WELCOME!
A Conference
in Honor of
ROBERT BRYANT
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
1 / 36
WELCOME!
A Conference
in Honor of
ROBERT BRYANT
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
2 / 36
A GEOMETRIC PERSPECTIVE
ON NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Reese Harvey
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
3 / 36
A GEOMETRIC PERSPECTIVE
ON NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Reese Harvey
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
4 / 36
Fully Nonlinear PDE’s
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Fully Nonlinear PDE’s
Standard Setting:
Ωopen ⊂ Rn
f (x, u, Du, D 2 u) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Fully Nonlinear PDE’s
Standard Setting:
Ωopen ⊂ Rn
f (x, u, Du, D 2 u) = 0
f (x, r , p, A)
Blaine Lawson
on
Ω × R × Rn × Sym2 (Rn )
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Fully Nonlinear PDE’s
Ωopen ⊂ Rn
Standard Setting:
f (x, u, Du, D 2 u) = 0
f (x, r , p, A)
on
Ω × R × Rn × Sym2 (Rn )
Example : The Linear Case.
f =
X
ij
Blaine Lawson
aij (x)
∂ 2 u(x) X
∂u(x)
+
bj (x)
+ c(x)u(x)
∂xi ∂xj
∂xj
j
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Fully Nonlinear PDE’s
Ωopen ⊂ Rn
Standard Setting:
f (x, u, Du, D 2 u) = 0
f (x, r , p, A)
on
Ω × R × Rn × Sym2 (Rn )
Example : The Linear Case.
f =
X
ij
aij (x)
∂ 2 u(x) X
∂u(x)
+
bj (x)
+ c(x)u(x)
∂xi ∂xj
∂xj
j
f (x, r , p, A) = ha(x), Ai + hb(x), pi + c(x)r
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Fully Nonlinear PDE’s
Ωopen ⊂ Rn
Standard Setting:
f (x, u, Du, D 2 u) = 0
f (x, r , p, A)
on
Ω × R × Rn × Sym2 (Rn )
Example : The Linear Case.
f =
X
ij
aij (x)
∂ 2 u(x) X
∂u(x)
+
bj (x)
+ c(x)u(x)
∂xi ∂xj
∂xj
j
f (x, r , p, A) = ha(x), Ai + hb(x), pi + c(x)r
Weak ellipticity: a ≥ 0
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Properness: c ≤ 0
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
5 / 36
Some Standard Examples
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
6 / 36
Some Standard Examples
Example 1. (Laplace).
tr(A) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
6 / 36
Some Standard Examples
Example 1. (Laplace).
tr(A) = 0
Example 2. (Monge-Ampère).
det(A) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
6 / 36
Some Standard Examples
Example 1. (Laplace).
tr(A) = 0
Example 2. (Monge-Ampère).
det(A) = 0
Example 3. (Elementary Symmetric Functions).
σk (A) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
6 / 36
Some Standard Examples
Example 4. (Special Lagrangian Potential).
tr(arctan A) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
7 / 36
Some Standard Examples
Example 4. (Special Lagrangian Potential).
tr(arctan A) = 0
Example 5. (Minimal Surface Equation).
(1 + |p|2 )trA − pt Ap = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
7 / 36
Some Standard Examples
Example 4. (Special Lagrangian Potential).
tr(arctan A) = 0
Example 5. (Minimal Surface Equation).
(1 + |p|2 )trA − pt Ap = 0
Example 6. (k -Laplacian 1 ≤ k ≤ ∞).
|p|2 tr(A) + (k − 2)pt Ap = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
7 / 36
Some Basic Points
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
1. Equations like det(A) = 0 can have many BRANCHES.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
1. Equations like det(A) = 0 can have many BRANCHES.
Given A ∈ Sym2 (Rn ) let
λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λn (A)
denote the ordered eigenvalues of A.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
1. Equations like det(A) = 0 can have many BRANCHES.
Given A ∈ Sym2 (Rn ) let
λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λn (A)
denote the ordered eigenvalues of A. Then
detA = λ1 (A) · · · λn (A) = 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
1. Equations like det(A) = 0 can have many BRANCHES.
Given A ∈ Sym2 (Rn ) let
λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λn (A)
denote the ordered eigenvalues of A. Then
detA = λ1 (A) · · · λn (A) = 0
The k th branch of this equation is
λk (A) = 0.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
1. Equations like det(A) = 0 can have many BRANCHES.
Given A ∈ Sym2 (Rn ) let
λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λn (A)
denote the ordered eigenvalues of A. Then
detA = λ1 (A) · · · λn (A) = 0
The k th branch of this equation is
λk (A) = 0.
(Classical one considers the primary branch λ1 (A) = 0, i.e., A ≥ 0.)
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
8 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
1. Cn = (R2n , J) and A ∈ Sym2 (R2n ).
Set
AC ≡
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1
2 (A
+ JAJ).
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
1. Cn = (R2n , J) and A ∈ Sym2 (R2n ).
Set
AC ≡
1
2 (A
+ JAJ).
Eigenspaces of AC are complex and there are ordered eigenvalues
C
λC
1 (A) ≤ · · · ≤ λn (A).
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
1. Cn = (R2n , J) and A ∈ Sym2 (R2n ).
Set
AC ≡
1
2 (A
+ JAJ).
Eigenspaces of AC are complex and there are ordered eigenvalues
C
λC
1 (A) ≤ · · · ≤ λn (A).
2. Hn = (R4n , I, J, K ) and A ∈ Sym2 (R4n ).
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
1. Cn = (R2n , J) and A ∈ Sym2 (R2n ).
Set
AC ≡
1
2 (A
+ JAJ).
Eigenspaces of AC are complex and there are ordered eigenvalues
C
λC
1 (A) ≤ · · · ≤ λn (A).
2. Hn = (R4n , I, J, K ) and A ∈ Sym2 (R4n ).
Set
AH ≡
Blaine Lawson
1
4 (A
+ IAI + JAJ + IKI).
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
Some Basic Points
2. Equations like
det(A) = 0,
σk (A) = 0, etc.
make sense in the complex and quaternionic worlds.
1. Cn = (R2n , J) and A ∈ Sym2 (R2n ).
Set
AC ≡
1
2 (A
+ JAJ).
Eigenspaces of AC are complex and there are ordered eigenvalues
C
λC
1 (A) ≤ · · · ≤ λn (A).
2. Hn = (R4n , I, J, K ) and A ∈ Sym2 (R4n ).
Set
AH ≡
1
4 (A
+ IAI + JAJ + IKI).
Eigenspaces of AH are quaternionic and there are ordered eigenvalues
H
λH
1 (A) ≤ · · · ≤ λn (A).
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
9 / 36
All the examples above
carry over to any riemannian manifold.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
10 / 36
All the examples above
carry over to any riemannian manifold.
Suppose X is a riemannian manifold with Levi-Civita connection ∇.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
10 / 36
All the examples above
carry over to any riemannian manifold.
Suppose X is a riemannian manifold with Levi-Civita connection ∇.
Definition. For f ∈ C 2 (X ), the riemannian hessian of f is the section
Hess f ∈ Sym2 (T ∗ X )
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
10 / 36
All the examples above
carry over to any riemannian manifold.
Suppose X is a riemannian manifold with Levi-Civita connection ∇.
Definition. For f ∈ C 2 (X ), the riemannian hessian of f is the section
Hess f ∈ Sym2 (T ∗ X )
defined on vector fields V , W by
(Hess f )(V , W ) ≡ V W f − (∇V W )f .
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
10 / 36
The Viscosity Approach (Crandall, Iishi, Lions, Evans).
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
11 / 36
The Viscosity Approach (Crandall, Iishi, Lions, Evans).
The idea: Extend the notion of “solution”
f (x, u, Du, D 2 u) = 0
to more general functions u.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
11 / 36
The Viscosity Approach (Crandall, Iishi, Lions, Evans).
The idea: Extend the notion of “solution”
f (x, u, Du, D 2 u) = 0
to more general functions u.
For this one introduces subsolutions
f (x, u, Du, D 2 u) ≥ 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
11 / 36
The Viscosity Approach (Crandall, Iishi, Lions, Evans).
The idea: Extend the notion of “solution”
f (x, u, Du, D 2 u) = 0
to more general functions u.
For this one introduces subsolutions
f (x, u, Du, D 2 u) ≥ 0
and supersolutions
f (x, u, Du, D 2 u) ≤ 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
11 / 36
We Concentrate on Subsolutions.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Two basic assumptions.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Two basic assumptions. Set
P ≡ {(0, 0, A) : A ≥ 0}
Blaine Lawson
and
N ≡ {(r , 0, 0) : r ≤ 0}
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Two basic assumptions. Set
P ≡ {(0, 0, A) : A ≥ 0}
and
N ≡ {(r , 0, 0) : r ≤ 0}
We will assume positivity (weak ellipticity)
F +P ⊂ F
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Two basic assumptions. Set
P ≡ {(0, 0, A) : A ≥ 0}
and
N ≡ {(r , 0, 0) : r ≤ 0}
We will assume positivity (weak ellipticity)
F +P ⊂ F
and negativity (properness)
F +N ⊂ F
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
We Concentrate on Subsolutions.
Replace the inequality
f (x, u, Du, D 2 u) ≥ 0
by the subset
F ≡ {f ≥ 0}
Two basic assumptions. Set
P ≡ {(0, 0, A) : A ≥ 0}
and
N ≡ {(r , 0, 0) : r ≤ 0}
We will assume positivity (weak ellipticity)
F +P ⊂ F
and negativity (properness)
F +N ⊂ F
F is called a SUBEQUATION.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
12 / 36
Viscosity Subsolutions.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
13 / 36
Viscosity Subsolutions.
Given X open ⊂ Rn , set
USC(X ) ≡ {u : X → [−∞, ∞) : u is upper-semicontinuous}
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
13 / 36
Viscosity Subsolutions.
Given X open ⊂ Rn , set
USC(X ) ≡ {u : X → [−∞, ∞) : u is upper-semicontinuous}
Definition. A C 2 -function ϕ is a test function for u ∈ USC(X ) at a point
x ∈ X if
u ≤ ϕ
near x
u = ϕ
Blaine Lawson
at x
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
13 / 36
Viscosity Subsolutions.
Given X open ⊂ Rn , set
USC(X ) ≡ {u : X → [−∞, ∞) : u is upper-semicontinuous}
Definition. A C 2 -function ϕ is a test function for u ∈ USC(X ) at a point
x ∈ X if
u ≤ ϕ
near x
u = ϕ
at x
Definition. A function u ∈ USC(X ) is F -subharmonic on X if for each x ∈ X
and each test function ϕ for u at x,
Jx2 (u) ∈ F .
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
13 / 36
Viscosity Subsolutions.
Given X open ⊂ Rn , set
USC(X ) ≡ {u : X → [−∞, ∞) : u is upper-semicontinuous}
Definition. A C 2 -function ϕ is a test function for u ∈ USC(X ) at a point
x ∈ X if
u ≤ ϕ
near x
u = ϕ
at x
Definition. A function u ∈ USC(X ) is F -subharmonic on X if for each x ∈ X
and each test function ϕ for u at x,
Jx2 (u) ∈ F .
F (X ) ≡ the set of such functions.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
13 / 36
Properties.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Properties.
F (X ) is closed under:
• Uniform limits
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Properties.
F (X ) is closed under:
• Uniform limits
• Decreasing limits
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Properties.
F (X ) is closed under:
• Uniform limits
• Decreasing limits
• Taking maximum (u, v ∈ F (X ) ⇒ max{u, v } ∈ F (X )).
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Properties.
F (X ) is closed under:
• Uniform limits
• Decreasing limits
• Taking maximum (u, v ∈ F (X ) ⇒ max{u, v } ∈ F (X )).
• Taking upper envelopes: F ⊂ F (X ) locally bounded above, then
∗
v ≡ sup u
u∈F
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Properties.
F (X ) is closed under:
• Uniform limits
• Decreasing limits
• Taking maximum (u, v ∈ F (X ) ⇒ max{u, v } ∈ F (X )).
• Taking upper envelopes: F ⊂ F (X ) locally bounded above, then
∗
v ≡ sup u
u∈F
Like subharmonic functions!
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
14 / 36
Examples.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Examples.
F ≡ {trA ≥ 0}.
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F (X ) = {the subharmonic functions}
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Examples.
F ≡ {trA ≥ 0}.
F ≡ {A ≥ 0}.
Blaine Lawson
F (X ) = {the subharmonic functions}
F (X ) = {the convex functions}
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Examples.
F ≡ {trA ≥ 0}.
F ≡ {A ≥ 0}.
F ≡ {AC ≥ 0}.
Blaine Lawson
F (X ) = {the subharmonic functions}
F (X ) = {the convex functions}
F (X ) = {the plurisubharmonic functions}
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Examples.
F ≡ {trA ≥ 0}.
F ≡ {A ≥ 0}.
F ≡ {AC ≥ 0}.
F ≡ {AH ≥ 0}.
Blaine Lawson
F (X ) = {the subharmonic functions}
F (X ) = {the convex functions}
F (X ) = {the plurisubharmonic functions}
F (X ) = {the H-plurisubharmonic functions}
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Examples.
F ≡ {trA ≥ 0}.
F ≡ {A ≥ 0}.
F ≡ {AC ≥ 0}.
F ≡ {AH ≥ 0}.
F (X ) = {the subharmonic functions}
F (X ) = {the convex functions}
F (X ) = {the plurisubharmonic functions}
F (X ) = {the H-plurisubharmonic functions}
Fk ≡ {λk (A) ≥ 0}.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
15 / 36
Analogues in Calibrated Geometry.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
Analogues in Calibrated Geometry.
Let
φ ∈ Λ p Rn
be a calibration,
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
Analogues in Calibrated Geometry.
Let
φ ∈ Λ p Rn
be a calibration, i.e.
φP ≤ volP
for all oriented p-planes P.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
Analogues in Calibrated Geometry.
Let
φ ∈ Λ p Rn
be a calibration, i.e.
φP ≤ volP
for all oriented p-planes P. Let
G
l (φ) ≡
Blaine Lawson
P : φP = volP
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
Analogues in Calibrated Geometry.
Let
φ ∈ Λ p Rn
be a calibration, i.e.
φP ≤ volP
for all oriented p-planes P. Let
G
l (φ) ≡
P : φP = volP
Definition.
F (φ) ≡
Blaine Lawson
A : tr AP ≥ 0 ∀ P ∈ G
l (φ)
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
Analogues in Calibrated Geometry.
Let
φ ∈ Λ p Rn
be a calibration, i.e.
φP ≤ volP
for all oriented p-planes P. Let
G
l (φ) ≡
P : φP = volP
Definition.
F (φ) ≡
A : tr AP ≥ 0 ∀ P ∈ G
l (φ)
φ-subharmonic functions.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
16 / 36
What about Supersolutions?
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
17 / 36
What about Supersolutions?
A PARALLEL STORY UNDER DUALITY
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
17 / 36
Duality
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Duality
Given a subequation F , define the dual of F by
e ≡ ∼ (−IntF) = −(∼ IntF)
F
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Duality
Given a subequation F , define the dual of F by
e ≡ ∼ (−IntF) = −(∼ IntF)
F
We make mild topological assumptions on F :
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Duality
Given a subequation F , define the dual of F by
e ≡ ∼ (−IntF) = −(∼ IntF)
F
We make mild topological assumptions on F :
(i) F = IntF
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(ii) Fx = Intx Fx
(iii) Intx F = (IntF ) ∩ Fx
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Duality
Given a subequation F , define the dual of F by
e ≡ ∼ (−IntF) = −(∼ IntF)
F
We make mild topological assumptions on F :
(i) F = IntF
(ii) Fx = Intx Fx
(iii) Intx F = (IntF ) ∩ Fx
Then
e is a subequation
F
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Duality
Given a subequation F , define the dual of F by
e ≡ ∼ (−IntF) = −(∼ IntF)
F
We make mild topological assumptions on F :
(i) F = IntF
(ii) Fx = Intx Fx
(iii) Intx F = (IntF ) ∩ Fx
Then
e is a subequation
F
e
e = F
F
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
18 / 36
Examples.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
19 / 36
Solutions – F -harmonicity
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
20 / 36
Solutions – F -harmonicity
Definition. A function u ∈ C(X ) is called F -harmonic if
u is F -subharmonic
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and
e -subharmonic
−u is F
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
20 / 36
The Dirichlet Problem – F -Boundary Convexity
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
21 / 36
The Dirichlet Problem – F -Boundary Convexity
Consider a domain Ω ⊂⊂ Rn with smooth boundary ∂Ω.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
21 / 36
The Dirichlet Problem – F -Boundary Convexity
Consider a domain Ω ⊂⊂ Rn with smooth boundary ∂Ω.
Let F ⊂ Sym2 (Rn ) be a constant coefficient pure second-order subequation
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
21 / 36
The Dirichlet Problem – F -Boundary Convexity
Consider a domain Ω ⊂⊂ Rn with smooth boundary ∂Ω.
Let F ⊂ Sym2 (Rn ) be a constant coefficient pure second-order subequation
Definition.
∂Ω is strictly F -convex at a point x ∈ ∂Ω
if its second fundamental form IIx (w.r.t. the interior normal) satisfes
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t
0
0
IIx
∈ IntF
for all t ≥ some t0 .
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
21 / 36
The Dirichlet Problem
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
22 / 36
The Dirichlet Problem
THEOREM. (Harvey+L.) Let F be a constant coefficient subequation and
e -convex.
suppose ∂Ω ⊂⊂ Rn is both strictly F and F
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July 11, 2013
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The Dirichlet Problem
THEOREM. (Harvey+L.) Let F be a constant coefficient subequation and
e -convex. Then for every
suppose ∂Ω ⊂⊂ Rn is both strictly F and F
ϕ ∈ C(∂Ω), there exists u ∈ C(Ω) such that
u Ω is F -harmonic, and
u ∂Ω = ϕ.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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The Dirichlet Problem
THEOREM. (Harvey+L.) Let F be a constant coefficient subequation and
e -convex. Then for every
suppose ∂Ω ⊂⊂ Rn is both strictly F and F
ϕ ∈ C(∂Ω), there exists u ∈ C(Ω) such that
u Ω is F -harmonic, and
u ∂Ω = ϕ.
Uniqueness holds if F is pure second-order,
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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The Dirichlet Problem
THEOREM. (Harvey+L.) Let F be a constant coefficient subequation and
e -convex. Then for every
suppose ∂Ω ⊂⊂ Rn is both strictly F and F
ϕ ∈ C(∂Ω), there exists u ∈ C(Ω) such that
u Ω is F -harmonic, and
u ∂Ω = ϕ.
Uniqueness holds if F is pure second-order, (or, more generally,
gradient-independent).
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July 11, 2013
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Removable Singularities – Monotonicity Cones
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
Definition. A closed subset E ⊂ X open ⊂ Rn is C ∞ M-polar if
E ⊂ {x : ψ(x) = −∞}
for some ψ ∈ M(X ) which is smooth outside of E.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
Definition. A closed subset E ⊂ X open ⊂ Rn is C ∞ M-polar if
E ⊂ {x : ψ(x) = −∞}
for some ψ ∈ M(X ) which is smooth outside of E.
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone M.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
Definition. A closed subset E ⊂ X open ⊂ Rn is C ∞ M-polar if
E ⊂ {x : ψ(x) = −∞}
for some ψ ∈ M(X ) which is smooth outside of E.
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone M.
Let E ⊂ X be a closed subset (with no interior) which is C ∞ M-polar.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
Definition. A closed subset E ⊂ X open ⊂ Rn is C ∞ M-polar if
E ⊂ {x : ψ(x) = −∞}
for some ψ ∈ M(X ) which is smooth outside of E.
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone M.
Let E ⊂ X be a closed subset (with no interior) which is C ∞ M-polar. Then
any u ∈ F (X − E) which is locally bounded across E
e ∈ F (X ).
extends canonically to a function u
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Monotonicity Cones
Definition. A monotonicity cone for a subequation F is a convex cone
subequation M such that
F + M ⊂ F.
Definition. A closed subset E ⊂ X open ⊂ Rn is C ∞ M-polar if
E ⊂ {x : ψ(x) = −∞}
for some ψ ∈ M(X ) which is smooth outside of E.
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone M.
Let E ⊂ X be a closed subset (with no interior) which is C ∞ M-polar. Then
any u ∈ F (X − E) which is locally bounded across E
e ∈ F (X ).
extends canonically to a function u
THEOREM. (Harvey+L.) Let F and M be as above. Then for u ∈ C(X )
u is F harmonic on X − E
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⇒
u is F harmonic on X .
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Removable Singularities – Example
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
The p-convexity subequation
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
The p-convexity subequation
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone Pp .
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
The p-convexity subequation
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone Pp .
Then every closed set E ⊂ X of locally finite Hausdorff (p − 2)-measure is
removable for F -subharmonics and F -harmonics as above.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
The p-convexity subequation
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone Pp .
Then every closed set E ⊂ X of locally finite Hausdorff (p − 2)-measure is
removable for F -subharmonics and F -harmonics as above.
The proof uses Riesz potentials
ψ ≡ Kp ∗ µ
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Example
Fix 1 ≤ p ≤ n and let
Pp ≡ {A : λ1 (A) + · · · + λ[p] (A) + (p − [p])λp+1 (A) ≥ 0}
The p-convexity subequation
THEOREM. (Harvey+L.) Let F be a subequation with monotonicity cone Pp .
Then every closed set E ⊂ X of locally finite Hausdorff (p − 2)-measure is
removable for F -subharmonics and F -harmonics as above.
The proof uses Riesz potentials
ψ ≡ Kp ∗ µ

1

− |x|p−2 2 < p ≤ n − 1
Kp (x) ≡ log|x|
p=2

 2−p
|x|
1≤p<2
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July 11, 2013
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Removable Singularities – Riesz Characteristic
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Removable Singularities – Riesz Characteristic
Let F ⊂ Sym2 (Rn ) be a pure second-order c.c. subequation.
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Removable Singularities – Riesz Characteristic
Let F ⊂ Sym2 (Rn ) be a pure second-order c.c. subequation.
Definition. The Riesz characteristic of F is the number
αF ≡ sup{α : I − αPe ∈ F ∀ e}
where Pe is orthogonal projection onto the e-line.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Removable Singularities – Riesz Characteristic
Let F ⊂ Sym2 (Rn ) be a pure second-order c.c. subequation.
Definition. The Riesz characteristic of F is the number
αF ≡ sup{α : I − αPe ∈ F ∀ e}
where Pe is orthogonal projection onto the e-line.
THEOREM. (Harvey+L.) Suppose F ⊂ Sym2 (Rn ) has Riesz characteristic α.
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Removable Singularities – Riesz Characteristic
Let F ⊂ Sym2 (Rn ) be a pure second-order c.c. subequation.
Definition. The Riesz characteristic of F is the number
αF ≡ sup{α : I − αPe ∈ F ∀ e}
where Pe is orthogonal projection onto the e-line.
THEOREM. (Harvey+L.) Suppose F ⊂ Sym2 (Rn ) has Riesz characteristic α.
Then every closed set E ⊂ X of locally finite Hausdorff (α − 2)-measure is
removable for F -subharmonics and F -harmonics as above.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Example – Elementary Symmetric Functions of D 2 u
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Example – Elementary Symmetric Functions of D 2 u
Let
Fk ≡ {A ∈ Sym2 (Rn ) : σ1 (A) ≥ 0, ..., σk (A) ≥ 0}
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Example – Elementary Symmetric Functions of D 2 u
Let
Fk ≡ {A ∈ Sym2 (Rn ) : σ1 (A) ≥ 0, ..., σk (A) ≥ 0}
Fk has Riesz characteristic
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A Geometric Perspective on Nonlinear PDE’s
n
k
July 11, 2013
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Example – Elementary Symmetric Functions of D 2 u
Let
Fk ≡ {A ∈ Sym2 (Rn ) : σ1 (A) ≥ 0, ..., σk (A) ≥ 0}
Fk has Riesz characteristic
n
k
THEOREM. Every closed E ⊂ X of locally finite Hausdorff ( kn − 2)-measure
is removable for Fk -subharmonics and Fk -harmonics.
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Another Example – Branches of the Complex
Monge-Ampère Equation
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Another Example – Branches of the Complex
Monge-Ampère Equation
In Cn
PkC ≡ {A ∈ Sym2 (R2n ) : λk (AC ) ≥ 0}
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July 11, 2013
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Another Example – Branches of the Complex
Monge-Ampère Equation
In Cn
PkC ≡ {A ∈ Sym2 (R2n ) : λk (AC ) ≥ 0}
A set E ⊂ Cn is pluripolar if E = {u = −∞} for some plurisubharmonic
function u.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Another Example – Branches of the Complex
Monge-Ampère Equation
In Cn
PkC ≡ {A ∈ Sym2 (R2n ) : λk (AC ) ≥ 0}
A set E ⊂ Cn is pluripolar if E = {u = −∞} for some plurisubharmonic
function u.
THEOREM. Any pluirpolar set is removable for all branches of the complex
Monge-Ampère equation.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Differential Equations on Manifolds.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Differential Equations on Manifolds.
Let X be any manifold.
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Differential Equations on Manifolds.
Let X be any manifold.
Definition. The 2-jet bundle of X is the vector bundle
J 2 (X ) −→ X
whose fibre at x ∈ X is
∞
Jx2 (X ) ≡ Cx∞ /Cx,3
∞
= germs of functions which vanish to order 3 at x.
where Cx,3
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Differential Equations on Manifolds.
Let X be any manifold.
Definition. The 2-jet bundle of X is the vector bundle
J 2 (X ) −→ X
whose fibre at x ∈ X is
∞
Jx2 (X ) ≡ Cx∞ /Cx,3
∞
= germs of functions which vanish to order 3 at x.
where Cx,3
There is a short exact sequence
0 → Sym2 (T∗ X) → J2 (X) → J1 (X) → 0.
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July 11, 2013
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All of our Definitions Transfer to this Setting.
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All of our Definitions Transfer to this Setting.
Definition. A second-order subequation on X is a closed subset
F ⊂ J 2 (X )
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All of our Definitions Transfer to this Setting.
Definition. A second-order subequation on X is a closed subset
F ⊂ J 2 (X )
satisfying the positivity, negativity and topological conditions above:
F +P ⊂ F
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F +N ⊂ F
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All of our Definitions Transfer to this Setting.
Definition. A second-order subequation on X is a closed subset
F ⊂ J 2 (X )
satisfying the positivity, negativity and topological conditions above:
F +P ⊂ F
F +N ⊂ F
and
(i) F = IntF
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(ii) Fx = Intx Fx
(iii) Intx F = (IntF ) ∩ Fx
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All of our Definitions Transfer to this Setting.
Definition. A second-order subequation on X is a closed subset
F ⊂ J 2 (X )
satisfying the positivity, negativity and topological conditions above:
F +P ⊂ F
F +N ⊂ F
and
(i) F = IntF
(ii) Fx = Intx Fx
(iii) Intx F = (IntF ) ∩ Fx
e , F -harmonic, etc. as before.
Concepts of F -subharmonic, F
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The Riemannian Hessian gives a Splitting
of the Sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → J 1 (X ) → 0.
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The Riemannian Hessian gives a Splitting
of the Sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → J 1 (X ) → 0.
so that
J 2 (X ) = R ⊕ T ∗ X ⊕ Sym2 (T ∗ X )
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July 11, 2013
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Jet-Equivalence.
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Jet-Equivalence.
Definition. An automorphism of J 2 (X ) is a bundle isomorphism
Φ : J 2 (X ) −→ J 2 (X )
which respects the exact sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → R ⊕ T ∗ X → 0.
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July 11, 2013
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Jet-Equivalence.
Definition. An automorphism of J 2 (X ) is a bundle isomorphism
Φ : J 2 (X ) −→ J 2 (X )
which respects the exact sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → R ⊕ T ∗ X → 0.
and there are bundle isomorphisms
g : T ∗X → T ∗X
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and
h : T ∗X → T ∗X
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Jet-Equivalence.
Definition. An automorphism of J 2 (X ) is a bundle isomorphism
Φ : J 2 (X ) −→ J 2 (X )
which respects the exact sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → R ⊕ T ∗ X → 0.
and there are bundle isomorphisms
g : T ∗X → T ∗X
and
h : T ∗X → T ∗X
so the restriction of Φ to Sym2 (T ∗ X ) is
Φ(A) = gAg t
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Jet-Equivalence.
Definition. An automorphism of J 2 (X ) is a bundle isomorphism
Φ : J 2 (X ) −→ J 2 (X )
which respects the exact sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → R ⊕ T ∗ X → 0.
and there are bundle isomorphisms
g : T ∗X → T ∗X
and
h : T ∗X → T ∗X
so the restriction of Φ to Sym2 (T ∗ X ) is
Φ(A) = gAg t
and the induced map on R ⊕ T ∗ X is Id⊕h.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Jet-Equivalence.
Definition. An automorphism of J 2 (X ) is a bundle isomorphism
Φ : J 2 (X ) −→ J 2 (X )
which respects the exact sequence
0 → Sym2 (T ∗ X ) → J 2 (X ) → R ⊕ T ∗ X → 0.
and there are bundle isomorphisms
g : T ∗X → T ∗X
and
h : T ∗X → T ∗X
so the restriction of Φ to Sym2 (T ∗ X ) is
Φ(A) = gAg t
and the induced map on R ⊕ T ∗ X is Id⊕h.
An affine automorphism is Ψ = J + Φ where Φ is an automorphism and J is
a section of J 2 (X ).
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Jet-Equivalence.
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Jet-Equivalence.
Definition. Two subequations F , F 0 are affinely jet-equivalent if there exists
an affine automorphism Ψ : J 2 (X ) → J 2 (X ) such that
Φ(F ) = F 0
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The Dirichlet Problem.
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The Dirichlet Problem.
Let F be a subequation on a manifold X .
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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The Dirichlet Problem.
Let F be a subequation on a manifold X .
Suppose there exists some strictly M-subharmonic function on X where M is
a monotonicity cone for F .
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July 11, 2013
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The Dirichlet Problem.
Let F be a subequation on a manifold X .
Suppose there exists some strictly M-subharmonic function on X where M is
a monotonicity cone for F .
THEOREM. (Harvey+L.) Suppose F is locally affinely jet-equivalent to a
constant coefficient subequation.
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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The Dirichlet Problem.
Let F be a subequation on a manifold X .
Suppose there exists some strictly M-subharmonic function on X where M is
a monotonicity cone for F .
THEOREM. (Harvey+L.) Suppose F is locally affinely jet-equivalent to a
constant coefficient subequation. Then for every domain Ω ⊂⊂ X whose
e convex,
boundary is strictly F and F
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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The Dirichlet Problem.
Let F be a subequation on a manifold X .
Suppose there exists some strictly M-subharmonic function on X where M is
a monotonicity cone for F .
THEOREM. (Harvey+L.) Suppose F is locally affinely jet-equivalent to a
constant coefficient subequation. Then for every domain Ω ⊂⊂ X whose
e convex, the Dirichlet problem for F -harmonic
boundary is strictly F and F
functions is uniquely solvable for all continuous boundary data ϕ ∈ ∂Ω.
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July 11, 2013
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Example: Manifolds with G-Structure.
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Example: Manifolds with G-Structure.
Fix a subgroup G ⊂ O(n)
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Example: Manifolds with G-Structure.
Fix a subgroup G ⊂ O(n) and let
F ⊂ J2 = R × Rn × Sym2 (Rn )
be a constant coefficient subequation which is G-invariant.
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July 11, 2013
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Example: Manifolds with G-Structure.
Fix a subgroup G ⊂ O(n) and let
F ⊂ J2 = R × Rn × Sym2 (Rn )
be a constant coefficient subequation which is G-invariant.
Then F determines a subequation F on every riemannian manifold with
G-structure (not necessarily integrable).
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Example: Manifolds with G-Structure.
Fix a subgroup G ⊂ O(n) and let
F ⊂ J2 = R × Rn × Sym2 (Rn )
be a constant coefficient subequation which is G-invariant.
Then F determines a subequation F on every riemannian manifold with
G-structure (not necessarily integrable). Furthermore, each such F is
locally jet-equivalent to the constant coefficient equation F.
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July 11, 2013
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Additional Applications.
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Additional Applications.
Jet Equivalence allows one to:
• Solve the Dirichlet Problem for inhomogeneous equations, e.g.,
λk (Hess u) = f (x).
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A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Additional Applications.
Jet Equivalence allows one to:
• Solve the Dirichlet Problem for inhomogeneous equations, e.g.,
λk (Hess u) = f (x).
• Solve the Dirichlet Problem for parabolic equations.
Blaine Lawson
A Geometric Perspective on Nonlinear PDE’s
July 11, 2013
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Additional Applications.
Jet Equivalence allows one to:
• Solve the Dirichlet Problem for inhomogeneous equations, e.g.,
λk (Hess u) = f (x).
• Solve the Dirichlet Problem for parabolic equations.
• Solve the Dirichlet Problem for equations with obstacles.
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July 11, 2013
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HAPPY BIRTHDAY ROBERT!!
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A Geometric Perspective on Nonlinear PDE’s
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