−ar2−b logr+ c

g satisfies: ∀Bε(s0)⊂ annulusffl

Bε(s0)g(s) ds = g(s0)− ε2

• Compute: Ex0σI,σ

zII

g(|xk− z|)+Cε2k | F x0

k−1

(x0 . . .xk−1)

≤ α2g(|xk−1−z|+ε)+α

2g(|xk−1−z|−ε)+β(g(|xk−1− z|)− ε2)+Cε2k

≤ g(|xk−1−z|)+Cε2(k−1) because g is increasing and concave.

20



Hence, Doob’s Thm yields: Ex0σI,σ

zII[g(|xτ∗−z|)+Cε2τ∗]≤ g(|x0−z|)

where τ∗ is the exit time into the inner ball Bδ(z). Clearly: τ≤ τ∗

Hence:

Ex0σI,σ

zII[Cε2τ]≤ Ex0

σI,σzII[Cε2τ∗]

≤ g(|x0− z|)−Ex0σI,σ

zII[g(|xτ∗− z|)]

≤ Lg dist(x0,∂Bδ(z))+Lgε≤ Lg(|x0− z|+ ε)

Recall: Ex0σI,σ

zII[|xτ− z|]≤ |x0− z|+Cε2Ex0

σI,σzII[τ] ≤Cδ(|x0−z|+ε)

• Concluding: |uε(x0)−uε(y)| ≤Cδ,Ω (|x0− y|+ ε)+CFδ

which gives exactly condition (ii)!

21

Proof of Thm 2: the limit satisfies ∆pv = 0(after Manfredi, Parviainen, Rossi 2012)

We proved that: uε ⇒ v in Ω. Clearly v = F on ∂Ω

Verification that v is a viscosity solution:Find xε→ x0 so that: (uε−φ)(xε)≤ (uε−φ)(x)+ ε3 ∀x

• uε(xε) =α2 supBε(xε)uε+

α2 infBε(xε)uε+β

fflBε(xε)

uε

• φ(xε) =α2 supBε(xε)φ+ α

2 infBε(xε)φ+βffl

Bε(xε)φ

−βε2

8 (∆φ(xε)+(p−2)∆∞φ(uε))+o(ε2)

Hence: (uε−φ)(xε)=α2(supuε−supφ)+α

2(infuε− infφ)+βffl(uε−φ)

+ βε2

81

|∇φ|p−2∆pφ(xε)+o(ε2)

≥ (uε−φ)(xε)+βε2

81

|∇φ|p−2∆pφ(xε)+o(ε2).

Consequently: 1|∇φ|p−2∆pφ(xε)≤ o(1).

Since ∇φ(x0) 6= 0, we get: ∆pφ(x0)≤ 0.

22

Harnack’s inequality for p-harmonious functions (2≤ p < ∞)(after Luiro, Parviainen, Saksman 2013)

THM 3 For each ε, let uε be a p-harmonious function in Ω. Then:

∀Bρ(z0)⊂ Br(z0)B2r(z0)⊂Ω

∃Cp ∀ε < ρ osc(uε,Bρ)≤ Cpρ

r osc(uε,Br)

Corollary Let uε be a positive p-harmonious function in B2r(z0).

Then there exists C (independent of ε) so that:

∀Bρ(x0)⊂ Br(z0) supBρuε ≤ C infBρ

uε

Corollary Harnack’s inequality for p-harmonic v: supBρv≤ C infBρ

v

Prior proofs of Harnack’s inequality:

• Ladyzhenskaya-Uraltseva 1968, Serrin 1967

• Nash-Moser iteration, DeGiorgi classes (DiBenedetto-Trudinger 1984)

• Here: completely different proof!

23

Proof of Thm 3: Harnack’s inequality(after Luiro, Parviainen, Saksman 2013)

Let x0,y0 ∈ Br(z0). Let z ∈ B2r: |x0− z|= mε = |y0− z|, m = b|x0−y0|εc.

Define the strategy σzII:

• player II cancels the earliest move of player I

• otherwise player II “pulls towards z” (by ε)

Position of token:xk+1 = x0+∑i∈Jk

Iwi+∑i∈Jk

IIwi+∑i∈Jk

randomwi

Define the stopping time τ∗II = mink; (i),(ii),(iii):(i) player II has won m times more than player I: |Jk

II|= |JkI |+m

(ii) player I has won d2rεe times more than player II: |Jk

I |= |JkII|+d

2rεe

(iii) total random moves are large: |∑i∈Jkrandom

wi| ≥ 2r.

In particular: xτ∗II∈ B6r(z0). Also: τ∗II ≤ τ, so (by lemma):

u(x0)−u(y0)≤ supσI,σII

(Ex0

σI,σzII[uτ∗II

]−Ey0σ

zI,σII

[uτ∗I]

)24

Proof of Thm 3: Harnack’s inequality(after Luiro, Parviainen, Saksman 2013)

Fix σI,σII. Then: Ex0σI,σ

zII[uτ∗II

]−Ey0σ

zI,σII

[uτ∗I]

= Ex0σI,σ

zII[uτ∗II

χgame endsby (i)

]+Ex0σI,σ

zII[uτ∗II

χgame endsby (ii) or (iii)

]

−Ey0σ

zI,σII

[uτ∗Iχ

game endsby (i)

]−Ey0σ

zI,σII

[uτ∗Iχ

game endsby (ii) or (iii)

]

= Ex0σI,σ

zII[uτ∗II

χgame endsby (ii) or (iii)

]−Ey0σ

zI,σII

[uτ∗Iχ

game endsby (ii) or (iii)

]

≤ P ( supB6r(z0)

u− infB6r(z0)

u)≤ Cp|x0−y0|

r ( supB6r(z0)

u− infB6r(z0)

u)

where P = probability that game ends by (ii) or (iii).

• Concluding: |u(x0)−u(y0)| ≤Cp|x0−y0|

r ( supB6r(z0)

u− infB6r(z0)

u)

⇒ osc(u,Bρ)≤Cpρ

6r(sup

B6r

u− infB6r

u)

25

Proof of Thm 3: P≤Cp|x0−y0|

r(after Luiro, Parviainen, Saksman 2013)

To estimate P, couple the game with a cylinder walk:

Rules of the walk:

• token initially at (z,mε)

• probability α2 → player I moves by (0,ε)

• probability α2 → player II moves by (0,−ε)

• probability β→ random move by (y,0), y ∈ Bε(0)

Lemma P = probability (the walk does not exit the cylinderthrough its bottom)

≤Cpmε+ ε

2r

26

The limiting case p→ ∞ (β = 0)

Theorem (Bhattacharya-DiBenedetto-Manfredi 1989, Jensen 1993)

Let vp be the unique solution to: ∆pvp = 0, vp|∂Ω= F.

Then vp ⇒ v∞ in Ω as p→∞, and v∞ is the unique viscosity solution to:

∆∞v∞ = 0, v∞|∂Ω = F

∇φ(x0) 6= 0⇒

〈∇2φ : ∇φ

|∇φ|⊗∇φ

|∇φ|〉(x0)≤ 0

∇φ(x0) 6= 0⇒ ∆∞φ(x0)≥ 0

Theorem (Aronsson, Aronsson-Crandall-Juutinen 1967, 2004)

v∞ ∈ Lip(Ω,R) has the uniquely defining property:

∀ open V ⊂Ω ∀v : V → Rv|∂V = v∞|∂V

Lip(v∞,V )≤ Lip(v,V )

• v∞ is AMLE (Absolutely Minimizing Lipschitz Extension)

“ ∆∞v = 0 is the Euler-Lagrange equation of ‖∇v‖∞”27

The limiting case p = ∞

• ∆∞v = 0 is a fully nonlinear equation,

only viscosity solution definition valid (no integration by parts)

• n = 2: v∞ ∈ C 1,α (Savin, Savin-Evans 2008),

n > 2: v∞ differentiable everywhere (Evans-Smart 2011)

Thm (Peres-Schramm-Sheffield-Wilson 2008)

uε ⇒ v∞ as ε→ 0 where uε are values of the Tug-of-War game:uε(x) = 1

2 supBε(x)uε+12 infBε(x)uε in Ω

uε = F on Γε

An AMLE extension

is inifinity harmonic!

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The limiting case p→ 1Thm (Peres-Sheffield 2008)Let 1 < p < ∞. Then the values uε of the following game converge,as ε→ 0 to v with: ∆pv = 0, v|∂Ω = F .

• token initially at x0 ∈Ω.

• the winning player (probability 12) moves token

by vector w (|w|< ε)

• token moved randomly by (p−1)−1/2|w|in the direction orthogonal to w

Game stops when token position in Γε. Player II pays F(xn) to player I.

Observe that the following structure is needed to define the game:

• p = ∞ metric (extensions to length spaces by PSSW)• 2≤ p < ∞ measure (extensions to Heisenberg groups

(Carnot-Caratheodory groups) by Ferrari-Liu-Manfredi)• 1 < p < 2 perpendicularity: Riemannian structure

(other extensions: Barron-Evans-Jenssen)29

The limiting case p→ 1Thm (Juutinen 2005, Sternberg-Williams-Ziemer 1992)

Let Ω be convex. Let vp be unique solutions to: ∆pvp = 0, vp|∂Ω=F.

Then vp ⇒ v1 in Ω as p→ 1, and v1 is the function of least gradient:

v1 ∈ BV (Ω)∩C (Ω) and v1 minimizes ‖∇v‖BV (Ω)

Moreover, v1 is a viscosity solution to ∆1:

∆1v1 = div( ∇v1|∇v1|

) = 0, v1|∂Ω = F

However, there exist viscosity solutions to ∆1, other than v1.

Motion by curvature and ∆1

Level set v = t is the image of ∂Ω under motion by curvature for time t

• curvature of the level set: κ =−div( ∇v|∇v|)

• |∇v|= |∇v ·~n|= ∂v∂~n ≈

∆t∆x

• κ = ∆x∆t ⇔ −∆1v = 1

|∇v||∇v|∆1v+1 = 0, v|∂Ω = 0

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Motion by mean curvature

The Paul/Carol game in R2: • token initially at x0 ∈ R2

• given t0 > 0, the length of game N = b t0ε2c steps

• player I (Paul) chooses wk, |wk|= 1

• player II (Carol) chooses bk =±1

• token moved to xk+1 = xk+√

2ε bkwk

After N steps, Paul pays u0(xN) to Carol.

Game value:

uε(x0, t0) = minw1

maxb1

. . . minwN

maxbN

u0(xN)

uε(x0,0) = u0(x0)

Thm (Kohn-Serfaty 2006) Let u0 ∈ Cc(R2). Then uε ⇒ v in R2× [0,∞)

locally uniformly as ε→ 0, and v is the unique viscosity solution of:vt−|∇v|∆1v = 0 in R2× (0,∞)

v(x,0) = u0(x) in R2

• existence/uniqueness: Evans-Spruck, Chen-Giga-Goto 1991.31

Conclusions

• ∃ connection between: potential theory/nonlinear PDEs,and probability theory/ stochastic tug-of-war games

• based on: harmonic/p−harmonic functions and martingalesshare a common cancellation property, via mean value property

• variety of PDEs and other problems:– Harnack inequality (Luiro-Parviainen-Saksman)– continuous uε, finite difference scheme (Armstrong-Smart)– obstacle problem (Manfredi-Rossi-Somersille, L-Manfredi)– Neumann boundary condition (Antunovic-Peres-Sheffield-Somersille)

• in fact: ∃ game interpretation for a large class of elliptic and parabolic,nonlinear equations (Kohn-Serfaty 2010)

• advantage: direct proofs by assigning “strategies”⇒ “inequalities”.Probability tools used for nonlinear equations.

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THANK YOU.

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Game Theoretical Methods in PDEs seminars... · 2020-03-11 · Linear PDEs (D) and...

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