Post on 12-Jun-2020
university-logo
Regolarità SBV delle soluzioni di viscositàdi equazioni di Hamilton-Jacobi in Rn.
Roger Robyr
Università di Zurigo - Istituto di Matematica.
Università di Roma "La Sapienza" – Gennaio 2011
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 1 / 32
university-logo
BV functions of several variables
DefinitionLet Ω ⊂ Rn be an open set. Let u ∈ L1(Ω); we say that u ∈ BV (Ω) ifthe distributional derivative of u, denoted by Du, is representable by afinite Radon measure in Ω, i.e. if∫
Ωu∂φ
∂xidx = −
∫ΩφdDiu ∀φ ∈ C∞c (Ω), i = 1, . . . ,n
for some Rn-valued measure Du = (D1u, . . . ,Dnu) in Ω. A functionu ∈ L1
loc(Ω) belongs to BVloc(Ω) if for each open set V ⊂⊂ Ω,u ∈ BV (V ).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 2 / 32
university-logo
BV functions are, roughly speaking, functions with a bound on the totalamount of oscillations.The total amount of the oscillations can be measured by |Du|, which isthe total variation of the Radon measure Du. Then,
We can characterize the BV space as the class of u ∈ L1(Ω) functionswith finite (total) variation, |Du|(Ω) <∞.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 3 / 32
university-logo
BV functions are, roughly speaking, functions with a bound on the totalamount of oscillations.The total amount of the oscillations can be measured by |Du|, which isthe total variation of the Radon measure Du. Then,
We can characterize the BV space as the class of u ∈ L1(Ω) functionswith finite (total) variation, |Du|(Ω) <∞.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 3 / 32
university-logo
Functions of bounded variation in 1− d
To give an intuitive idea of BV functions let us consider theone-dimensional case.Remarks:
I If u is differentiable and u′ ∈ L1([a,b]), its total variation is
|Du|(I) =
∫ b
a|u′(t)|d t . <∞
I If u(x) =
0, on ]− a,0];1, on [0,a[.
Then, |Du|(I) = 1 <∞.
I L∞(R) is not a subspace of BV (R): for instance consideru(x) = sin(1/x) on R+.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 4 / 32
university-logo
Functions of bounded variation in 1− d
To give an intuitive idea of BV functions let us consider theone-dimensional case.Remarks:
I If u is differentiable and u′ ∈ L1([a,b]), its total variation is
|Du|(I) =
∫ b
a|u′(t)|d t . <∞
I If u(x) =
0, on ]− a,0];1, on [0,a[.
Then, |Du|(I) = 1 <∞.
I L∞(R) is not a subspace of BV (R): for instance consideru(x) = sin(1/x) on R+.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 4 / 32
university-logo
Cantor function
The Cantor function c on [0,1] is a monotone increasing continuousfunction such that
2c(x/3) = c(x) and c(x) + c(1− x) = 1.
I c ∈ BV ([0,1]), i.e. c is a function of bounded variation!I It is continuous but not absolutely continuous.I Its derivative Dc is a singular nonatomic measure, which contains
a fractal structure (the construction of c is indeed based on theclassical Cantor ternary set).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 5 / 32
university-logo
Decomposition of the derivative
Let u ∈ BV (R).By the Radon-Nikodym Theorem we split the measure Du:
Du = Dau + Dsu = Du (Ω\S) + Du S
whereS :=
x ∈ Ω : lim
ρ↓0
|Du|(Bρ(x))
ρ=∞
.
I absolute continuous part Dau (with respect to the Lebesguemeasure L1)
I singular part DsuRemark: for the singular part the slope of u is equal to∞: jumps,Cantor.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 6 / 32
university-logo
Decomposition of the singular part Dsu = Dju + Dcu
Moreover, if A denotes the set of atoms of Du, i.e. x ∈ A if and only ifDu(x) 6= 0, we have
Du = Dau + Dju + Dcu︸ ︷︷ ︸Dsu
= Du (Ω\S) + Du A + Du (S\A).
I if u ∈ BV (R) the jump set A = Ju consists of countably manypoints.
I Dju = Du A is purely atomic.I Dcu = Du (S\A) is the diffusive part (i.e. without atoms)I The above decomposition is unique and the three measures
Dau,Dju,Dcu are mutually singular.We have
|Du|(Ω) = |Dau|(Ω) + |Dju|(Ω) + |Dcu|(Ω)
=
∫Ω\S|u ′|dt +
∑t∈A
|u(t+)− u(t−)|+ |Dcu|(Ω).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 7 / 32
university-logo
The space SBV
DefinitionLet u ∈ BV (Ω), then u is a special function of bounded variation, andwe write u ∈ SBV (Ω), if
Dcu = 0,
i.e. if the measure Du has no Cantor part.
RemarkThe classical example of a 1-dimensional continuous function whichbelongs to BV but not to SBV is the Cantor staircase.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 8 / 32
university-logo
The space SBV in n-dimensions
Rough intuitive picture: Lipschitz functions whose gradient ispiecewise smooth, undergoing jump discontinuities along a rectifiableset of codimension 1. We exclude that the derivative can have acomplicated fractal behaviour.
Let Ω ⊂ Rn and u ∈ BV (Ω). Then, u ∈ SBV (Ω) if Dcu = 0:
Du = Dau + Dju = ∇uLn + (u+ − u−)νuHn−1 Ju.
where (u+,u−) are the one sided limits and νu the normal to the jumpset Ju.
I 1− d : SBV (Ω) = W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, L1 a.e..I n − d : W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, Ln a.e.(SBV (Ω).
Example: u : R2\S → R, u =√ρ sin(θ/2) ∈ SBVloc(R2) where S is
the negative x-axis.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 9 / 32
university-logo
The space SBV in n-dimensions
Rough intuitive picture: Lipschitz functions whose gradient ispiecewise smooth, undergoing jump discontinuities along a rectifiableset of codimension 1. We exclude that the derivative can have acomplicated fractal behaviour.
Let Ω ⊂ Rn and u ∈ BV (Ω). Then, u ∈ SBV (Ω) if Dcu = 0:
Du = Dau + Dju = ∇uLn + (u+ − u−)νuHn−1 Ju.
where (u+,u−) are the one sided limits and νu the normal to the jumpset Ju.
I 1− d : SBV (Ω) = W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, L1 a.e..I n − d : W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, Ln a.e.(SBV (Ω).
Example: u : R2\S → R, u =√ρ sin(θ/2) ∈ SBVloc(R2) where S is
the negative x-axis.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 9 / 32
university-logo
The space SBV in n-dimensions
Rough intuitive picture: Lipschitz functions whose gradient ispiecewise smooth, undergoing jump discontinuities along a rectifiableset of codimension 1. We exclude that the derivative can have acomplicated fractal behaviour.
Let Ω ⊂ Rn and u ∈ BV (Ω). Then, u ∈ SBV (Ω) if Dcu = 0:
Du = Dau + Dju = ∇uLn + (u+ − u−)νuHn−1 Ju.
where (u+,u−) are the one sided limits and νu the normal to the jumpset Ju.
I 1− d : SBV (Ω) = W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, L1 a.e..I n − d : W 1,1(Ω)⊕ u ∈ SBV : ∇u = 0, Ln a.e.(SBV (Ω).
Example: u : R2\S → R, u =√ρ sin(θ/2) ∈ SBVloc(R2) where S is
the negative x-axis.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 9 / 32
university-logo
Hamilton-Jacobi equations
Evolutionary case:
∂tu + H(Dxu) = 0, on R+ × Rn.
Stationary case:H(Dxu) = 0, x ∈ Rn.
PrincipleLooking at the classical mechanics or at optimal control problemsanother powerful principle comes into the play: the least action (orHamiltonian) principle.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 10 / 32
university-logo
An example of stationary HJ-equation: Eikonalequation
I 1-dimensional case:|u′(x)| = 1, x ∈ (a,b) ⊂ R;u(a) = u(b) = 0, boundary data.
I n-dimensional case|∇u(x)| = 1, x ∈ Ω ⊂ Rn;u|∂Ω = 0, boundary data.
A non-smooth solution is given by the distance function:
u(x) = dist(x , ∂Ω).
The weak non-smooth solutions are not unique.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 11 / 32
university-logo
An example of stationary HJ-equation: Eikonalequation
I 1-dimensional case:|u′(x)| = 1, x ∈ (a,b) ⊂ R;u(a) = u(b) = 0, boundary data.
I n-dimensional case|∇u(x)| = 1, x ∈ Ω ⊂ Rn;u|∂Ω = 0, boundary data.
A non-smooth solution is given by the distance function:
u(x) = dist(x , ∂Ω).
The weak non-smooth solutions are not unique.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 11 / 32
university-logo
An example of stationary HJ-equation: Eikonalequation
I 1-dimensional case:|u′(x)| = 1, x ∈ (a,b) ⊂ R;u(a) = u(b) = 0, boundary data.
I n-dimensional case|∇u(x)| = 1, x ∈ Ω ⊂ Rn;u|∂Ω = 0, boundary data.
A non-smooth solution is given by the distance function:
u(x) = dist(x , ∂Ω).
The weak non-smooth solutions are not unique.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 11 / 32
university-logo
If H ∈ C2(Rn) uniformly convex (i.e. ∃c > 0, c−1Id ≤ D2H ≤ cId) forthe Hamilton-Jacobi equation:
∂tu + H(Dxu) = 0 or H(Dxu) = 0, x ∈ Rn
we have:I Singularities appear for the gradient ∇u(x) of the solution.I Weak solutions are not unique.
But as for Conservation laws we can:I use the space BV as working space, i.e. the derivative Dxu is a
function of bounded variation.I introduce the concept of viscosity solution which single out unique
solution!
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 12 / 32
university-logo
If H ∈ C2(Rn) uniformly convex (i.e. ∃c > 0, c−1Id ≤ D2H ≤ cId) forthe Hamilton-Jacobi equation:
∂tu + H(Dxu) = 0 or H(Dxu) = 0, x ∈ Rn
we have:I Singularities appear for the gradient ∇u(x) of the solution.I Weak solutions are not unique.
But as for Conservation laws we can:I use the space BV as working space, i.e. the derivative Dxu is a
function of bounded variation.I introduce the concept of viscosity solution which single out unique
solution!
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 12 / 32
university-logo
If H ∈ C2(Rn) uniformly convex (i.e. ∃c > 0, c−1Id ≤ D2H ≤ cId) forthe Hamilton-Jacobi equation:
∂tu + H(Dxu) = 0 or H(Dxu) = 0, x ∈ Rn
we have:I Singularities appear for the gradient ∇u(x) of the solution.I Weak solutions are not unique.
But as for Conservation laws we can:I use the space BV as working space, i.e. the derivative Dxu is a
function of bounded variation.I introduce the concept of viscosity solution which single out unique
solution!
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 12 / 32
university-logo
If H ∈ C2(Rn) uniformly convex (i.e. ∃c > 0, c−1Id ≤ D2H ≤ cId) forthe Hamilton-Jacobi equation:
∂tu + H(Dxu) = 0 or H(Dxu) = 0, x ∈ Rn
we have:I Singularities appear for the gradient ∇u(x) of the solution.I Weak solutions are not unique.
But as for Conservation laws we can:I use the space BV as working space, i.e. the derivative Dxu is a
function of bounded variation.I introduce the concept of viscosity solution which single out unique
solution!
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 12 / 32
university-logo
Definition (Viscosity solution)A bounded, uniformly continuous function u is called a viscositysolution of
∂tu + H(Dxu) = 0, in Ω ⊂ [0,T ]× Rn ,
provided that1. u is a viscosity subsolution: for each v ∈ C∞(Ω) such that u − v
has a maximum at (t0, x0)
vt (t0, x0) + H(Dxv(t0, x0)) ≤ 0
2. u is a viscosity supersolution: for each v ∈ C∞(Ω) such that u − vhas a minimum at (t0, x0)
vt (t0, x0) + H(Dxv(t0, x0)) ≥ 0
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 13 / 32
university-logo
Viscosity solution as vanishing viscosity limit
A viscosity solution is the uniform limit, as ε→ 0, of solutions to theparabolic problem:
uεt + H(Dxuε)− ε4uε = 0 on R+ × Rn.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 14 / 32
university-logo
Hopf-Lax formula
The unique viscosity solution is given by:
u(x , t) := miny∈R
t · L
(x − yt
)+ u0(y)
.
whereL = H∗ := sup
q∈Rp · q − H(q)
is the Legendre transformation.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 15 / 32
university-logo
SBV Regularity for scalar conservation laws.
L.AMBROSIO & C.DE LELLIS, 2004.
TheoremLet u(x) ∈ L∞(Ω) be an entropy solution of the following scalarconservation law:
Dtu(x , t) + Dx [f (u(x , t))] = 0 in an open set Ω ⊂ R2,
where f ∈ C2(R) and f ′′ > 0 (locally uniformly convex).
⇒ the entropy solutions u(x , t) ∈ SBVloc(Ω).
Moreover: there exists a set S ⊂ R at most countable such that∀τ ∈ R\S the following holds:
u(., τ) ∈ SBVloc(Ωτ ) with Ωτ := x ∈ R : (x , τ) ∈ Ω.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 16 / 32
university-logo
SBV Regularity for scalar conservation laws.
L.AMBROSIO & C.DE LELLIS, 2004.
TheoremLet u(x) ∈ L∞(Ω) be an entropy solution of the following scalarconservation law:
Dtu(x , t) + Dx [f (u(x , t))] = 0 in an open set Ω ⊂ R2,
where f ∈ C2(R) and f ′′ > 0 (locally uniformly convex).
⇒ the entropy solutions u(x , t) ∈ SBVloc(Ω).
Moreover: there exists a set S ⊂ R at most countable such that∀τ ∈ R\S the following holds:
u(., τ) ∈ SBVloc(Ωτ ) with Ωτ := x ∈ R : (x , τ) ∈ Ω.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 16 / 32
university-logo
Hamilton-Jacobi equations and scalar conservationlaws
If u : R+ × R→ R is a viscosity solution of a Hamilton-Jacobi equation,then ux is an entropy solution of a scalar conservation law.
In particular for these Hamilton-Jacobi equations by L.AMBROSIO &C.DE LELLIS, 2004, for viscosity solutions u ∈W 1,∞ we have that:
if H ∈ C2(R) is a uniformly convex Hamiltonian, then Dtu,Dxu ∈ SBVloc .
In particular, the "middle state" of the gradient given by the Cantorfunction will be transformed in jump.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 17 / 32
university-logo
Hamilton-Jacobi equations and scalar conservationlaws
If u : R+ × R→ R is a viscosity solution of a Hamilton-Jacobi equation,then ux is an entropy solution of a scalar conservation law.
In particular for these Hamilton-Jacobi equations by L.AMBROSIO &C.DE LELLIS, 2004, for viscosity solutions u ∈W 1,∞ we have that:
if H ∈ C2(R) is a uniformly convex Hamiltonian, then Dtu,Dxu ∈ SBVloc .
In particular, the "middle state" of the gradient given by the Cantorfunction will be transformed in jump.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 17 / 32
university-logo
Hamilton-Jacobi equations and scalar conservationlaws
If u : R+ × R→ R is a viscosity solution of a Hamilton-Jacobi equation,then ux is an entropy solution of a scalar conservation law.
In particular for these Hamilton-Jacobi equations by L.AMBROSIO &C.DE LELLIS, 2004, for viscosity solutions u ∈W 1,∞ we have that:
if H ∈ C2(R) is a uniformly convex Hamiltonian, then Dtu,Dxu ∈ SBVloc .
In particular, the "middle state" of the gradient given by the Cantorfunction will be transformed in jump.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 17 / 32
university-logo
Hamilton-Jacobi equations in Rn
Theorem (S.Bianchini, C.De Lellis & R.Robyr)
Let u be a viscosity solution of
∂tu + H(Dxu) = 0 in Ω ⊂ [0,T ]× Rn.
coupled with a Lipschitz continuous and bounded initial datau0 : Rn → R. Moreover, assume
H ∈ C2(Rn) and c−1H Idn ≤ D2H ≤ cH Idn for some cH > 0.
and set Ωt := x ∈ Rn : (t , x) ∈ Ω. Then, the set of times
S := t : Dxu(t , .) /∈ SBVloc(Ωt )
is at most countable. In particular Dxu, ∂tu ∈ SBVloc(Ω).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 18 / 32
university-logo
What is new?: SBV-Regularity for any dimension n ∈ N:Hamilton-Jacobi equations in Rn.
Key idea: Find a monotone geometric functional which jumps everytime that a Cantor part appears in the derivative of the solution.
Difficulty: In general the derivative Dxu(t , x) is not continuous and theprojection of the jump set (of the derivative Dxu(t , .)) on Rn × 0 hasnow a complex geometric structure.
Solution: We use the Hopf-Lax formula and we work with the regularpoints, i.e. the points for which the derivative Dxu(t , x) is continuous.
Starting Remark: Viscosity solutions are semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 19 / 32
university-logo
What is new?: SBV-Regularity for any dimension n ∈ N:Hamilton-Jacobi equations in Rn.
Key idea: Find a monotone geometric functional which jumps everytime that a Cantor part appears in the derivative of the solution.
Difficulty: In general the derivative Dxu(t , x) is not continuous and theprojection of the jump set (of the derivative Dxu(t , .)) on Rn × 0 hasnow a complex geometric structure.
Solution: We use the Hopf-Lax formula and we work with the regularpoints, i.e. the points for which the derivative Dxu(t , x) is continuous.
Starting Remark: Viscosity solutions are semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 19 / 32
university-logo
What is new?: SBV-Regularity for any dimension n ∈ N:Hamilton-Jacobi equations in Rn.
Key idea: Find a monotone geometric functional which jumps everytime that a Cantor part appears in the derivative of the solution.
Difficulty: In general the derivative Dxu(t , x) is not continuous and theprojection of the jump set (of the derivative Dxu(t , .)) on Rn × 0 hasnow a complex geometric structure.
Solution: We use the Hopf-Lax formula and we work with the regularpoints, i.e. the points for which the derivative Dxu(t , x) is continuous.
Starting Remark: Viscosity solutions are semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 19 / 32
university-logo
What is new?: SBV-Regularity for any dimension n ∈ N:Hamilton-Jacobi equations in Rn.
Key idea: Find a monotone geometric functional which jumps everytime that a Cantor part appears in the derivative of the solution.
Difficulty: In general the derivative Dxu(t , x) is not continuous and theprojection of the jump set (of the derivative Dxu(t , .)) on Rn × 0 hasnow a complex geometric structure.
Solution: We use the Hopf-Lax formula and we work with the regularpoints, i.e. the points for which the derivative Dxu(t , x) is continuous.
Starting Remark: Viscosity solutions are semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 19 / 32
university-logo
What is new?: SBV-Regularity for any dimension n ∈ N:Hamilton-Jacobi equations in Rn.
Key idea: Find a monotone geometric functional which jumps everytime that a Cantor part appears in the derivative of the solution.
Difficulty: In general the derivative Dxu(t , x) is not continuous and theprojection of the jump set (of the derivative Dxu(t , .)) on Rn × 0 hasnow a complex geometric structure.
Solution: We use the Hopf-Lax formula and we work with the regularpoints, i.e. the points for which the derivative Dxu(t , x) is continuous.
Starting Remark: Viscosity solutions are semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 19 / 32
university-logo
Let Ω ⊂ Rn be an open set.
DefinitionWe say that a continuous function u : Ω→ R is semiconcave if, for anyconvex K ⊂⊂ Ω, there exists CK > 0 such that
u(x + h) + u(x − h)− 2u(x) ≤ CK |h|2,
for all x ,h ∈ Rn with x , x − h, x + h ∈ K .
ExampleLet Ω = (−1,1). The function
u(x) := dist(x , ∂Ω),
is semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 20 / 32
university-logo
Let Ω ⊂ Rn be an open set.
DefinitionWe say that a continuous function u : Ω→ R is semiconcave if, for anyconvex K ⊂⊂ Ω, there exists CK > 0 such that
u(x + h) + u(x − h)− 2u(x) ≤ CK |h|2,
for all x ,h ∈ Rn with x , x − h, x + h ∈ K .
ExampleLet Ω = (−1,1). The function
u(x) := dist(x , ∂Ω),
is semiconcave.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 20 / 32
university-logo
DefinitionLet u : Ω→ R be a measurable function. The set ∂u(x), called thesuperdifferential of u at point x ∈ Ω, is defined as
∂u(x) :=
p ∈ Rn : lim supy→x
u(y)− u(x)− p · (y − x)
|y − x |≤ 0
.
ExampleLet Ω = (−1,1) and consider the semiconcave functionu(x) := dist(x , ∂Ω), thus
I ∂u(x) = 1 for x ∈ (−1,0) and ∂u(x) = −1 for x ∈ (0,1).I ∂u(x) = [−1,1] for x = 0.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 21 / 32
university-logo
DefinitionLet u : Ω→ R be a measurable function. The set ∂u(x), called thesuperdifferential of u at point x ∈ Ω, is defined as
∂u(x) :=
p ∈ Rn : lim supy→x
u(y)− u(x)− p · (y − x)
|y − x |≤ 0
.
ExampleLet Ω = (−1,1) and consider the semiconcave functionu(x) := dist(x , ∂Ω), thus
I ∂u(x) = 1 for x ∈ (−1,0) and ∂u(x) = −1 for x ∈ (0,1).I ∂u(x) = [−1,1] for x = 0.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 21 / 32
university-logo
By this Proposition:I To every semiconcave function we can associate a concave
function!I Concave functions have monotonicity properties.
Proposition
Let Ω ⊂ Rn be open and K ⊂ Ω a compact convex set. Let u : Ω→ Rbe a semiconcave function with semiconcavity constant CK ≥ 0. Then,the function
u : x 7→ u(x)− CK
2|x |2 is concave in K .
In particular, for any given x , y ∈ K , p ∈ ∂u(x) and q ∈ ∂u(y) we havethat
〈q − p, y − x〉 ≤ 0.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 22 / 32
university-logo
Following the work of G.ALBERTI & L.AMBROSIO, 1999
Definition
Let B : Rn → Rn be a multifunction, then1. B is a monotone function if
〈y1 − y2, x1 − x2〉 ≤ 0 ∀xi ∈ Rn, yi ∈ B(xi), i = 1,2.
2. A monotone function B is called maximal when it is maximal withrespect to the inclusion in the class of monotone functions, i.e. ifthe following implication holds:
A(x) ⊃ B(x) for all x ,A monotone ⇒ A = B.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 23 / 32
university-logo
Helpful remarks:
Let u be the unique viscosity solution of the Hamilton-Jacobi equation(with H ∈ C2(Rn) uniformly convex).
I u is semiconcave and ∂u(x) = Du(x) for a.e. x , and at anypoint where ∂u is single-valued, Du is continuous.
I To every semiconcave function we can associate a concavefunction!
I The supergradient ∂u of a concave function is a maximalmonotone function.
I Hille-Yosida Approximation: for every ε and for every maximalmonotone function B on Rn there exists a 1/ε-Lipschitz maximalmonotone function Bε := (εId − B−1)−1.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 24 / 32
university-logo
Helpful remarks:
Let u be the unique viscosity solution of the Hamilton-Jacobi equation(with H ∈ C2(Rn) uniformly convex).
I u is semiconcave and ∂u(x) = Du(x) for a.e. x , and at anypoint where ∂u is single-valued, Du is continuous.
I To every semiconcave function we can associate a concavefunction!
I The supergradient ∂u of a concave function is a maximalmonotone function.
I Hille-Yosida Approximation: for every ε and for every maximalmonotone function B on Rn there exists a 1/ε-Lipschitz maximalmonotone function Bε := (εId − B−1)−1.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 24 / 32
university-logo
Helpful remarks:
Let u be the unique viscosity solution of the Hamilton-Jacobi equation(with H ∈ C2(Rn) uniformly convex).
I u is semiconcave and ∂u(x) = Du(x) for a.e. x , and at anypoint where ∂u is single-valued, Du is continuous.
I To every semiconcave function we can associate a concavefunction!
I The supergradient ∂u of a concave function is a maximalmonotone function.
I Hille-Yosida Approximation: for every ε and for every maximalmonotone function B on Rn there exists a 1/ε-Lipschitz maximalmonotone function Bε := (εId − B−1)−1.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 24 / 32
university-logo
Helpful remarks:
Let u be the unique viscosity solution of the Hamilton-Jacobi equation(with H ∈ C2(Rn) uniformly convex).
I u is semiconcave and ∂u(x) = Du(x) for a.e. x , and at anypoint where ∂u is single-valued, Du is continuous.
I To every semiconcave function we can associate a concavefunction!
I The supergradient ∂u of a concave function is a maximalmonotone function.
I Hille-Yosida Approximation: for every ε and for every maximalmonotone function B on Rn there exists a 1/ε-Lipschitz maximalmonotone function Bε := (εId − B−1)−1.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 24 / 32
university-logo
We recall the Hopf-Lax formula:
u(x , t) := miny∈R
t · L
(x − yt
)+ u0(y)
.
which is the unique (semiconcave) viscosity solution.
We have:I ∂u = Du(x) is single-valued, if x is a continuous point for the
derivative: then we have a unique backward characteristicI ∂u is not single-valued, if x is a discontinuous point for the
derivative: several backward characteristic with bounded speed.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 25 / 32
university-logo
Definition of the geometric functional
We consider the projection of the points for which the derivativeDxu(t , x) is single valued (or continuous):
for such a regular point (t , x) ∈ t × Rn the minimum in the Hopf–Laxformula is obtained for a unique y ∈ 0 × Rn and
χt ,0(x) := x − t DH(Dxu(t , x)) = y .
The function χt ,0 is defined a.e. on Ωt := x ∈ Rn : (t , x) ∈ Ω and wedefine the nonincreasing functional:
F (t) := |χt ,0(Ωt )|.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 26 / 32
university-logo
Definition of the geometric functional
We consider the projection of the points for which the derivativeDxu(t , x) is single valued (or continuous):
for such a regular point (t , x) ∈ t × Rn the minimum in the Hopf–Laxformula is obtained for a unique y ∈ 0 × Rn and
χt ,0(x) := x − t DH(Dxu(t , x)) = y .
The function χt ,0 is defined a.e. on Ωt := x ∈ Rn : (t , x) ∈ Ω and wedefine the nonincreasing functional:
F (t) := |χt ,0(Ωt )|.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 26 / 32
university-logo
Definition of the geometric functional
We consider the projection of the points for which the derivativeDxu(t , x) is single valued (or continuous):
for such a regular point (t , x) ∈ t × Rn the minimum in the Hopf–Laxformula is obtained for a unique y ∈ 0 × Rn and
χt ,0(x) := x − t DH(Dxu(t , x)) = y .
The function χt ,0 is defined a.e. on Ωt := x ∈ Rn : (t , x) ∈ Ω and wedefine the nonincreasing functional:
F (t) := |χt ,0(Ωt )|.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 26 / 32
university-logo
3 steps:
I Step 1: By the dynamic programming principle we prove that thefunctional is nonincreasing.
I Step 2a: For every t ∈ R+, there exists Borel set E ⊂ Ωt with:
|E | = 0, and |D2c ut |(Ωt \ E) = 0.
I Step 2b: For every (t , x) ∈ E the function χt ,0(x) is single valued(since Dxu continuous on E).
I Step 2c: χt ,0(E) ∩ χt+δ,0(Ωt+δ) = ∅.I Step 3: When |D2
c ut | 6= 0, the functional jumps
F (t) > F (t + δ).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 27 / 32
university-logo
3 steps:
I Step 1: By the dynamic programming principle we prove that thefunctional is nonincreasing.
I Step 2a: For every t ∈ R+, there exists Borel set E ⊂ Ωt with:
|E | = 0, and |D2c ut |(Ωt \ E) = 0.
I Step 2b: For every (t , x) ∈ E the function χt ,0(x) is single valued(since Dxu continuous on E).
I Step 2c: χt ,0(E) ∩ χt+δ,0(Ωt+δ) = ∅.I Step 3: When |D2
c ut | 6= 0, the functional jumps
F (t) > F (t + δ).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 27 / 32
university-logo
3 steps:
I Step 1: By the dynamic programming principle we prove that thefunctional is nonincreasing.
I Step 2a: For every t ∈ R+, there exists Borel set E ⊂ Ωt with:
|E | = 0, and |D2c ut |(Ωt \ E) = 0.
I Step 2b: For every (t , x) ∈ E the function χt ,0(x) is single valued(since Dxu continuous on E).
I Step 2c: χt ,0(E) ∩ χt+δ,0(Ωt+δ) = ∅.I Step 3: When |D2
c ut | 6= 0, the functional jumps
F (t) > F (t + δ).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 27 / 32
university-logo
Remarks:I the function χt ,0 is injective, i.e. for x 6= y ⇒ χt ,0(x) ∩ χt ,0(y) = ∅.I we approximate the maximal monotone functions ∂u = ∂u − CK x
with Lipschitz continuous function to prove some estimates. TheLipschitz regularity is important to use the area formula.
I Key estimate: For t ∈]0,T ], any δ ∈ [0, t ] and any Borel setE ⊂ Ωt := (s, x) ∈ Ω : s = t we have:
|Xt ,δ(E)| ≥ (t − δ)n
tn |Xt ,0(E)|;
where for t < s we define Xt ,s(x) := x − (t − s)DH(∂ut (x)) andut : Rn → Rn with ut (x) := u(t , x).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 28 / 32
university-logo
Remarks:I the function χt ,0 is injective, i.e. for x 6= y ⇒ χt ,0(x) ∩ χt ,0(y) = ∅.I we approximate the maximal monotone functions ∂u = ∂u − CK x
with Lipschitz continuous function to prove some estimates. TheLipschitz regularity is important to use the area formula.
I Key estimate: For t ∈]0,T ], any δ ∈ [0, t ] and any Borel setE ⊂ Ωt := (s, x) ∈ Ω : s = t we have:
|Xt ,δ(E)| ≥ (t − δ)n
tn |Xt ,0(E)|;
where for t < s we define Xt ,s(x) := x − (t − s)DH(∂ut (x)) andut : Rn → Rn with ut (x) := u(t , x).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 28 / 32
university-logo
Remarks:I the function χt ,0 is injective, i.e. for x 6= y ⇒ χt ,0(x) ∩ χt ,0(y) = ∅.I we approximate the maximal monotone functions ∂u = ∂u − CK x
with Lipschitz continuous function to prove some estimates. TheLipschitz regularity is important to use the area formula.
I Key estimate: For t ∈]0,T ], any δ ∈ [0, t ] and any Borel setE ⊂ Ωt := (s, x) ∈ Ω : s = t we have:
|Xt ,δ(E)| ≥ (t − δ)n
tn |Xt ,0(E)|;
where for t < s we define Xt ,s(x) := x − (t − s)DH(∂ut (x)) andut : Rn → Rn with ut (x) := u(t , x).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 28 / 32
university-logo
On the functionals:
We observe that in the one-dimensional case, the functional defined inthe proof of Ambrosio & De Lellis for the scalar conservation laws andfor the Hamilton-Jacobi equations are similar:
F HJ(.) is exactly the "complementary" functional of F CL(.).
I F HJ(.) measures the projection of regular points.I F CL(.) measures the projection of singular points.
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 29 / 32
university-logo
Overview: results on the SBV regularity forHamilton-Jacobi equations:
I 1997 - P.CANNARSA, A. MENNUCCI & C.SINESTRARI.Under strong regularity assumption: for a fixed integer R letu0 ∈W 1,∞(Rn) ∩ CR+1(Rn) and consider:
Dtu(x , t) + H(Dxu(x , t), x , t) = 0, t ∈ R+, x ∈ Rn;u(x ,0) = u0(x), x ∈ Rn.
where H(., x , t) is strictly convex. Then, Du belongs to the classSBVloc , i.e D2u is a measure with no Cantor part.
I 2004 - L.AMBROSIO & C.DE LELLIS.2-dimensional case: Let H ∈ C2(R2) be locally uniformly convexand let u ∈W 1,∞(Ω) be a viscosity solution of H(Du) = 0. ThenDu ∈ SBVloc(Ω).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 30 / 32
university-logo
I 2007 - R.ROBYR.Let H(p, x , t) ∈ C2(R× R× R+) be locally uniformly convex in p,i.e. DppH > 0. If u ∈W 1,∞(Ω) is a viscosity solution of
Dtu(x , t) + H(Dxu(x , t), x , t) = 0, (1)
then Du ∈ SBVloc(Ω).I 2010 - S.BIANCHINI, C.DE LELLIS & R.ROBYR
n-dimensional case: Under the convexity assumption that
H ∈ C2(Rn) and c−1H Idn ≤ D2H ≤ cH Idn for some cH > 0,
the gradient of any viscosity solution u of
H(Dxu) = 0 in Ω ⊂ Rn,
belongs to SBVloc(Ω).
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 31 / 32
university-logo
Thank you!!!!!
Roger Robyr (UZH) Regolarità SBV per HJ in Rn 17 Gennaio 2011 32 / 32