Improved regularity for elliptic equations in the double-divergence - - PowerPoint PPT Presentation

Improved regularity for elliptic equations in the double-divergence form

Edgard A. Pimentel PUC-Rio, Rio de Janeiro

Swedish Summer PDEs KTH, Stockholm 26 August 2019

General overview

General overview

• 1. Elliptic equations in the double-divergence form

1

General overview

• 1. Elliptic equations in the double-divergence form;
• 2. Motivation: former developments and applications

1

General overview

• 1. Elliptic equations in the double-divergence form;
• 2. Motivation: former developments and applications;
• 3. Regularity in H¨
• lder spaces

1

General overview

• 1. Elliptic equations in the double-divergence form;
• 2. Motivation: former developments and applications;
• 3. Regularity in H¨
• lder spaces;
• 4. Regularity transmission by approximation methods.

1

The double-divergence setting

The double-divergence equation

We are interested in the following PDE: ∂2

xixj

• aij(x)u(x)
• = 0

in B1

2

The double-divergence equation

We are interested in the following PDE: ∂2

xixj

• aij(x)u(x)
• = 0

in B1, where

• 1. the matrix (aij)d

i,j=1 is H¨

• lder continuous in B1

2

The double-divergence equation

We are interested in the following PDE: ∂2

xixj

• aij(x)u(x)
• = 0

in B1, where

• 1. the matrix (aij)d

i,j=1 is H¨

• lder continuous in B1; that is

aij ∈ Cβ

loc(B1)

for every i, j = 1, . . . , d

2

The double-divergence equation

We are interested in the following PDE: ∂2

xixj

• aij(x)u(x)
• = 0

in B1, where

• 1. the matrix (aij)d

i,j=1 is H¨

• lder continuous in B1; that is

aij ∈ Cβ

loc(B1)

for every i, j = 1, . . . , d.

• 2. Uniform ellipticity - there exist contants 0 < λ < Λ such that

λId ≤ (aij(x))d

i, j=1 ≤ ΛId, 2

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes

3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x) 3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x)

have KFP given by L∗v(x) := ∂2

xixj

• aij(x)v(x)
• = 0

3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x)

have KFP given by L∗v(x) := ∂2

xixj

• aij(x)v(x)
• = 0;

− →

Anisotropic diffusion (e.g., green noise) 3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x)

have KFP given by L∗v(x) := ∂2

xixj

• aij(x)v(x)
• = 0;

− →

Anisotropic diffusion (e.g., green noise)

− →

Bogachev, Krylov, many others 3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x)

have KFP given by L∗v(x) := ∂2

xixj

• aij(x)v(x)
• = 0;

− →

Anisotropic diffusion (e.g., green noise)

− →

Bogachev, Krylov, many others

Solutions describe the distribution of the population driven by the process

3

Motivation: stochastic analysis

Stochastic analysis: Kolmogorov-Fokker-Planck equation associated with certain stochastic processes; In particular, diffusions with infinitesimal generators of the form Lv(x) := aij(x)∂2

xixjv(x)

have KFP given by L∗v(x) := ∂2

xixj

• aij(x)v(x)
• = 0;

− →

Anisotropic diffusion (e.g., green noise)

− →

Bogachev, Krylov, many others

Solutions describe the distribution of the population driven by the process. − →

Bogachev, Krylov, R¨

• ckner, Shaposhnikov

3

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1

4

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1, where F : S(d) → R is a (λ, Λ)-elliptic operator and F ij(M) stands for the derivative of F with respect to the entry mi,j of M

4

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1, where F : S(d) → R is a (λ, Λ)-elliptic operator and F ij(M) stands for the derivative of F with respect to the entry mi,j of M; Introduced in the works of J.-M. Lasry and P.-L. Lions circa 2006

4

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1, where F : S(d) → R is a (λ, Λ)-elliptic operator and F ij(M) stands for the derivative of F with respect to the entry mi,j of M; Introduced in the works of J.-M. Lasry and P.-L. Lions circa 2006; further developed by several authors − →

Bensoussan, Frehse, Yam; Cardaliaguet; Gomes, P., Voskanyan 4

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1, where F : S(d) → R is a (λ, Λ)-elliptic operator and F ij(M) stands for the derivative of F with respect to the entry mi,j of M; Introduced in the works of J.-M. Lasry and P.-L. Lions circa 2006; further developed by several authors − →

Bensoussan, Frehse, Yam; Cardaliaguet; Gomes, P., Voskanyan

Recent developments: existence and regularity theory for fully nonlinear MFG

4

Motivation: mean-field games

Toy-model of the form    F(D2V ) = g(u) in B1 ∂2

xixj

• F ij(D2V )u(x)
• = 0

in B1, where F : S(d) → R is a (λ, Λ)-elliptic operator and F ij(M) stands for the derivative of F with respect to the entry mi,j of M; Introduced in the works of J.-M. Lasry and P.-L. Lions circa 2006; further developed by several authors − →

Bensoussan, Frehse, Yam; Cardaliaguet; Gomes, P., Voskanyan

Recent developments: existence and regularity theory for fully nonlinear MFG; − →

Jointly w/ P. Andrade 4

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω)

5

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω). The gradient graph of u is Γu := {(x, Du(x)) , x ∈ Ω}

5

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω). The gradient graph of u is Γu := {(x, Du(x)) , x ∈ Ω} . The volume of Γu is given by FΩ(u) =

• Ω
• det(I + (D2u)TD2u)

1/2 d x

5

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω). The gradient graph of u is Γu := {(x, Du(x)) , x ∈ Ω} . The volume of Γu is given by FΩ(u) =

• Ω
• det(I + (D2u)TD2u)

1/2 d x. − → Choose u as to minimize the volume of the gradient graph

5

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω). The gradient graph of u is Γu := {(x, Du(x)) , x ∈ Ω} . The volume of Γu is given by FΩ(u) =

• Ω
• det(I + (D2u)TD2u)

1/2 d x. − → Choose u as to minimize the volume of the gradient graph Critical points satisfy the (Euler) equation ∂2

xjxl

• det ggijδkluxixk
• = 0

in Ω

5

Motivation: Hamiltonian stationary Lagrangian manifolds

Fix Ω ⊂ Rd and consider u ∈ C ∞(Ω). The gradient graph of u is Γu := {(x, Du(x)) , x ∈ Ω} . The volume of Γu is given by FΩ(u) =

• Ω
• det(I + (D2u)TD2u)

1/2 d x. − → Choose u as to minimize the volume of the gradient graph Critical points satisfy the (Euler) equation ∂2

xjxl

• det ggijδkluxixk
• = 0

in Ω, where g := I + (D2u)TD2u is the induced metric.

5

Previous developments

Maximum principles for supersolutions of the double-divergence equation

6

Previous developments

Maximum principles for supersolutions of the double-divergence equation; − → Littman (59)

6

Previous developments

Maximum principles for supersolutions of the double-divergence equation; − → Littman (59) A potential theory, with analogous properties to the case of aij( · )∂2

xi xj 6

Previous developments

Maximum principles for supersolutions of the double-divergence equation; − → Littman (59) A potential theory, with analogous properties to the case of aij( · )∂2

xi xj;

− → Herv´

e (62) 6

Previous developments

Maximum principles for supersolutions of the double-divergence equation; − → Littman (59) A potential theory, with analogous properties to the case of aij( · )∂2

xi xj;

− → Herv´

e (62)

Improved maximum principles and preliminary approximation schemes

6

Previous developments

Maximum principles for supersolutions of the double-divergence equation; − → Littman (59) A potential theory, with analogous properties to the case of aij( · )∂2

xi xj;

− → Herv´

e (62)

Improved maximum principles and preliminary approximation schemes; − → Littman (63)

6

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1) 7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1;

− → Fabes-Stroock, Duke Mathematical Journal (84)

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1;

− → Fabes-Stroock, Duke Mathematical Journal (84) Study of the density of the solutions, in the sense of Radon-Nykodim derivatives

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1;

− → Fabes-Stroock, Duke Mathematical Journal (84) Study of the density of the solutions, in the sense of Radon-Nykodim derivatives; − → Bogachev-Krylov-R¨

• ckner, Comm. Partial Differential Equations (01)

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1;

− → Fabes-Stroock, Duke Mathematical Journal (84) Study of the density of the solutions, in the sense of Radon-Nykodim derivatives; − → Bogachev-Krylov-R¨

• ckner, Comm. Partial Differential Equations (01)

− → Bogachev-Krylov-R¨

• ckner (15)

7

Previous developments

Regularity theory in H¨

• lder spaces provided aij ∈ Cβ

loc(B1);

− → Sj¨

• gren (73)

Properties of the associated Green’s function; gains of integrability

• f the type L1 =

⇒ L

d d−1;

− → Fabes-Stroock, Duke Mathematical Journal (84) Study of the density of the solutions, in the sense of Radon-Nykodim derivatives; − → Bogachev-Krylov-R¨

• ckner, Comm. Partial Differential Equations (01)

− → Bogachev-Krylov-R¨

• ckner (15)

− → Bogachev-Shaposhnikov (17)

7

A (very) distinctive feature

Regularity of the coefficients acts as an upper bound for the regularity of the solutions

8

A (very) distinctive feature

Regularity of the coefficients acts as an upper bound for the regularity of the solutions; (a(x)u(x))xx = 0 in ] − 1, 1[ Let ℓ(x) be an affine function and set v(x) := ℓ(x)/a(x)

8

A (very) distinctive feature

Regularity of the coefficients acts as an upper bound for the regularity of the solutions; (a(x)u(x))xx = 0 in ] − 1, 1[ Let ℓ(x) be an affine function and set v(x) := ℓ(x)/a(x). Then, 1

−1

a(x)ℓ(x) a(x)φxx(x) d x = 0 for every φ ∈ C2

0(] − 1, 1[) 8

A (very) distinctive feature

Regularity of the coefficients acts as an upper bound for the regularity of the solutions; (a(x)u(x))xx = 0 in ] − 1, 1[ Let ℓ(x) be an affine function and set v(x) := ℓ(x)/a(x). Then, 1

−1

a(x)ℓ(x) a(x)φxx(x) d x = 0 for every φ ∈ C2

0(] − 1, 1[); therefore, v is a solution 8

A (very) distinctive feature

Regularity of the coefficients acts as an upper bound for the regularity of the solutions; (a(x)u(x))xx = 0 in ] − 1, 1[ Let ℓ(x) be an affine function and set v(x) := ℓ(x)/a(x). Then, 1

−1

a(x)ℓ(x) a(x)φxx(x) d x = 0 for every φ ∈ C2

0(] − 1, 1[); therefore, v is a solution;

− → Were a(x) discontinuous, so would be v(x).

8

Our program and main results

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary Regularity transmission by approximation methods

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary Regularity transmission by approximation methods − → Caffarelli, Ann. Math. (89)

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary Regularity transmission by approximation methods − → Caffarelli, Ann. Math. (89) − → Teixeira, Math. Ann. (14)

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary Regularity transmission by approximation methods − → Caffarelli, Ann. Math. (89) − → Teixeira, Math. Ann. (14) − → Teixeira (15)

9

Our program

Regularity theory along zero level-sets; focus on x0 ∈ {u = 0}; − → nonphysical free boundary Regularity transmission by approximation methods − → Caffarelli, Ann. Math. (89) − → Teixeira, Math. Ann. (14) − → Teixeira (15) Are there gains of regularity, as solutions approach their zero level-sets?

9

Our program

Import information from the well-understood non-divergence prob- lem Tr

• aijD2u
• = 0

in B1, for which a richer regularity theory is available

10

Our program

Import information from the well-understood non-divergence prob- lem Tr

• aijD2u
• = 0

in B1, for which a richer regularity theory is available The regularity of the coefficients is an upper bound for the regularity

• f the solutions ‘in the large’.

Therefore, we look for regularity improvements at x0 ∈ {u = 0}.

10

H¨

• lder continuity

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

11

H¨

• lder continuity

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2

11

H¨

• lder continuity

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix 11

H¨

• lder continuity

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix.

Then, u ∈ C1−

loc (B1 ∩ ∂{u > 0}) 11

H¨

• lder continuity

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix.

Then, u ∈ C1−

loc (B1 ∩ ∂{u > 0}) and for every α ∈ (0, 1) there

exists Cα > 0 such that sup

Br (x0)

|u(x) − u(x0)| ≤ Cαr α, for every 0 < r ≪ 1/2 and x0 ∈ ∂{u > 0}.

11

A few remarks

Gains of regularity are independent of the data

12

A few remarks

Gains of regularity are independent of the data; aij ∈ Cε

loc(B1)

= ⇒ u ∈ Cε

loc(B1) 12

A few remarks

Gains of regularity are independent of the data; aij ∈ Cε

loc(B1)

= ⇒ u ∈ Cε

loc(B1);

however, at the free boundary u is of class C1−

12

A few remarks

Gains of regularity are independent of the data; aij ∈ Cε

loc(B1)

= ⇒ u ∈ Cε

loc(B1);

however, at the free boundary u is of class C1−; Our results extend to equations involving lower-order terms ∂2

xixj

• aij(x)u(x)
• + ∂xi
• bi(x)u(x)
• + c(x)u(x) = 0

in B1

12

A few remarks

Gains of regularity are independent of the data; aij ∈ Cε

loc(B1)

= ⇒ u ∈ Cε

loc(B1);

however, at the free boundary u is of class C1−; Our results extend to equations involving lower-order terms ∂2

xixj

• aij(x)u(x)
• + ∂xi
• bi(x)u(x)
• + c(x)u(x) = 0

in B1, provided bi, c : B1 → R are well-prepared.

12

Strategy of the proof

GOAL: to control the oscillation of the solutions within balls of radius 0 < r ≪ 1, centered at points x0 ∈ ∂{u > 0}

13

Strategy of the proof

GOAL: to control the oscillation of the solutions within balls of radius 0 < r ≪ 1, centered at points x0 ∈ ∂{u > 0}; Main ingredients are

13

Strategy of the proof

GOAL: to control the oscillation of the solutions within balls of radius 0 < r ≪ 1, centered at points x0 ∈ ∂{u > 0}; Main ingredients are: − → Preliminary (uniform) compactness

13

Strategy of the proof

GOAL: to control the oscillation of the solutions within balls of radius 0 < r ≪ 1, centered at points x0 ∈ ∂{u > 0}; Main ingredients are: − → Preliminary (uniform) compactness; − → Approximation results preserving zero level-sets

13

Strategy of the proof

GOAL: to control the oscillation of the solutions within balls of radius 0 < r ≪ 1, centered at points x0 ∈ ∂{u > 0}; Main ingredients are: − → Preliminary (uniform) compactness; − → Approximation results preserving zero level-sets; − → Iteration mechanism through scaling techniques.

13

Zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

14

Zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε

14

Zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε,

• ne can find h ∈ C1,1

loc (B1) satisfying

u − hL∞(B9/10) < δ

14

Zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε,

• ne can find h ∈ C1,1

loc (B1) satisfying

u − hL∞(B9/10) < δ. Moreover, h(x0) = 0 for every x0 ∈ ∂{u > 0}.

14

First-order oscillation control

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

15

First-order oscillation control

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1).

For every α ∈ (0, 1) there exists ε > 0 and ρ ∈ (0, 1/2) such that, if sup

B9/10

|aij(x) − aij| < ε

15

First-order oscillation control

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1).

For every α ∈ (0, 1) there exists ε > 0 and ρ ∈ (0, 1/2) such that, if sup

B9/10

|aij(x) − aij| < ε, then sup

Bρ(x0)

|u(x)| ≤ ρα

15

First-order oscillation control

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1).

For every α ∈ (0, 1) there exists ε > 0 and ρ ∈ (0, 1/2) such that, if sup

B9/10

|aij(x) − aij| < ε, then sup

Bρ(x0)

|u(x)| ≤ ρα, for every x0 ∈ ∂{u > 0}.

15

Oscillation control at discrete scales

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

16

Oscillation control at discrete scales

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

sup

B9/10

|aij(x) − aij| < ε

16

Oscillation control at discrete scales

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

sup

B9/10

|aij(x) − aij| < ε. Then sup

Bρn(x0)

|u(x)| ≤ ρnα

16

Oscillation control at discrete scales

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ Cβ

loc(B1) satisfies

sup

B9/10

|aij(x) − aij| < ε. Then sup

Bρn(x0)

|u(x)| ≤ ρnα, for every x0 ∈ ∂{u > 0} and every n ∈ N.

16

H¨

• lder regularity of the gradient

Impose further conditions on the coefficients to unlock a similar analysis at the level of the gradient Du; in principle, to ask for aij ∈ W 2,p

loc (B1)

for every i, j = 1, ..., d.

17

H¨

• lder regularity of the gradient

Impose further conditions on the coefficients to unlock a similar analysis at the level of the gradient Du; in principle, to ask for aij ∈ W 2,p

loc (B1)

for every i, j = 1, ..., d. Consider a suitable zero level-set in this context: {u = |Du| = 0}.

17

H¨

• lder continuity of the gradient

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

18

H¨

• lder continuity of the gradient

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2

18

H¨

• lder continuity of the gradient

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix 18

H¨

• lder continuity of the gradient

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix.

Then, u ∈ C1,1−

loc (B1 ∩ ∂{u > 0}) 18

H¨

• lder continuity of the gradient

Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1) satisfies

aij( · ) − aijL∞(B1) ≪ 1/2, where (aij)d

i, j=1 is a constant matrix.

Then, u ∈ C1,1−

loc (B1 ∩ ∂{u > 0}) and there exists Cα > 0 such that

sup

Br(x0)

|Du(x) − Du(x0)| ≤ Cαr α, for every 0 < r ≪ 1/2, and x0 ∈ ∂{u > 0} ∩ ∂{|Du| > 0} and α ∈ (0, 1).

18

Key: First-order zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

equation

19

Key: First-order zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε

19

Key: First-order zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε,

• ne can find h ∈ C1,1

loc (B1) satisfying

u − hL∞(B9/10) < δ

19

Key: First-order zero level-set approximation lemma

Proposition Let u ∈ L1

loc(B1) be a weak solution to the double-divergence

• equation. Suppose aij ∈ W 2,p

loc (B1). For every δ > 0 there exists

ε > 0 such that, if sup

B9/10

|aij(x) − aij| < ε,

• ne can find h ∈ C1,1

loc (B1) satisfying

u − hL∞(B9/10) < δ. Moreover, h(x0) = 0 and Dh(x0) = 0 for every x0 ∈ ∂{u > 0} ∩ ∂{|Du| > 0}.

19

Further directions and open problems

• 1. Extrapolate improved regularity to the interior

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation; − → not reasonable, given the regularity of the coefficients

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation; − → not reasonable, given the regularity of the coefficients.

• 2. Non-homogeneous setting:

∂2

xixj

• aij(x)u(x)
• = f

in B1, for a well-prepared source term f : B1 → R

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation; − → not reasonable, given the regularity of the coefficients.

• 2. Non-homogeneous setting:

∂2

xixj

• aij(x)u(x)
• = f

in B1, for a well-prepared source term f : B1 → R;

• 3. Parabolic setting

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation; − → not reasonable, given the regularity of the coefficients.

• 2. Non-homogeneous setting:

∂2

xixj

• aij(x)u(x)
• = f

in B1, for a well-prepared source term f : B1 → R;

• 3. Parabolic setting:

− → scaling arguments

20

Further directions and open problems

• 1. Extrapolate improved regularity to the interior;

− → adding a constant changes the equation; − → not reasonable, given the regularity of the coefficients.

• 2. Non-homogeneous setting:

∂2

xixj

• aij(x)u(x)
• = f

in B1, for a well-prepared source term f : B1 → R;

• 3. Parabolic setting:

− → scaling arguments; − → preliminary compactness.

20

Thank you very much!

20