1
Rates of convergence for approximations
of viscosity solutions and homogenization
Panagiotis E. Souganidis
The University of Texas at Austin
Nonlinear approximation techniques using L1
Texas A&M University
May 2008
2
F(D2 u, Du, u, x) = 0 in U
3
F(D2 u, Du, u, x) = 0 in U
numerical approximations
monotone, stable, consistent approximations
Fh (D2h uh , Dh uh , uh , x) = 0 in Uh
uh → u
and
mesh size h, numerical nonlinearity Fh ,
solution uh , domain Uh = U ∩ ZNh
uh − u = O(σ(h))
4
F(D2 u, Du, u, x) = 0 in U
numerical approximations
monotone, stable, consistent approximations
Fh (D2h uh , Dh uh , uh , x) = 0 in Uh
uh → u
and
uh − u = O(σ(h))
mesh size h, numerical nonlinearity Fh ,
solution uh , domain Uh = U ∩ ZNh
homogenization
x
F(D2 uε , Duε , uε , , ω) = 0 in U
ε
uε → u0
and
F0 (D2 u0 , Du0 , u0 ) = 0 in U
uε − u0 = O(σ(ε))
5
NUMERICAL APPROXIMATIONS
6
NUMERICAL APPROXIMATIONS
monotone schemes
convergence
uh −→ u
rate of convergence
ku − uh k = O(hα )
first-order (Hamilton-Jacobi)
α = 1/2
Crandall-Lions, Souganidis
α = 1/27
α = 1/5
α = 1/2
α small
Krylov
second-order
convex, deg. elliptic
uniformly elliptic
h→0
Crandall-Lions, Souganidis
Barles-Souganidis
Barles-Jakobsen
Krylov
Caffarelli-Souganidis
7
NUMERICAL APPROXIMATIONS
monotone schemes
convergence
uh −→ u
rate of convergence
ku − uh k = O(hα )
first-order (Hamilton-Jacobi)
α = 1/2
Crandall-Lions, Souganidis
α = 1/27
α = 1/5
α = 1/2
α small
Krylov
second-order
convex, deg. elliptic
uniformly elliptic
h→0
Crandall-Lions, Souganidis
Barles-Souganidis
Barles-Jakobsen
Krylov
Caffarelli-Souganidis
non-monotone schemes
few convergence results for non monotone schemes
TVD filtered and higher-order schemes
Osher-Tadmor
Lions-Souganidis
8
STOCHASTIC HOMOGENIZATION
F(D2 uε , Duε , εx , ω) = 0
F
uniformly elliptic stationary ergodic*
* stationarity: f (y, ω) is stationary if µ({ω : f (y, ω) > α}) independent of y
ergodicity:
all translation invariant quantities are constant a.s. in ω
9
STOCHASTIC HOMOGENIZATION
F(D2 uε , Duε , εx , ω) = 0
convergence
F
uniformly elliptic stationary ergodic*
uε (·, ω) → u0 a.s.
&
F0 (D2 u0 , Du0 ) = 0
linear
Papanicolaou-Varadhan, Kozlov
nonlinear
Caffarell-Souganidis-Wang
* stationarity: f (y, ω) is stationary if µ({ω : f (y, ω) > α}) independent of y
ergodicity:
all translation invariant quantities are constant a.s. in ω
10
STOCHASTIC HOMOGENIZATION
F(D2 uε , Duε , εx , ω) = 0
convergence
F
uniformly elliptic stationary ergodic*
uε (·, ω) → u0 a.s.
F0 (D2 u0 , Du0 ) = 0
&
linear
Papanicolaou-Varadhan, Kozlov
nonlinear
Caffarell-Souganidis-Wang
rates of convergence
strongly mixing media with algebraic rate
linear
kuε − u0 k = O(εγ ) a.s.
nonlinear
Yurinskii
−c| ln ε|1/2
kuε − u0 k = O(e
) off
−c| ln ε|1/2
a set with probability O(e
)
* stationarity: f (y, ω) is stationary if µ({ω : f (y, ω) > α}) independent of y
ergodicity:
all translation invariant quantities are constant a.s. in ω
Caffarelli-Souganidis
11
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
12
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
13
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
approximation scheme
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
S([uh ]x , uh (x), x, h) = 0
14
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
approximation scheme
monotone
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
S([uh ]x , uh (x), x, h) = 0
u≥v
=⇒
S([u]x , s, x, h) ≦ S([v]x , s, x, h)
15
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
approximation scheme
monotone
stable
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
S([uh ]x , uh (x), x, h) = 0
u≥v
=⇒
kuh k ≦ C
S([u]x , s, x, h) ≦ S([v]x , s, x, h)
independent of h
16
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
approximation scheme
monotone
stable
consistent
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
S([uh ]x , uh (x), x, h) = 0
u≥v
=⇒
S([u]x , s, x, h) ≦ S([v]x , s, x, h)
kuh k ≦ C
independent of h
S([φ + ξ]x , φ(y) + ξ, y, h) −→ F(D2 φ(x), Dφ(x), φ(x), x)
h→0
y→x
ξ→0
(φ smooth)
17
CONVERGENCE OF MONOTONE APPROXIMATIONS
(F) F(D2 u, Du, u, x) = 0
F degenerate elliptic
approximation scheme
monotone
stable
consistent
Theorem:
(X ≦ Y =⇒ F(X, p, r, x) ≧ F(Y, p, r, x))
S([uh ]x , uh (x), x, h) = 0
u≥v
S([u]x , s, x, h) ≦ S([v]x , s, x, h)
=⇒
kuh k ≦ C
independent of h
S([φ + ξ]x , φ(y) + ξ, y, h) −→ F(D2 φ(x), Dφ(x), φ(x), x)
h→0
y→x
ξ→0
uh −→ u
h→0
u solution of (F)
(φ smooth)
18
Proof
19
Proof
S([uh ]x , uh (x), x, h) = 0
20
Proof
S([uh ]x , uh (x), x, h) = 0
u∗ (x) = lim
uh (y)
y→x
stability
=⇒
h→0
u∗ (x) = lim uh (y)
y→0
h→0
exist
21
Proof
S([uh ]x , uh (x), x, h) = 0
u∗ (x) = lim
uh (y)
y→x
stability
h→0
=⇒
exist
u∗ (x) = lim uh (y)
y→0
h→0
u∗ subsolution
monotonicity
=⇒
consistency
of
u∗ supersolution
F(D2 u, Du, u, x) = 0
22
Proof
S([uh ]x , uh (x), x, h) = 0
u∗ (x) = lim
uh (y)
y→x
stability
h→0
=⇒
exist
u∗ (x) = lim uh (y)
y→0
h→0
u∗ subsolution
monotonicity
=⇒
consistency
comparison for
definition of
of
F(D2 u, Du, u, x) = 0
u∗ supersolution
(F) =⇒
u∗ ≦ u∗
u∗ , u∗ =⇒
u∗ ≦ u∗
)
u∗ = u∗ = u solution of (F)
and
=⇒
uh −→ u
h→0
23
∗
u subsolution
iff
(
for all smooth φ and all max x of u∗ − φ
F(D2 φ(x), Dφ(x), u∗ (x), x) ≦ 0
24
∗
u subsolution
iff
(
for all smooth φ and all max x of u∗ − φ
F(D2 φ(x), Dφ(x), u∗ (x), x) ≦ 0
fix φ smooth
x0
max of u∗ − φ and
u∗ (x0 ) = φ(x0 )
25
∗
u subsolution
iff
(
for all smooth φ and all max x of u∗ − φ
F(D2 φ(x), Dφ(x), u∗ (x), x) ≦ 0
fix φ smooth
x0
max of u∗ − φ and
“uh → u∗ ” =⇒
u∗ (x0 ) = φ(x0 )
xh max of
uh − φ
=⇒
uh ≦ φ − ξh
xh → x0 and ξh = uh (xh ) − φ(xh ) → 0 as h → 0
26
∗
u subsolution
iff
(
for all smooth φ and all max x of u∗ − φ
F(D2 φ(x), Dφ(x), u∗ (x), x) ≦ 0
fix φ smooth
x0
max of u∗ − φ and
“uh → u∗ ” =⇒
u∗ (x0 ) = φ(x0 )
xh max of
uh − φ
=⇒
uh ≦ φ − ξh
xh → x0 and ξh = uh (xh ) − φ(xh ) → 0 as h → 0
uh ≦ φ + ξh
w
w

monotonicity
S([φ + ξh ]x , φ(xh ) + ξh , xh , h) ≦ 0 = S([uh ]x , uh (xh ), xh , h)
27
∗
u subsolution
iff
(
for all smooth φ and all max x of u∗ − φ
F(D2 φ(x), Dφ(x), u∗ (x), x) ≦ 0
fix φ smooth
x0
max of u∗ − φ and
“uh → u∗ ” =⇒
u∗ (x0 ) = φ(x0 )
xh max of
uh − φ
=⇒
uh ≦ φ − ξh
xh → x0 and ξh = uh (xh ) − φ(xh ) → 0 as h → 0
uh ≦ φ + ξh
w
w

monotonicity
S([φ + ξh ]x , φ(xh ) + ξh , xh , h) ≦ 0 = S([uh ]x , uh (xh ), xh , h)
w
w

consistency
xh → 0
ξh → 0
F(D2 φ(x0 ), Dφ(x0 ), u∗ (x0 ), x0 ) ≦ 0
28
Examples
•
Hamilton-Jacobi equation
ut + H(ux ) = 0
29
Examples
•
Hamilton-Jacobi equation
ut + H(ux ) = 0
r − u(x, t − h)
h
» “
–
”
θ u(x + λh) − 2u(x) + u(x − λh)
u(x + λh) − u(x − λh)
−
−h H
2λh
λ
λh
S([u]x , r, x, t, h) =
Lax-Friedrichs
 “ n
n
n
n ff
n ”
Uj+1 − Uj−1
θ Uj+1 − 2Uj + Uj−1
−
Ujn+1 = Ujn − ∆t H
2∆x
λ
∆x
(CFL)
2θ − λkH ′ k ≧ 0
=⇒
monotonicity
(λ =
∆t
)
∆x
30
•
Isaacs-Bellman equation
ut + F(D2 u, Du, x) = 0
31
•
Isaacs-Bellman equation
ut + F(D2 u, Du, x) = 0
F(X, p, x) = max min[−tr (aα,β (x)X) − bα,β (x) · p]
α
β
stochastic differential games
32
•
Isaacs-Bellman equation
ut + F(D2 u, Du, x) = 0
F(X, p, x) = max min[−tr (aα,β (x)X) − bα,β (x) · p]
α
β
8
u(x + ei h) − u(x)
>
>
<
h
h,α,β
u=
uxi ≈ Di
> u(x − hei ) − u(x)
>
:
h
stochastic differential games
if
bα,β
(x) ≧ 0
i
if
bα,β
(x) < 0
i
33
•
Isaacs-Bellman equation
ut + F(D2 u, Du, x) = 0
F(X, p, x) = max min[−tr (aα,β (x)X) − bα,β (x) · p]
α
β
8
u(x + ei h) − u(x)
>
>
<
h
h,α,β
u=
uxi ≈ Di
> u(x − hei ) − u(x)
>
:
h
uxi xj
stochastic differential games
if
bα,β
(x) ≧ 0
i
if
bα,β
(x) < 0
i
8
u(x + hei ) − 2u(x) + u(x − hei )
>
if i = j
>
>
>
h2
>
>
>
8
>
>
>
<>
>
>
h,αβ
>
≈ Dij u(x) = >
if aα,β
≧0
ij
><
>
>
if i 6= j
>
>
>
>
α,β
>
>
>
>
if
a
<
0
ij
>
>
>>
:
:
34
S([u]h , r, x, t, h) =
r − u(x, t − h)
h
h
i
h,α,β
−h max min − aα,β
u(x, t − h) − bα,β
(x)Dh,α,β
u(x, t − h)
ij (x) · Dij
i
i
α
β
35
S([u]h , r, x, t, h) =
r − u(x, t − h)
h
h
i
h,α,β
−h max min − aα,β
u(x, t − h) − bα,β
(x)Dh,α,β
u(x, t − h)
ij (x) · Dij
i
i
α
β
aα,β diagonally dominant
aα,β
ii (x) −
X
j6=i
w
w

S monotone
|aα,β
ij (x)| ≧ 0 for all i, α, β, x
36
RATES OF CONVERGENCE – Hamilton-Jacobi
37
RATES OF CONVERGENCE – Hamilton-Jacobi
F(Du, u, x) = 0
S([uh ]x , uh (x), x, h) = 0
38
RATES OF CONVERGENCE – Hamilton-Jacobi
F(Du, u, x) = 0
S([uh ]x , uh (x), x, h) = 0
monotonicity
u≦v
and
m≧0
=⇒
S([u + m]x , r + m, x, h) ≧ λm + S([v]x , r, x, h)
39
RATES OF CONVERGENCE – Hamilton-Jacobi
F(Du, u, x) = 0
S([uh ]x , uh (x), x, h) = 0
monotonicity
u≦v
stability
and
m≧0
=⇒
kuh k ≦ C
S([u + m]x , r + m, x, h) ≧ λm + S([v]x , r, x, h)
independent of h
40
RATES OF CONVERGENCE – Hamilton-Jacobi
F(Du, u, x) = 0
S([uh ]x , uh (x), x, h) = 0
monotonicity
u≦v
stability
consistency
and
m≧0
=⇒
kuh k ≦ C
S([u + m]x , r + m, x, h) ≧ λm + S([v]x , r, x, h)
independent of h
|S([ψ]x , ψ(x), x, h) − F(Dψ(x), ψ(x), x)| ≦ C(1 + |D2 ψ|)h (all ψ smooth)
41
RATES OF CONVERGENCE – Hamilton-Jacobi
F(Du, u, x) = 0
S([uh ]x , uh (x), x, h) = 0
monotonicity
u≦v
stability
consistency
Theorem:
and
m≧0
=⇒
kuh k ≦ C
S([u + m]x , r + m, x, h) ≧ λm + S([v]x , r, x, h)
independent of h
|S([ψ]x , ψ(x), x, h) − F(Dψ(x), ψ(x), x)| ≦ C(1 + |D2 ψ|)h (all ψ smooth)
u Lipschitz solution of F = 0
|u − uh | ≦ Kh1/2
F convex
−Kh≦u−uh ≦Kh1/2
=⇒
(K = K(F, kDuk))
Capuzzo-Dolcetta-Ishii
42
proof of
uh ≦ u + Kh1/2
43
inf - and sup - convolution
regularizations
44
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
45
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
• ūε , uε
Lipschitz continuous
and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
ūε ↓ u,
uε ↑ u
46
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
• ūε , uε
Lipschitz continuous
and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
ūε ↓ u,
uε ↑ u
ūε (x) = u(y(x)) − (2ε)
−1
|x − y(x)|
2
ūε (x + hχ) − 2ūε (x) + ūε (x − hχ)
•
ūε
uε
semi-convex
semi-concave
(D ūε ≧ − εI )
(D2 uε ≦ εI )
≧ u(y(x)) − (2ε)
2
(−2ε)
−2u(y(x)) + (2ε)
(−2ε)
−1
2
−1
−1
−1
|x − y(x) + hχ|
2
2
|x − y(x)| + u(y(x))
|x − y(x) − hχ|2 =
2
2
−1 2
[|x−y(x)+hχ| −2|x−y(x)| +|x−y(x)−hχ| ] = −2
h
47
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
• ūε , uε
Lipschitz continuous
and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
ūε ↓ u,
uε ↑ u
ūε (x) = u(y(x)) − (2ε)
−1
|x − y(x)|
2
ūε (x + hχ) − 2ūε (x) + ūε (x − hχ)
•
ūε
uε
semi-convex
semi-concave
(D ūε ≧ − εI )
(D2 uε ≦ εI )
(−2ε)
• ūε , uε
≧ u(y(x)) − (2ε)
2
twice differentiable a.e.
−2u(y(x)) + (2ε)
(−2ε)
−1
2
−1
−1
−1
|x − y(x) + hχ|
2
2
|x − y(x)| + u(y(x))
|x − y(x) − hχ|2 =
2
2
−1 2
[|x−y(x)+hχ| −2|x−y(x)| +|x−y(x)−hχ| ] = −2
h
48
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
• ūε , uε
Lipschitz continuous
and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
ūε ↓ u,
uε ↑ u
ūε (x) = u(y(x)) − (2ε)
−1
|x − y(x)|
2
ūε (x + hχ) − 2ūε (x) + ūε (x − hχ)
•
ūε
uε
semi-convex
semi-concave
(D ūε ≧ − εI )
(D2 uε ≦ εI )
(−2ε)
• ūε , uε
≧ u(y(x)) − (2ε)
2
−2u(y(x)) + (2ε)
(−2ε)
−1
−1
−1
−1
|x − y(x) + hχ|
2
2
|x − y(x)| + u(y(x))
|x − y(x) − hχ|2 =
2
2
2
−1 2
[|x−y(x)+hχ| −2|x−y(x)| +|x−y(x)−hχ| ] = −2
twice differentiable a.e.
• u Lipschitz continuous ⇒ kūε − uk ≦ kDukε, kuε − uk ≦ kDukε
h
49
inf - and sup - convolution
regularizations
u bounded, continuous
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] and
• ūε , uε
Lipschitz continuous
and
uε (x) = inf[u(y) + (2ε)−1 |x − y|2 ]
ūε ↓ u,
uε ↑ u
ūε (x) = u(y(x)) − (2ε)
−1
|x − y(x)|
2
ūε (x + hχ) − 2ūε (x) + ūε (x − hχ)
•
ūε
uε
semi-convex
semi-concave
(D ūε ≧ − εI )
(D2 uε ≦ εI )
(−2ε)
• ūε , uε
≧ u(y(x)) − (2ε)
2
−2u(y(x)) + (2ε)
(−2ε)
−1
−1
−1
−1
|x − y(x) + hχ|
2
2
|x − y(x)| + u(y(x))
|x − y(x) − hχ|2 =
2
2
2
−1 2
[|x−y(x)+hχ| −2|x−y(x)| +|x−y(x)−hχ| ] = −2
twice differentiable a.e.
• u Lipschitz continuous ⇒ kūε − uk ≦ kDukε, kuε − uk ≦ kDukε
• F(D2 u, Du, u, x) ≦ 0
⇒
“F(D2 ūε , Dūε , ūε , x) ≦ ε”
• F(D2 u, Du, u, x) ≧ 0
⇒
“F(D2 uε , Duε , uε , x) ≧ −ε”
h
50
u(x) = −|x|
uε (x) = sup[−|y| − (2ε)−1 |x − y|2 ]
8
1
>
<− |x|2 for |x| ≦ ε
ε
2ε
u (x) =
>
:−|x| + ε for |x| ≧ ε
3
uε (x) = inf[−|y| + (2ε)−1 |x − y|2 ]
uε (x) = −|x| −
3ε
2
51
parallel surface regularization
52
parallel surface regularization
d(x, y) = dist((x, y), graph u) (x ∈ RN , y ∈ R)
d(x, u(x)) = 0
graph ūε = {(x, y) ∈ RN+1 : d(x, y) = ε and y ≧ u(x)}
fi d(x, ū (x)) = ε
ε
graph uε = {(x, y) ∈ RN+1 : d(x, y) = ε and y ≦ u(x)}
fi d(x, u (x)) = ε
ε
• ūε
ūε ≧ u
uε ≦ u
“F(D2 ūε , Dūε , ūε , x) ≦ ε”
semi-concave,
ūε ↓ u ,
semi-convex,
uε ↑ u , “F(D2 uε , Duε , uε , x) ≧ −ε”
Lip. continuous,
• uε
53
proof of
uh ≦ u + Kh1/2
54
proof of
•
u Lipschitz
=⇒
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
55
proof of
•
u Lipschitz
•
compare
=⇒
uh
and uε
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
56
proof of
•
u Lipschitz
•
compare
•
use
uε
=⇒
uh
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
and uε
as a test function
m = max(uh − uε )
=⇒
uh ≦ uε + m
57
proof of
•
u Lipschitz
•
compare
•
use
•
consistency
uε
=⇒
uh
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
and uε
as a test function
m = max(uh − uε )
=⇒
uh ≦ uε + m
“
1”
F(Duε , uε , x) ≧ 0 =⇒ S([uε ], uε (x), x, h) ≧ −C(1 + |D2 uε |)h ≧ −C 1 +
h
ε
58
proof of
•
u Lipschitz
•
compare
•
use
•
consistency
uε
=⇒
uh
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
and uε
as a test function
m = max(uh − uε )
=⇒
uh ≦ uε + m
“
1”
F(Duε , uε , x) ≧ 0 =⇒ S([uε ], uε (x), x, h) ≧ −C(1 + |D2 uε |)h ≧ −C 1 +
h
ε
•
monotonicity
S([uε ]x , uε (x), x, h) ≦ S([uh − m]x , uh (x) − m, x, h) ≦ −λm + S([uh ]x , uh (x), x, h)
”
“
=⇒
λm ≦ C 1 + 1ε h
59
proof of
•
u Lipschitz
•
compare
•
use
•
consistency
uε
=⇒
uh
uh ≦ u + Kh1/2
|u − uε | ≦ kDukε
and uε
as a test function
m = max(uh − uε )
=⇒
uh ≦ uε + m
“
1”
F(Duε , uε , x) ≧ 0 =⇒ S([uε ], uε (x), x, h) ≧ −C(1 + |D2 uε |)h ≧ −C 1 +
h
ε
•
monotonicity
S([uε ]x , uε (x), x, h) ≦ S([uh − m]x , uh (x) − m, x, h) ≦ −λm + S([uh ]x , uh (x), x, h)
”
“
=⇒
λm ≦ C 1 + 1ε h
”
“
•
total error
kDukε + C 1 + 1ε h ≈ Kh1/2
60
RATES OF CONVERGENCE – uniformly elliptic equations
61
RATES OF CONVERGENCE – uniformly elliptic equations
F(D2 u, Du, x) = 0
S([uh ]x , uh (x), x, h) = 0
consistency
|S([φ]x , φ(x), x, h) − F(D2 φ(x), Dφ(x), x)| ≦ K(1 + |D3 φ|)h
62
RATES OF CONVERGENCE – uniformly elliptic equations
F(D2 u, Du, x) = 0
S([uh ]x , uh (x), x, h) = 0
consistency
|S([φ]x , φ(x), x, h) − F(D2 φ(x), Dφ(x), x)| ≦ K(1 + |D3 φ|)h
problem
no regularization of viscosity solutions
controlling “third-derivatives” and “preserving” equation
63
RATES OF CONVERGENCE – uniformly elliptic equations
F(D2 u, Du, x) = 0
S([uh ]x , uh (x), x, h) = 0
consistency
|S([φ]x , φ(x), x, h) − F(D2 φ(x), Dφ(x), x)| ≦ K(1 + |D3 φ|)h
problem
F convex
no regularization of viscosity solutions
controlling “third-derivatives” and “preserving” equation
stochastic control representation, special schemes
pde-switching systems
Krylov
Barles-Jakobsen
64
RATES OF CONVERGENCE – uniformly elliptic equations
F(D2 u, Du, x) = 0
S([uh ]x , uh (x), x, h) = 0
consistency
|S([φ]x , φ(x), x, h) − F(D2 φ(x), Dφ(x), x)| ≦ K(1 + |D3 φ|)h
problem
F convex
no regularization of viscosity solutions
controlling “third-derivatives” and “preserving” equation
stochastic control representation, special schemes
pde-switching systems
F uniformly elliptic
new regularity, δ-solutions
Krylov
Barles-Jakobsen
Caffarelli-Souganidis
65
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
on
in
U
F uniformly elliptic
∂U
66
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
• δ-viscosity solutions
on
in
U
F uniformly elliptic
∂U
67
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
• δ-viscosity solutions
• new regularity result
on
in
U
F uniformly elliptic
∂U
68
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
on
in
U
F uniformly elliptic
∂U
• δ-viscosity solutions
• new regularity result
• Theorem A: u ∈ C0,1 (Ū) solves (∗), u± δ-sub- (super-) solution of (∗) and
ku± − uk = O(δ η ) on ∂U,
then
there exists a universal θ > 0
st
ku − u± k = O(δ θ ) in U.
69
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
on
in
U
F uniformly elliptic
∂U
• δ-viscosity solutions
• new regularity result
• Theorem A: u ∈ C0,1 (Ū) solves (∗), u± δ-sub- (super-) solution of (∗) and
ku± − uk = O(δ η ) on ∂U,
then
there exists a universal θ > 0
• Theorem B:
st
ku − u± k = O(δ θ ) in U.
numerical approximations are δ-solutions for
δ = δ(h).
70
GENERAL STRATEGY
(∗)
8
<F(D2 u) = f
:u = g
on
in
U
F uniformly elliptic
∂U
• δ-viscosity solutions
• new regularity result
• Theorem A: u ∈ C0,1 (Ū) solves (∗), u± δ-sub- (super-) solution of (∗) and
ku± − uk = O(δ η ) on ∂U,
then
there exists a universal θ > 0
• Theorem B:
st
ku − u± k = O(δ θ ) in U.
numerical approximations are δ-solutions for
δ = δ(h).
• Theorem C: oscillatory solutions are δ-solutions for δ = δ(ε) off a set of ω’s
with probability less than δ.
71
δ-viscosity solution of F(D2 u) = f in U
72
δ-viscosity solution of F(D2 u) = f in U
u viscosity subsolution
iff
(
for all x ∈ U and all quadratics P touching u from above at x,
F(D2 P) ≦ f (x)
P touches u from above at x
u(y) ≦ u(x) + P(y − x) + o(|y − x|2 ) ≦ u(x) + (P + εI)(x − y) in B(x, δ(ε))
73
δ-viscosity solution of F(D2 u) = f in U
u viscosity subsolution
iff
(
for all x ∈ U and all quadratics P touching u from above at x,
F(D2 P) ≦ f (x)
P touches u from above at x
u(y) ≦ u(x) + P(y − x) + o(|y − x|2 ) ≦ u(x) + (P + εI)(x − y) in B(x, δ(ε))
u δ-viscosity subsolution
iff
8
>
<for all B(x, δ) ⊂ U and all quadratics P such that
u ≦ P in B(x, δ) , u(x) = P(x) and D2 P = O(δ −α ) for some α > 0
>
:
F(D2 P) ≦ f (x)
74
δ-viscosity solution of F(D2 u) = f in U
u viscosity subsolution
iff
(
for all x ∈ U and all quadratics P touching u from above at x,
F(D2 P) ≦ f (x)
P touches u from above at x
u(y) ≦ u(x) + P(y − x) + o(|y − x|2 ) ≦ u(x) + (P + εI)(x − y) in B(x, δ(ε))
u δ-viscosity subsolution
iff
8
>
<for all B(x, δ) ⊂ U and all quadratics P such that
u ≦ P in B(x, δ) , u(x) = P(x) and D2 P = O(δ −α ) for some α > 0
>
:
F(D2 P) ≦ f (x)
subsolutions are always δ-subsolutions
δ-subsolutions are not always subsolutions
75
Lemma:
Any monotone, consistent approximation uh of
F(D2 u) = f is an h-solution of F(D2 w) = f ± Kh.
8
< all B(x, δ) ⊂ U and all quadratics P such that
u δ-viscosity subsolution iff u ≦ P in B(x, δ), u(x) = P(x) and |D2 P| = O(δ −α ) (α > 0)
:
F(D2 P) ≦ f (x)
76
Lemma:
Any monotone, consistent approximation uh of
F(D2 u) = f is an h-solution of F(D2 w) = f ± Kh.
8
< all B(x, δ) ⊂ U and all quadratics P such that
u δ-viscosity subsolution iff u ≦ P in B(x, δ), u(x) = P(x) and |D2 P| = O(δ −α ) (α > 0)
:
F(D2 P) ≦ f (x)
Proof:
uh ≦ Q
monotonicity
=⇒
S([Q]x , Q(x), x, h) ≦ S([uh ]x , uh (x), x, h) = 0
consistency
=⇒
S([Q]x , Q(x), x, h) ≧ F(D2 Q) − f − Kh
Theorem:
kuh − uk = O(hα )
in B(x, δ),
uh (x) = Q(x)
α ∈ (0, 1)
77
HOMOGENIZATION
x
F(D2 uε , , ω) = 0 in U
ε
uε −→ u0 a.s.
ε→0
and
F0 (D2 u0 ) = 0 in U
F uniformly elliptic, stationary ergodic
78
HOMOGENIZATION
x
F(D2 uε , , ω) = 0 in U
ε
uε −→ u0 a.s.
ε→0
and
F uniformly elliptic, stationary ergodic
F0 (D2 u0 ) = 0 in U
For each Q ∈ SN , F0 (Q) is the unique constant st
•
8
<F(D2 uε , εx , ω) = F0 (Q) in B1
, then kuε (·, ω) − QkC(B̄1 ) → 0 a.s.
if
:
uε = Q on ∂B1
if
8
!
<F(D2 uε , x, ω) = F0 (Q) in B1/ε , then kε2 uε (·, ω) − QkC(B̄1/ε ) → 0 a.s.
:
uε = Q on
∂B1/ε
uε (x) = ε2 uε ( εx )
Caffarelli-Souganidis-Wang
79
•
Lemma: strongly mixing media with algebraic rate
∃ Aε ⊂ Ω st
P(Aε ) ≦ Ce−c| ln ε|
1/2
and
kuε (·, ω) − QkC(B̄1 ) ≦ C(1 + kQk)e−c| ln ε|
1/2
in Acε
80
•
Lemma: strongly mixing media with algebraic rate
∃ Aε ⊂ Ω st
P(Aε ) ≦ Ce−c| ln ε|
1/2
and
kuε (·, ω) − QkC(B̄1 ) ≦ C(1 + kQk)e−c| ln ε|
•
Lemma: If
uε
F(D2 uε , εx , ω) = 0 in U,
is e−c| ln ε|
1/2
– solution off a set
1/2
in Acε
then
Aε ∈ Ω st P(Aε ) ≦ Ce−c| ln ε|
1/2
81
back to sup- and inf-convolutions
some key properties of the regularizations
of Lipschitz sub- and super-solutions
82
back to sup- and inf-convolutions
some key properties of the regularizations
of Lipschitz sub- and super-solutions
F(D2 u, Du, u, x) = 0
graph ūε = {(x, y) ∈ RN+1 : d(x, y) = ε
and y ≧ u(x)}
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ]
y(x) “maximizer” for x
ūε (x) = u(y(x)) − (2ε)
−1
F uniformly elliptic
y(x) “maximizer” for x
2
|x − y(x)|
d((x, ūε (x)), (y(x), u(y(x))) = ε
83
∃ C > 0 (depending ONLY on ellipticity constants and N) st
• |x1 − x2 | ≦ C|y(x1 ) − y(x2 )|
(Jacobian of
y 7→ y−1 (x)
is bdd)
• if a quadratic P touches ūε from above at x, then u is touched
at y(x) from above by a quadratic Pε
and
2
D ūε (x) ≧ D u(y(x)) + Cε2 |D2 u(y(x))|2
• ∃ t0 , σ
st
for t ≧ t0
2
∃ Aεt
st
|Aεt | ≦ t−σ
and
uε has a second order expansion from above with error of size t in Aε,c
t
ūε has a second order expansion from below with error of size t in Aε,c
t
84
>
>
u
85
>
>
u
x
>
>
D2 uε (x) ≧ − εc
u
86
>
>
u
x
>
>
>
y(x)
u
D2 uε (x) ≧ − εc
87
>
>
u
x
>
>
>
y(x)
u
>
D2 uε (x) ≧ − C
ε
2
|D u(y(x))| ≦ C
ε
88
>
>
u
|A*t ,c | <= C|Atc |
A*t
x
>
>
>
y(x)
u
>
At
D2 uε (x) ≧ − C
ε
2
|D u(y(x))| ≦ C
ε
89
Sketch of proof
90
Sketch of proof
•
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] = u(y(x)) − (2ε)−1 |x − y(x)|2
91
Sketch of proof
•
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] = u(y(x)) − (2ε)−1 |x − y(x)|2
P
•
ūε touched from above by a quadratic P at x
⇒
x
y(x)
u−ε
u
P(z) − P(x) ≧ ūε (z) − ūε (x) ≧ u(z) − (2ε)−1 |z − y|2 − (u(y(x)) − (2ε)−1 |x − y(x)|2 )
⇒
u(y) ≦ u(y(x)) − (2ε)−1 [|z − y|2 − |x − y(x)|2 ]
⇒
u is touched from above at y(x) by a quadratic of opening ε−1
92
Sketch of proof
•
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] = u(y(x)) − (2ε)−1 |x − y(x)|2
P
•
ūε touched from above by a quadratic P at x
⇒
x
y(x)
u−ε
u
P(z) − P(x) ≧ ūε (z) − ūε (x) ≧ u(z) − (2ε)−1 |z − y|2 − (u(y(x)) − (2ε)−1 |x − y(x)|2 )
⇒
u(y) ≦ u(y(x)) − (2ε)−1 [|z − y|2 − |x − y(x)|2 ]
⇒
u is touched from above at y(x) by a quadratic of opening ε−1
P
•
u solves u uniformly elliptic equation
x
y(x)
u−ε
u
Harnack inequality
=⇒
u is also touched from below at y(x) by a quadratic of opening c/ε
93
Sketch of proof
•
ūε (x) = sup[u(y) − (2ε)−1 |x − y|2 ] = u(y(x)) − (2ε)−1 |x − y(x)|2
P
•
ūε touched from above by a quadratic P at x
⇒
x
y(x)
u−ε
u
P(z) − P(x) ≧ ūε (z) − ūε (x) ≧ u(z) − (2ε)−1 |z − y|2 − (u(y(x)) − (2ε)−1 |x − y(x)|2 )
⇒
u(y) ≦ u(y(x)) − (2ε)−1 [|z − y|2 − |x − y(x)|2 ]
⇒
u is touched from above at y(x) by a quadratic of opening ε−1
P
•
u solves u uniformly elliptic equation
x
y(x)
u−ε
u
Harnack inequality
=⇒
u is also touched from below at y(x) by a quadratic of opening c/ε
u is differentiable at y(x) and has C1 -contact from above and below with convex
and concave envelops of paraboloids with opening c/ε
•
x = y(x) − εDu(y(x)) =⇒ |Du(y(x1 )) − Du(y(x2 ))| ≦ cε−1 |y(x1 ) − y(x2 )|
=⇒
|x1 − x2 | ≦ (1 + c)|y(x1 ) − y(x2 )|
94
(NEW) REGULARITY RESULT
F(D2 u) = f
in U
95
(NEW) REGULARITY RESULT
F(D2 u) = f
in U
• F uniformly elliptic, u, f Lip, U = B1 =⇒
∃ t0 , σ depending on ellipticity and N st for t ≧ t0
∃ At ⊂ B1 st |(B1 \ At ) ∩ B1/2 | ≦ t−σ , and for all x0 ∈ At ∩ B1/2
∃ quadratic Qt,x0 such that F(D2 Qt,x0 ) = f (x0 ), |D2 Qt,x0 | ≦ t, and
u(x) = u(x0 ) + Qt,x0 (x − x0 ) + O(t|x − x0 |3 ) in
B1
Caffarelli
96
(NEW) REGULARITY RESULT
F(D2 u) = f
in U
• F uniformly elliptic, u, f Lip, U = B1 =⇒
∃ t0 , σ depending on ellipticity and N st for t ≧ t0
∃ At ⊂ B1 st |(B1 \ At ) ∩ B1/2 | ≦ t−σ , and for all x0 ∈ At ∩ B1/2
∃ quadratic Qt,x0 such that F(D2 Qt,x0 ) = f (x0 ), |D2 Qt,x0 | ≦ t, and
u(x) = u(x0 ) + Qt,x0 (x − x0 ) + O(t|x − x0 |3 ) in
B1
Caffarelli
•• F uniformly elliptic, u, f Lip, U = B1
u±
ε sup, inf-convolution
∃ t0 , σ depending on ellipticity and N st for t ≧ t0
∃ Aεt ⊂ B1 st |(B1 \ Aεt ) ∩ B1/2 | ≦ t−σ and for all x0 ∈ Aεt ∩ B1/2
∃ quadratic Qεt,x0 ∈ SN such that F(D2 Qεt,x0 ) ≈ f (x0 ), |D2 Qεt,x0 | ≦ t and
ε
3
u±
ε (x) ≈ uε (x0 ) + Qt,x0 (x − x0 ) + O(t|x − x0 | )
in
B1
97
Proof of regularity result
F(D2 u) = f in B1
98
Proof of regularity result
F(D2 u) = f in B1
tr Dx FD2 uxi = fxi in B1
∃ “universal” t0 , σ st, for all t ≥ t0 , v = uxi is touched from above and below
i,t
in At ∩ B1/2 by quadratics Pi,t
x0 and P̄x0 with opening t and
C
−σ
|At | ≦ C(kDuk∞ + kDx f kN )t
=⇒
i,t
uxi differentiable in At ∩ B1/2 , Duxi (x0 ) = DPi,t
x0 (x0 ) = DPx0 (x0 ) and
|Du(x) − Du(x0 ) − D2 u(x0 )(x − x0 )| ≦ Ct|x − x0 |2
99
Sketch of proof of
u solution, v δ-supersolution of F(D2 w) = f in U
u ≦ v + O(δ γ ) on ∂U
ff
=⇒ u ≦ v + cδ θ
• several approximations/regularizations
u −→ u subsolution
F(D2 w) = f − δ β
2
v −→ v δ-supersolution F(D v) = f
D2 u ≧ −δ −2ζ I
D2 v ≦ δ 2ζ I
(θ > 0)
100
Sketch of proof of
u solution, v δ-supersolution of F(D2 w) = f in U
u ≦ v + O(δ γ ) on ∂U
ff
=⇒ u ≦ v + cδ θ
• several approximations/regularizations
u −→ u subsolution
F(D2 w) = f − δ β
2
v −→ v δ-supersolution F(D v) = f
• Γw
concave envelope
D2 u ≧ −δ −2ζ I
D2 v ≦ δ 2ζ I
w=u−v
of
|D2 Γw | ≦ cδ 2ζ on contact set
ABP-estimate
=⇒
sup w ≦ cδ −2ζ |{Γw = w}|1/N
(θ > 0)
101
U
• if δ θ ≦ sup w ,
Uδ
(∗∗)
{Γw =w}
then
δ
(θ+2ζ)N
≦ c|{Γw = w}|
102
U
• if δ θ ≦ sup w ,
Uδ
(∗∗)
then
{Γw =w}
δ (θ+2ζ)N ≦ c|{Γw = w}|
• covering argument and (∗∗) =⇒
∃ B(xi , δ γ ) st |B(xi , 21 δ γ ) ∩ {Γw = w}| ≧ cδ (θ+2ζ+γ)N
B(xi, 2-1δγ)
103
U
• if δ θ ≦ sup w ,
Uδ
(∗∗)
then
{Γw =w}
δ (θ+2ζ)N ≦ c|{Γw = w}|
• covering argument and (∗∗) =⇒
∃ B(xi , δ γ ) st |B(xi , 21 δ γ ) ∩ {Γw = w}| ≧ cδ (θ+2ζ+γ)N
1
• apply regularity result to B(xi , δ γ ) with t = δ − σ (θ+2ζ+γ)N
∃ x0 ∈ contact set ∩ B(xi , 2−1 δ γ ) and
2
|D Qt | ≦ t,
2
F(D Qt ) ≦ f − δ
β
quadratic Qt
and
u(x) = u(x0 ) + Qt (x − x0 ) + O(t|x − x0 |3 )
x0
At
such that
γ
B(xi, 2-1δ )
104
U
• if δ θ ≦ sup w ,
Uδ
(∗∗)
then
{Γw =w}
δ (θ+2ζ)N ≦ c|{Γw = w}|
• covering argument and (∗∗) =⇒
∃ B(xi , δ γ ) st |B(xi , 21 δ γ ) ∩ {Γw = w}| ≧ cδ (θ+2ζ+γ)N
1
• apply regularity result to B(xi , δ γ ) with t = δ − σ (θ+2ζ+γ)N
∃ x0 ∈ contact set ∩ B(xi , 2−1 δ γ ) and
2
|D Qt | ≦ t,
2
F(D Qt ) ≦ f − δ
β
quadratic Qt
and
u(x) = u(x0 ) + Qt (x − x0 ) + O(t|x − x0 |3 )
• v
δ-supersolution
u≦ v+ℓ
=⇒
ℓ linear
F(D2 Qt − tδ) ≧ f (x0 )
x0
At
such that
γ
B(xi, 2-1δ )
105
U
• if δ θ ≦ sup w ,
Uδ
(∗∗)
then
{Γw =w}
δ (θ+2ζ)N ≦ c|{Γw = w}|
• covering argument and (∗∗) =⇒
∃ B(xi , δ γ ) st |B(xi , 21 δ γ ) ∩ {Γw = w}| ≧ cδ (θ+2ζ+γ)N
1
• apply regularity result to B(xi , δ γ ) with t = δ − σ (θ+2ζ+γ)N
∃ x0 ∈ contact set ∩ B(xi , 2−1 δ γ ) and
2
|D Qt | ≦ t,
2
F(D Qt ) ≦ f − δ
β
quadratic Qt
and
u(x) = u(x0 ) + Qt (x − x0 ) + O(t|x − x0 |3 )
• v
δ-supersolution
u≦ v+ℓ
=⇒
ℓ linear
F(D2 Qt − tδ) ≧ f (x0 )
1
• uniform ellipticity =⇒ δ β ≦ cδ 1− σ (θ+2ζ+γ)N
a contradiction if σ − (θ + 2ζ + γ) > σβ
x0
At
such that
γ
B(xi, 2-1δ )
106
HOMOGENIZATION
8
<F(D2 uε , εx , ω) = 0 in U
:
uε = g on ∂U
F uniformly elliptic, stationary, ergodic
107
HOMOGENIZATION
8
<F(D2 uε , εx , ω) = 0 in U
:
uε = g on ∂U
∃ F0 uniformly elliptic st
uε → u0 in C(Ū)
ε→0
and a.s.
F uniformly elliptic, stationary, ergodic
8
<F0 (D2 u0 ) = 0 in U
:
u0 = g on ∂U
Caffarelli-Souganidis-Wang
108
HOMOGENIZATION
8
<F(D2 uε , εx , ω) = 0 in U
:
uε = g on ∂U
∃ F0 uniformly elliptic st
uε → u0 in C(Ū)
ε→0
rate of convergence
and a.s.
F uniformly elliptic, stationary, ergodic
8
<F0 (D2 u0 ) = 0 in U
:
u0 = g on ∂U
kuε − u0 k = O(σ(ε))
Caffarelli-Souganidis-Wang
109
HOMOGENIZATION
8
<F(D2 uε , εx , ω) = 0 in U
:
uε = g on ∂U
∃ F0 uniformly elliptic st
uε → u0 in C(Ū)
and a.s.
ε→0
rate of convergence
F uniformly elliptic, stationary, ergodic
8
<F0 (D2 u0 ) = 0 in U
Caffarelli-Souganidis-Wang
:
u0 = g on ∂U
kuε − u0 k = O(σ(ε))
strongly mixing with algebraic rate*
F linear
σ(ε) = εγ
Yurinskii
* F strongly mixing with rate φ
E[F(x, ·)F(y, ·) − EF(x, ·)EF(y, ·)] ≦ φ(|x − y|)(E(F(x, ·))2 E(F(y, ·))2 )1/2
110
HOMOGENIZATION
8
<F(D2 uε , εx , ω) = 0 in U
:
uε = g on ∂U
∃ F0 uniformly elliptic st
uε → u0 in C(Ū)
and a.s.
ε→0
rate of convergence
F uniformly elliptic, stationary, ergodic
8
<F0 (D2 u0 ) = 0 in U
Caffarelli-Souganidis-Wang
:
u0 = g on ∂U
kuε − u0 k = O(σ(ε))
strongly mixing with algebraic rate*
F linear
F nonlinear
σ(ε) = εγ
σ(ε) = e−c| ln ε|
Yurinskii
1/2
Caffarelli-Souganidis
* F strongly mixing with rate φ
E[F(x, ·)F(y, ·) − EF(x, ·)EF(y, ·)] ≦ φ(|x − y|)(E(F(x, ·))2 E(F(y, ·))2 )1/2
111
•
rate for quadratic data
=⇒ rate for general data
F(D2 uε , εx , ω) = 0 in U =⇒ uε is δ-subsolution of F0 (D2 w) = −δ α
δ = Ce−| ln ε|
1/2
112
•
rate for quadratic data
=⇒ rate for general data
F(D2 uε , εx , ω) = 0 in U =⇒ uε is δ-subsolution of F0 (D2 w) = −δ α
δ = Ce−| ln ε|
1/2
• P quadratic st |D2 P| ≦ δ −σ , uε ≦ P in B(x0 , δ), uε (x0 ) = P(x0 )
P
• assume
uε
F0 (D2 P) < −δ α
x0
Bδ(x0)
113
•
=⇒ rate for general data
rate for quadratic data
F(D2 uε , εx , ω) = 0 in U =⇒ uε is δ-subsolution of F0 (D2 w) = −δ α
δ = Ce−| ln ε|
1/2
• P quadratic st |D2 P| ≦ δ −σ , uε ≦ P in B(x0 , δ), uε (x0 ) = P(x0 )
•
P
F0 (D2 P) < −δ α
• assume
“lower” P to Pδ (x) = P(x) − ηδ α (δ 2 − |x − x0 |2 )
uε (x0 ) = P(x0 )
⇒
Pδ
Pδ (x0 ) − uε (x0 ) = ηδ 2+α
uniform ellipticity of F0
⇒
F0 (D2 Pδ ) < 0
x0
Bδ(x0)
uε
114
•
=⇒ rate for general data
rate for quadratic data
F(D2 uε , εx , ω) = 0 in U =⇒ uε is δ-subsolution of F0 (D2 w) = −δ α
δ = Ce−| ln ε|
1/2
• P quadratic st |D2 P| ≦ δ −σ , uε ≦ P in B(x0 , δ), uε (x0 ) = P(x0 )
•
α
uεδ
2
Pδ
2
“lower” P to Pδ (x) = P(x) − ηδ (δ − |x − x0 | )
uε (x0 ) = P(x0 )
⇒
⇒
F0 (D2 Pδ ) < 0
(
F(D2 uδε , εx , ω) = F0(D2 Pδ ) < 0 in B(x0 , δ)
uδε = Pδ on ∂B(x0 , δ)
uε
Pδ (x0 ) − uε (x0 ) = ηδ 2+α
uniform ellipticity of F0
•
P
F0 (D2 P) < −δ α
• assume
•
⇒
•
x0
Bδ(x0)
uδε ≧ uε
kuδε −Pδ k ≦ Cδ 2 (1+δ −σ )e−c| ln(ε/δ)|
1/2
in Aεδ
115
•
=⇒ rate for general data
rate for quadratic data
F(D2 uε , εx , ω) = 0 in U =⇒ uε is δ-subsolution of F0 (D2 w) = −δ α
δ = Ce−| ln ε|
1/2
• P quadratic st |D2 P| ≦ δ −σ , uε ≦ P in B(x0 , δ), uε (x0 ) = P(x0 )
•
α
uεδ
2
Pδ
2
“lower” P to Pδ (x) = P(x) − ηδ (δ − |x − x0 | )
uε (x0 ) = P(x0 )
⇒
uε
Pδ (x0 ) − uε (x0 ) = ηδ 2+α
uniform ellipticity of F0
•
P
F0 (D2 P) < −δ α
• assume
⇒
F0 (D2 Pδ ) < 0
(
F(D2 uδε , εx , ω) = F0(D2 Pδ ) < 0 in B(x0 , δ)
uδε = Pδ on ∂B(x0 , δ)
•
⇒
•
x0
Bδ(x0)
uδε ≧ uε
kuδε −Pδ k ≦ Cδ 2 (1+δ −σ )e−c| ln(ε/δ)|
• 0 ≦ uδε (x0 ) − uε (x0 ) = uδε (x0 ) − P(x0 ) + P(x0 ) − uε (x0 )
≦ Cδ 2 (1 + δ −σ )e−c| ln(ε/δ)|
1/2
− ηδ α+2 < 0
for δ = Ce−c| ln ε|
1/2
1/2
in Aεδ
116
• rate for quadratic data
8
<F(D2 uε , εx , ω) = F0 (D2 P) in Q1
:
uε = P on ∂Q1
uε −→ P a.s.
ε→0
8
<F0 (D2 u0 ) = F0 (D2 P) in Q1
:u = P on ∂Q
0
1
Q1 unit cube
117
• rate for quadratic data
8
<F(D2 uε , εx , ω) = F0 (D2 P) in Q1
:
uε = P on ∂Q1
•
estimate
uε −→ P a.s.
ε→0
uε − P
8
<F0 (D2 u0 ) = F0 (D2 P) in Q1
:u = P on ∂Q
0
1
Q1 unit cube
118
• rate for quadratic data
8
<F(D2 uε , εx , ω) = F0 (D2 P) in Q1
:
uε = P on ∂Q1
•
estimate
or
(better)
uε −→ P a.s.
ε→0
uε − P
8
<F0 (D2 u0 ) = F0 (D2 P) in Q1
:u = P on ∂Q
0
1
Q1 unit cube
v±
ε −P
x
(F(D2 P, εx , ω)−ℓ)+ in Q1
F(D2 v+
ε , ε , ω) = ℓ+1{v+
ε =P}
obstacle problem with
obstacle P from below
x
F(D2 v−
(F(D2 P, εx , ω)−ℓ)− in Q1
ε , ε , ω) = ℓ−1{v−
ε =P}
obstacle problem with
obstacle P from above
−
v+
ε = vε = P on ∂Q1
(ℓ = F0 (D2 P))
119
• rate for quadratic data
8
<F(D2 uε , εx , ω) = F0 (D2 P) in Q1
:
uε = P on ∂Q1
•
estimate
or
(better)
uε −→ P a.s.
ε→0
uε − P
8
<F0 (D2 u0 ) = F0 (D2 P) in Q1
:u = P on ∂Q
0
1
Q1 unit cube
v±
ε −P
x
(F(D2 P, εx , ω)−ℓ)+ in Q1
F(D2 v+
ε , ε , ω) = ℓ+1{v+
ε =P}
obstacle problem with
obstacle P from below
x
F(D2 v−
(F(D2 P, εx , ω)−ℓ)− in Q1
ε , ε , ω) = ℓ−1{v−
ε =P}
obstacle problem with
obstacle P from above
−
v+
ε = vε = P on ∂Q1
•
(ℓ = F0 (D2 P))
N
sup(v±
ε − P) is controlled by L -norm of rhs (total mass)
120
REVIEW OF KEY FACTS ABOUT UNIFORMLY ELLIPTIC PDE
121
REVIEW OF KEY FACTS ABOUT UNIFORMLY ELLIPTIC PDE
• linearization
F(D2 u, x) = 0
F(D2 v, x) = 0
=⇒
w=u−v
−aij (x)wxi xj = 0
aij bdd meas
122
REVIEW OF KEY FACTS ABOUT UNIFORMLY ELLIPTIC PDE
• linearization
F(D2 u, x) = 0
F(D2 v, x) = 0
=⇒
w=u−v
−aij (x)wxi xj = 0
aij bdd meas
• Alexandrov-Bakelman-Pucci (ABP)-estimate
−aij wxi xj = f in B1
=⇒
sup w+ ≦ sup w+ + Ckf+ kLN
B1
∂B1
(C universal)
123
REVIEW OF KEY FACTS ABOUT UNIFORMLY ELLIPTIC PDE
• linearization
F(D2 u, x) = 0
F(D2 v, x) = 0
=⇒
w=u−v
−aij (x)wxi xj = 0
aij bdd meas
• Alexandrov-Bakelman-Pucci (ABP)-estimate
−aij wxi xj = f in B1
=⇒
sup w+ ≦ sup w+ + Ckf+ kLN
B1
(C universal)
∂B1
• Fabes-Stroock (FS)-estimate
(
−aij wxi xj = f in B1 0<f <1
=⇒ w|B1/2 ≧ Ckf kLM
w = 0 on ∂B1
(C, M universal, M large)
• obstacle problem
u smallest st
• u = Q on ∂B1
8
2
<F(D2 u, x) ≧ 0 in B1=⇒ • F(D u, x) = 0 in {u > Q}
• 0 ≦ (u−Q)(y) ≦ C|x − y|2 (u(x) = Q(x))
:u ≧ Q in B1
• F(D2 u, x) = 1{u=Q} F(Q, x)+
124
•
rate for quadratic data P – wlog F0 (D2 P) = 0
v±
ε solution of obstacle problem from above (below) obstacle P
uε solution
125
•
rate for quadratic data P – wlog F0 (D2 P) = 0
v±
ε solution of obstacle problem from above (below) obstacle P
uε solution
•
−
enough to find rate for kv+
ε − Pk and kvε − Pk
±
kv±
ε − Pk ≦ Chε (ω)
R
y
2
N
1/N
Λ±
h±
ε contact set
ε (ω) = [ Q ∩Λ± [F(D P, ε , ω)± ] dy]
1
ε
126
•
rate for quadratic data P – wlog F0 (D2 P) = 0
v±
ε solution of obstacle problem from above (below) obstacle P
uε solution
•
−
enough to find rate for kv+
ε − Pk and kvε − Pk
•
±
kv±
ε − Pk ≦ Chε (ω)
R
y
2
N
1/N
Λ±
h±
ε contact set
ε (ω) = [ Q ∩Λ± [F(D P, ε , ω)± ] dy]
1
ε
estimate
− 2
2
Hk = E(h+
k ) E(hk )
(ε = 3−k )
127
KEY STEP
Assume
h±
k (ω) ≥ θ > 0
FS
k
± N
+
−
N
−
⇒ v+
F(D2 v±
k , 3 x, ω) = (hk ) ⇒ −aij (vk − vk ) ≧ 2θ
k − vk |Q
monotonicity of obstacle problem
⇒
1/2
≧ 2θMN
±
±
±
Λ±
k+ℓ ⊂ Λ ⇒ if vk+ℓ = P ⇒ vk = P
k - scale
each cell of k + ℓ-scale has diameter 3−k−ℓ
−
MN
v+ − P + P − v− = v+
k − vk ≧ 2θ
|k {z } | {z k}
≧0
≧0
either must have
if
k + l scale
MN
v+
k −P ≧ θ
Λ+
k+ℓ ∩ B1/2 6= ∅
6⇒
or
MN
P − v−
k ≧ θ
3−2(k+ℓ) C ≧ θMN impossible for ℓ large!
• need to go to k + ℓ instead of k + 1 leads to slow rate