Noncoercive Hamiltonians, Absolute Minimizers and ......P. Soravia Absolute minimizers and Aronsson...
Transcript of Noncoercive Hamiltonians, Absolute Minimizers and ......P. Soravia Absolute minimizers and Aronsson...
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P. Soravia Absolute minimizers and Aronsson equation – 1 / 33
Noncoercive Hamiltonians, AbsoluteMinimizers and Aronsson Equation
Pierpaolo SoraviaUniversita di Padova (Italy)
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The problem
The problem
The AronssonEquation
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 2 / 33
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The problem
The AronssonEquation
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 3 / 33
For f : Rn × A → Rn, h : Rn × A → R bounded from
below, we have a Hamiltonian
H(x, p) = supa∈A
−f(x, a) · p− h(x, a)(≥ 0).
A may be unbounded. H is continuous but in generalnot coercive.Given Ω ⊂ R
n and g : ∂Ω → R we want to minimize
supx∈Ω
H(x,Du(x)),
subject to u(x) = g(x), for x ∈ ∂Ω.
Question. Choose the correct class of functions.
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The problem
The AronssonEquation
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 4 / 33
If H(x, ·) is coercive (H(x, p) → +∞ as |p| → +∞) thenit is natural to consider u ∈ W 1,+∞
loc (Ω) ∩ C(Ω) so that
ess sup H(x,Du(x)) → min .
A classical problem: Lipschitz extension: if g ∈ Lip(∂Ω),
|g(x)− g(y)| ≤ L|x− y|, x, y ∈ ∂Ω,
find u ∈ Lip(Ω), u = g on ∂Ω such that
|u(x)− u(y)| ≤ L|x− y|, x, y ∈ Ω.
Equivalently, minimize
ess sup |Du(x)|, (or ess sup|Du(x)|2
2).
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The problem
The AronssonEquation
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 5 / 33
The Lipschitz extension problem has possibly manysolutions, and the extremal ones are explicit.It is a nonstandard variational problem versus
∫
Ω
|Du(x)|p dx → min
(worst case analysis) and its solutions do not have ingeneral the property of being local minima in order to beable to derive an Euler equation.
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The Aronsson Equation
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 6 / 33
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 7 / 33
We will introduce the notion of Absolute minimizer andthe Aronsson equation
−D (H(x,Du(x))) ·DpH(x,Du(x)) = 0, x ∈ Ω (AE)u(x) = g(x)∈ C(∂Ω),
where Ω ⊂ RN is open and connected, H = H(x, p),
H : Ω× RN → [0,+∞) and H(x, ·) is convex and in
general not coercive, u : Ω → R.
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 7 / 33
We will introduce the notion of Absolute minimizer andthe Aronsson equation
−D (H(x,Du(x))) ·DpH(x,Du(x)) = 0, x ∈ Ω (AE)u(x) = g(x)∈ C(∂Ω),
where Ω ⊂ RN is open and connected, H = H(x, p),
H : Ω× RN → [0,+∞) and H(x, ·) is convex and in
general not coercive, u : Ω → R.
Classical case: H ∈ C1, u ∈ C2. (AE) is a quasilineardegenerate elliptic pde.Taking derivatives AE becomes
−Tr(DpH ⊗DpH D2u)−DxH ·DpH = 0
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 8 / 33
Recall that:
H(x, p) = supa∈A
−f(x, a) · p− h(x, a),
the problem can be reduced to H ≥ 0, total controllability of the vectogram f(x,A) in every
connected and open subdomain is a crucialingredient.
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 9 / 33
Included is the case of a Carnot-Caratheodory structure:
H(x, p) = H(x, σt(x)p) (1)
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 9 / 33
Included is the case of a Carnot-Caratheodory structure:
H(x, p) = H(x, σt(x)p) (1)
σ : Ω → RN×M , M ≤ N , is a matrix valued function
whose columns are a family of vector fields σjj=1,...,M ,satisfying Hormander’s finite rank condition andH : Ω× R
M → [0,+∞) is coercive.
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Infinity Laplace equation
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 10 / 33
When in particular (for A = σσt, N ×N matrix,)
H(x, p) =1
2A(x)p · p =
1
2|σt(x)Du(x)|2,
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Infinity Laplace equation
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 10 / 33
When in particular (for A = σσt, N ×N matrix,)
H(x, p) =1
2A(x)p · p =
1
2|σt(x)Du(x)|2,
we obtain the infinity-Laplace equation with respect tothe family of vector fields,
−∆σ∞u(x) = −D2
σu(x) Dσu(x) ·Dσu(x) = 0, x ∈ D,
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Infinity Laplace equation
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 10 / 33
When in particular (for A = σσt, N ×N matrix,)
H(x, p) =1
2A(x)p · p =
1
2|σt(x)Du(x)|2,
we obtain the infinity-Laplace equation with respect tothe family of vector fields,
−∆σ∞u(x) = −D2
σu(x) Dσu(x) ·Dσu(x) = 0, x ∈ D,
where Dσu(x) = σt(x)Du(x) are the directionalderivatives with respect to the family of vector fields(horizontal gradient) and the horizontal hessianD2
σu(x) =(
12(Dσ
j (Dσi u(x)) +Dσ
i (Dσj u(x)))
)
i,j.
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Variational theory
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 11 / 33
Aronsson’s Lipschitz extension problem (1965): giveng ∈ Lip(∂Ω), find u : Ω → R such that u|∂Ω = g with thesame best Lipschitz constant.Equivalently minimize the following functional
‖Du‖L∞(Ω) → min
with prescribed boundary conditions.
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Variational theory
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 11 / 33
Aronsson’s Lipschitz extension problem (1965): giveng ∈ Lip(∂Ω), find u : Ω → R such that u|∂Ω = g with thesame best Lipschitz constant.Equivalently minimize the following functional
‖Du‖L∞(Ω) → min
with prescribed boundary conditions.The problem has minima and they are nonunique. Toderive a Euler-Lagrange equation, define AbsoluteMinimizers, i.e. minima in every relatively compact opensubdomain.
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 12 / 33
Aronsson proves that u ∈ C2 is an absolute minimizer iffit solves the infinity-Laplace equation. Existence ofabsolute minimizers is obtained through approximationwith more classical variational problems
∫
Ω
|Du|pdx → min,
as p → +∞.
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 12 / 33
Aronsson proves that u ∈ C2 is an absolute minimizer iffit solves the infinity-Laplace equation. Existence ofabsolute minimizers is obtained through approximationwith more classical variational problems
∫
Ω
|Du|pdx → min,
as p → +∞.In general given a Hamiltonian H(x, p), for u : Ω → R ,the problem
ess supx∈Ω H(x,Du(x)) → min
corresponds to the Aronsson equation.
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More references
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 13 / 33
Jensen (1993): obtains Aronsson’s results for theinfinity-Laplacian without the regularity assumptionon u;
Juutinen (1998/2002): extends Jensen to a class ofstrictly convex functionals and existence of theabsolutely minimizing Lipschitz extension in lengthspaces;
Barron-Jensen-Wang (2001): derivation of AE forgeneral Hamiltonian and existence of absoluteminimizers via approximation with Lp minimization;
Barles-Busca (2001): uniqueness proof for a classof quasilinear x-independent equations;
Crandall (2003)/Crandall-Wang-Yu (2009) generalderivation of AE for C1 Hamiltonians.
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The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 14 / 33
Bieske (2005) equivalence of AM and solutions ofAE for Riemannian metrics and Grushin;
Bieske-Capogna (2005)/Wang (2007): derivation ofAE in Carnot-Caratheodory spaces, uniqueness inCarnot groups;
Peres-Schramm-Sheffield-Wilson (2009) Tug of warapproach, existence and uniqueness of theabsolutely minimizing Lipschitz extension in lengthspaces;
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towards a pde approach:
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 15 / 33
Crandall-Evans-Gariepy (2001): for infinity-Laplace,comparison with cones, continuity estimates;
Aronsson-Crandall-Juutinen (2004): a complete pdeapproach for the infinity-Laplace equation(euclidean), comparison with cones, Perron’smethod, Harnack inequality.
Champion-De Pascale (2007): equivalence of AMand comparison with cones in length spaces.
Jensen-Wang-Yu (2009) results on uniqueness andnon uniqueness, x−independent;
Savin (2000) C1 regularity of infinitely harmonicfunctions in dimention 2.
Evans-Savin (2008) C1,α regularity of infinitelyharmonic functions in dimention 2.
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AE and optimal control
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 16 / 33
There are important relationships with solutions of theHJ equations (for us H(x, · ) is not symmetric)
H(x,Du(x)) = k > 0, H(x,−Du(x)) = k, x ∈ Ω.
(recall that H(x, p) = supa∈A−f(x, a) · p− h(x, a).)
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AE and optimal control
The problem
The AronssonEquation
Infinity Laplaceequation
Variational theory
More references
towards a pdeapproach:
AE and optimalcontrol
Cone functions
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 16 / 33
There are important relationships with solutions of theHJ equations (for us H(x, · ) is not symmetric)
H(x,Du(x)) = k > 0, H(x,−Du(x)) = k, x ∈ Ω.
(recall that H(x, p) = supa∈A−f(x, a) · p− h(x, a).)These are value functions of optimal control problems,thus we consider the control system
y(t) = f(y(t), a(t)), t > 0,y(0) = xo,
(2)
or its backward version y(t) = −f(y(t), a(t)), for suitablecontrol functions a(·) ∈ A.
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Cone functions
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 17 / 33
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The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 18 / 33
We assume Total controllability. For all x, z ∈ D ⊂⊂ Ω,D open and connected, the set
ADx,z = a(·) ∈ A : y(0, a) = x, y(tx,z; a) = z, tx,z ≥ 0y(t, a) ∈ D, t ∈ (0, tx,z) 6= ∅.
(3)
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The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 18 / 33
We assume Total controllability. For all x, z ∈ D ⊂⊂ Ω,D open and connected, the set
ADx,z = a(·) ∈ A : y(0, a) = x, y(tx,z; a) = z, tx,z ≥ 0y(t, a) ∈ D, t ∈ (0, tx,z) 6= ∅.
(3)
For any (large) k > 0, we therefore define the followingfunction (generalized cone). For convenience J1 ≡ J .
JDk (x, z) = inf
a(·)∈Ax,z
∫ tx,z
0
(h(y(t), a(t)) + k)dt < +∞,
JDk (x, z) = lim inf
D×D∋(x,z)→(x,z)JDk (x, z), (x, z) ∈ D ×D.
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Basic setting
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 19 / 33
Each Jk is a semi-distance in D (cfr. PL Lions’ book, itlacks symmetry) and satisfies
J(x, z) ≥ λK |x− z|(≥ 0) for J(x, z) ≤ K.
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Basic setting
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 19 / 33
Each Jk is a semi-distance in D (cfr. PL Lions’ book, itlacks symmetry) and satisfies
J(x, z) ≥ λK |x− z|(≥ 0) for J(x, z) ≤ K.
We assume it is ”uniformly continuous” and satisfieslocal estimates in D of the form
J(x, z) ≤ ω(|x− z|),
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Basic setting
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 19 / 33
Each Jk is a semi-distance in D (cfr. PL Lions’ book, itlacks symmetry) and satisfies
J(x, z) ≥ λK |x− z|(≥ 0) for J(x, z) ≤ K.
We assume it is ”uniformly continuous” and satisfieslocal estimates in D of the form
J(x, z) ≤ ω(|x− z|),
Thus
H(x,DxJk(x, z)) = k, x ∈ D\z, Jk(z, z) = 0.H(z,−DzJk(x, z)) = k, z ∈ D\x, Jk(x, x) = 0.
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Distances and AE equation
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 20 / 33
Although V (x) = Jk(x, z) satisfies the HJ equation:
H(x,DV (x)) = k, x ∈ D\z, (HJ),
we have thatTheorem. V (x) = Jk(x, z), k > 0, is a viscositysupersolution of the Aronsson equation
−Dx (H(x,DV (x))) ·DpH(x,DV (x)) = 0, x ∈ D\z,
and it is a viscosity solution iff it is a bilateral solution of(HJ) (see Barron-Jensen, Barles, Pi. Sor.). In this caseV is also unique as an absolute minimizer.
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A-priori regularity
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 21 / 33
An important feature of the distances is that theydetermine the regularity of subsolutions of the HJequation: if u is a viscosity solution of H(x,Du(x)) ≤ kin Ω, then locally (C depends on sup |u|)
|u(x)− u(y)| ≤ C J(x, y).
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A-priori regularity
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 21 / 33
An important feature of the distances is that theydetermine the regularity of subsolutions of the HJequation: if u is a viscosity solution of H(x,Du(x)) ≤ kin Ω, then locally (C depends on sup |u|)
|u(x)− u(y)| ≤ C J(x, y).
In the CC case H(x, σt(x)p), subsolutions are locallyLipschitz continuous with respect to the CC distance.
dDCC(x, z) = infa∈AD
x,z
tx,z.
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A-priori regularity
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 21 / 33
An important feature of the distances is that theydetermine the regularity of subsolutions of the HJequation: if u is a viscosity solution of H(x,Du(x)) ≤ kin Ω, then locally (C depends on sup |u|)
|u(x)− u(y)| ≤ C J(x, y).
In the CC case H(x, σt(x)p), subsolutions are locallyLipschitz continuous with respect to the CC distance.
dDCC(x, z) = infa∈AD
x,z
tx,z.
We can use the well established theory of Sobolevspaces for CC metrics (e.g. Franchi-Serapioni-SerraCassano, Garofalo-Nhieu, Franchi-Hajlasz-Koskela,Bonfiglioli-Lanconelli-Uguzzoni).
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The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 22 / 33
In particular:Proposition. (Pi. Sor. 2010) For u : Ω → R we havethat
H(x, σt(x)Du(x)) ≤ k
in the viscosity sense if and only if
H(x,Xu(x)) ≤ k, a.e. x ∈ Ω.
Here the differential operator X = σt(x)D = Dσ has tobe interpreted in the sense of distributions.This holds in particular if σ(x) ≡ In.
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The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 23 / 33
On Jk we also assume
Jk(x, z) ≥ κD1 (J(x, z))κ
D2 (k), for all z ∈ D, (4)
if the set x : J(x, z) ≤ r ⊂ Ω. Here
κD1 , κ
D2 : (0,+∞) → (0,+∞) are increasing and κD
2 issurjective.This happens for instance for:
A bounded ; A unbounded if H(x, ·) is positively homogeneous.
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Generalized cone functions
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 24 / 33
A (generalized) cone with positive slope k > 0 andvertex z ∈ R
n, possibly constrained in D with z ∈ ∂aD isa function (b ∈ R)
C(x) = Jk(x, z) + b.
A (generalized) cone with negative slope k < 0 andvertex z ∈ R
n is a function
C(x) = −J|k|(z, x) + b.
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Viscosity absolute minimizers
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 25 / 33
Definition. Given a Hamiltonian H ∈ C(Ω× Rn), H(x, ·)
convex for all x, we say that u ∈ C(U), U ⊂ Ω open, isan absolute minimizer (in the viscosity sense) for H if forany open, bounded subset D ⊂⊂ U we have thatwhenever v ∈ C(D) is such that u(x) = v(x) in ∂D and
H(x,Dv(x)) ≤ k, ∀x ∈ D
in the viscosity sense, then
H(x,Du(x)) ≤ k, ∀x ∈ D, (5)
in the viscosity sense.
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A delicate property of the HJB
The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 26 / 33
For the forward and backward Hamiltonian it holds that ifu ∈ C(Ω), then
H(x,Du(x)) ≤ k, x ∈ Ω
in the viscosity sense, if and only if
H(x,−D(−u)(x)) ≤ k, x ∈ Ω.
Note: u absolute minimizer for H implies that −u isabsolute minimizer for H(x,−p).Also the variational problem may be seen as
sup(x,p)∈D×(D+u(x)∪D−u(x))
H(x, p) → min .
(Recall here that D+(−u) = −D−u.)
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The problem
The AronssonEquation
Cone functions
Basic setting
Distances and AEequation
A-priori regularity
Generalized conefunctions Viscosity absoluteminimizers A delicateproperty of the HJB
Perron’s method
P. Soravia Absolute minimizers and Aronsson equation – 27 / 33
A locally dCC-Lipschitz continuous function u : U → R, isan absolute minimizer for H if and only if for any open,bounded subset D, D ⊂⊂ U we have that wheneverv ∈ C(D) ∩W 1,∞
X (D) is such that u(x) = v(x) in ∂D and
H(x,Xv(x)) ≤ k, a.e. x ∈ D,
then
H(x,Xu(x)) ≤ k, a.e. x ∈ D. (6)
This is as saying that u is a local minimizer in U for thevariational problem
ess supH(x,Xv(x)) → min
among all locally dCC-Lipschitz continuous functions v.
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Perron’s method
The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 28 / 33
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Comparison with cones
The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 29 / 33
We have the following characterization.Theorem. (Pi. Sor., cfr. Champions-De Pascale)u ∈ C(U) is an absolute minimizer if and only if:
u ∈ CCA(U): for any open, connected and boundedset V ⊂⊂ U , k > 0, z /∈ V and cone C(x) (possiblyrelative to V ) with slope k and vertex z we have that
u(x)− C(x) ≤ supw∈∂V
u(w)− C(w), for all x ∈ V ;
u ∈ CCB(U) i.e. for k < 0, z /∈ V we have that
u(x)− C(x) ≥ infw∈∂V
u(w)− C(w), for all x ∈ V.
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Properties of functions in CCA andCCB
The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 30 / 33
Propositions.
(local Lipschitz/Holder continuity) If u ∈ CCA(D)and E ⊂⊂ D, then we can find a constant L,depending only on ‖u‖∞, κE
1 , κE2 and infy∈E d(y, ∂D),
such that
|u(x)− u(y)| ≤ LJ(x, y), ∀z ∈ E,x, y ∈ BR(z), 3R < infz∈E d(z, ∂D).
Uniformly bounded families of functions in CCA(D)are locally equi-Lipschitz continuous with respect tothe distance J induced by the Hamiltonian;
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The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 31 / 33
(strong maximum principle) if u ∈ CCA(D) has alocal maximum, then it is locally constant;
(existence of functions in CCA) a cone C withnegative slope satisfies C ∈ CCA (and it is aviscosity subsolution of the (AE));
(Harnack inequality) If H(x, ·) is positiveyhomogeneous, u ≥ 0, u ∈ CCB(D), then for z ∈ D
maxJ(x,z)≤R
u(x) ≤ 3 minJ(y,z)≤R
u(y), R <d(z, ∂D)
4;
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The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 32 / 33
Let ∅ 6= F ⊂ CCA(D) and
h(x) = supv∈F
v(x).
If h is locally bounded from above thenh ∈ CCA(D) ∩ C(D).
We couple the last property with the more standard(in viscosity solutions theory): Let ∅ 6= F ⊂ C(D) bea family of viscosity subsolutions of (AE),
h(x) = supv∈F
v(x).
If h ∈ C(D) then h is a viscosity subsolution of the(AE).
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Existence
The problem
The AronssonEquation
Cone functions
Perron’s method Comparison withcones Properties offunctions in CCAand CCB
Existence
P. Soravia Absolute minimizers and Aronsson equation – 33 / 33
Theorem. (Pi.Sor. 2010) Let D ⊂ Ω open, g ∈ C(∂D),b−, b+ ∈ R, k− < 0, k+ > 0, z ∈ ∂D:
C−(x) = −J|k−|(z, x) + b− ≤ g(x) ≤ Jk+(x, z) + b+ = C+(x),
for all x ∈ ∂D. Then there exists u ∈ C(D) an absoluteminimizer, such that u = g on ∂D and
C−(x) ≤ u(x) ≤ C+(x), x ∈ D.
If moreover H ∈ C1(D × RN ) then there exists an
absolute minimizer that is in addition a viscosity solutionof the AE.NB. Perron’s method cannot prove that AE is satisfiedby all of the absolute minimizers (unless comparisonprinciple holds).