Regularity of the singular set in the fully nonlinear obstacle - - PowerPoint PPT Presentation

Regularity of the singular set in the fully nonlinear obstacle problem

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia)

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

The fully nonlinear obstacle problem

• F(D2u) = χ{u>0}

in Ω ⊂ Rd, u ≥ 0 in Ω. F is uniformly elliptic. F(0) = 0. F is convex and C 1.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

The fully nonlinear obstacle problem

• F(D2u) = χ{u>0}

in Ω ⊂ Rd, u ≥ 0 in Ω. F is uniformly elliptic. F(0) = 0. F is convex and C 1.

• Eg. F(D2u) = ∆u.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

The fully nonlinear obstacle problem

• F(D2u) = χ{u>0}

in Ω ⊂ Rd, u ≥ 0 in Ω. F is uniformly elliptic. F(0) = 0. F is convex and C 1.

• Eg. F(D2u) = ∆u.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Ki-Ahm Lee ’98) u ∈ C 1,1

loc (Ω).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Ki-Ahm Lee ’98) u ∈ C 1,1

loc (Ω).

For 0 ∈ ∂{u > 0} ∩ Ω and r > 0 small, define ur(x) = 1

r2 u(rx).

Up to a subseqn of r → 0, ur → u0 in C 1,α

loc (Rd).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Ki-Ahm Lee ’98) u ∈ C 1,1

loc (Ω).

For 0 ∈ ∂{u > 0} ∩ Ω and r > 0 small, define ur(x) = 1

r2 u(rx).

Up to a subseqn of r → 0, ur → u0 in C 1,α

loc (Rd).

Thm (Lee ’98) Either u0(x) = γ

2 max{x · e, 0}2 for an e ∈ Sd−1 and

F(γe ⊗ e) = 1 (half-space solutions);

• r u0(x) = 1

2x · Ax for an A ≥ 0 and F(A) = 1 (parabola

solutions). Furthermore, this ‘type’ is independent of the subseqn of r → 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

(Indirect) consequences:

• 1. The decomposition ∂{u > 0} ∩ Ω = Reg(u) ∪ Sing(u).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

(Indirect) consequences:

• 1. The decomposition ∂{u > 0} ∩ Ω = Reg(u) ∪ Sing(u).
• 2. As r → 0, D2ur ≥ −o(1) in B1.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

(Indirect) consequences:

• 1. The decomposition ∂{u > 0} ∩ Ω = Reg(u) ∪ Sing(u).
• 2. As r → 0, D2ur ≥ −o(1) in B1.

Assuming 0 ∈ Sing(u),

• 3. As r → 0, we can find parabola solutions pr such that

ur − prL∞(B1) → 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

(Indirect) consequences:

• 1. The decomposition ∂{u > 0} ∩ Ω = Reg(u) ∪ Sing(u).
• 2. As r → 0, D2ur ≥ −o(1) in B1.

Assuming 0 ∈ Sing(u),

• 3. As r → 0, we can find parabola solutions pr such that

ur − prL∞(B1) → 0.

• 4. As r → 0, |{u=0}∩Br|

rd

→ 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

(Indirect) consequences:

• 1. The decomposition ∂{u > 0} ∩ Ω = Reg(u) ∪ Sing(u).
• 2. As r → 0, D2ur ≥ −o(1) in B1.

Assuming 0 ∈ Sing(u),

• 3. As r → 0, we can find parabola solutions pr such that

ur − prL∞(B1) → 0.

• 4. As r → 0, |{u=0}∩Br|

rd

→ 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Lee ’98) Reg(u) ∩ Ω is relatively open in ∂{u > 0} ∩ Ω and is locally a C 1,α-hypersurface.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Lee ’98) Reg(u) ∩ Ω is relatively open in ∂{u > 0} ∩ Ω and is locally a C 1,α-hypersurface. Idea: Near a regular point, ur ∼ 1

2 max{x1, 0}2.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Lee ’98) Reg(u) ∩ Ω is relatively open in ∂{u > 0} ∩ Ω and is locally a C 1,α-hypersurface. Idea: Near a regular point, ur ∼ 1

2 max{x1, 0}2. =

⇒ ∂{u > 0} ∩ Br ∼ {x1 = 0} ∩ Br.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Lee ’98) Reg(u) ∩ Ω is relatively open in ∂{u > 0} ∩ Ω and is locally a C 1,α-hypersurface. Idea: Near a regular point, ur ∼ 1

2 max{x1, 0}2. =

⇒ ∂{u > 0} ∩ Br ∼ {x1 = 0} ∩ Br. Sing(u)?

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Lee ’98) Reg(u) ∩ Ω is relatively open in ∂{u > 0} ∩ Ω and is locally a C 1,α-hypersurface. Idea: Near a regular point, ur ∼ 1

2 max{x1, 0}2. =

⇒ ∂{u > 0} ∩ Br ∼ {x1 = 0} ∩ Br. Sing(u)?

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F(D2u) = ∆u, monotonicity formulae = ⇒ |u − p| ≤ σ(r)r2 in Br for a modulos of continuity σ. (Previously we had: As r → 0, we can find parabola solutions pr such that ur − prL∞(B1) → 0.)

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F(D2u) = ∆u, monotonicity formulae = ⇒ |u − p| ≤ σ(r)r2 in Br for a modulos of continuity σ. (Previously we had: As r → 0, we can find parabola solutions pr such that ur − prL∞(B1) → 0.) No monotonicity formulae for fully nonlinear F.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F(D2u) = ∆u, monotonicity formulae = ⇒ |u − p| ≤ σ(r)r2 in Br for a modulos of continuity σ. (Previously we had: As r → 0, we can find parabola solutions pr such that ur − prL∞(B1) → 0.) No monotonicity formulae for fully nonlinear F. Observation: Instability of ∂{u > 0} near Sing(u)

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F(D2u) = ∆u, monotonicity formulae = ⇒ |u − p| ≤ σ(r)r2 in Br for a modulos of continuity σ. (Previously we had: As r → 0, we can find parabola solutions pr such that ur − prL∞(B1) → 0.) No monotonicity formulae for fully nonlinear F. Observation: Instability of ∂{u > 0} near Sing(u) = ⇒ Rigidity of the solution.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Savin-Y. ’19) Sing(u) = ∪d−1

k=0Σk(u).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Savin-Y. ’19) Sing(u) = ∪d−1

k=0Σk(u).

Σd−1(u) is locally covered by a C 1,α-hypersurface.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Savin-Y. ’19) Sing(u) = ∪d−1

k=0Σk(u).

Σd−1(u) is locally covered by a C 1,α-hypersurface. For k ≤ d − 2, Σk(u) is locally covered by a k-dim C 1,logε-manifold.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Goal: Find a parabola solution p such that for all r > 0, |u − p| ≤ σ(r)r2 in Br.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Goal: Find a parabola solution p such that for all r > 0, |u − p| ≤ σ(r)r2 in Br. One step in a discretized version: |u − p| < ε in B1 for some ε very small. = ⇒

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Goal: Find a parabola solution p such that for all r > 0, |u − p| ≤ σ(r)r2 in Br. One step in a discretized version: |u − p| < ε in B1 for some ε very small. = ⇒ We can find p′ such that |u − p′| < ε′r2

0 in Br0 for some r0 ∈ (0, 1)

and ε′ < ε.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume

• ∆u = χ{u>0}

in B1 ⊂ R2, u ≥ 0 in B1, 0 ∈ Sing(u), and |u − 1

2x2 1| < ε in B1.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume

• ∆u = χ{u>0}

in B1 ⊂ R2, u ≥ 0 in B1, 0 ∈ Sing(u), and |u − 1

2x2 1| < ε in B1.

Normalize ˆ uε = 1 ε(u − 1 2x2

1), ˆ

Oε = 1 ε(−1 2x2

1).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume

• ∆u = χ{u>0}

in B1 ⊂ R2, u ≥ 0 in B1, 0 ∈ Sing(u), and |u − 1

2x2 1| < ε in B1.

Normalize ˆ uε = 1 ε(u − 1 2x2

1), ˆ

Oε = 1 ε(−1 2x2

1).

The strategy: 1. ˆ uε → ˆ u0 as ε → 0. 2. ˆ u0 is C 2 at 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume

• ∆u = χ{u>0}

in B1 ⊂ R2, u ≥ 0 in B1, 0 ∈ Sing(u), and |u − 1

2x2 1| < ε in B1.

Normalize ˆ uε = 1 ε(u − 1 2x2

1), ˆ

Oε = 1 ε(−1 2x2

1).

The strategy: 1. ˆ uε → ˆ u0 as ε → 0. 2. ˆ u0 is C 2 at 0. u = 1 2x2

1 + εˆ

uε = 1 2x2

1 + εˆ

u0 + εo(1) = 1 2x2

1 + εq + εo(r2) + εo(1) in Br

if q = the second order Taylor expansion of ˆ u0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ uε = 1

ε(u − 1 2x2 1), ˆ

Oε = 1

ε(− 1 2x2 1).

|ˆ uε| ≤ 1. ˆ uε ≥ ˆ Oε.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ uε = 1

ε(u − 1 2x2 1), ˆ

Oε = 1

ε(− 1 2x2 1).

|ˆ uε| ≤ 1. ˆ uε ≥ ˆ Oε. In {ˆ uε > ˆ Oε}, ∆ˆ uε = 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ uε = 1

ε(u − 1 2x2 1), ˆ

Oε = 1

ε(− 1 2x2 1).

|ˆ uε| ≤ 1. ˆ uε ≥ ˆ Oε. In {ˆ uε > ˆ Oε}, ∆ˆ uε = 0. In {ˆ uε = ˆ Oε}, ∆ˆ uε = −1/ε = ∆ ˆ Oε.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ uε = 1

ε(u − 1 2x2 1), ˆ

Oε = 1

ε(− 1 2x2 1).

|ˆ uε| ≤ 1. ˆ uε ≥ ˆ Oε. In {ˆ uε > ˆ Oε}, ∆ˆ uε = 0. In {ˆ uε = ˆ Oε}, ∆ˆ uε = −1/ε = ∆ ˆ Oε. That is, ˆ uε solves the obstacle problem with ˆ Oε as the obstacle.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ uε = 1

ε(u − 1 2x2 1), ˆ

Oε = 1

ε(− 1 2x2 1).

|ˆ uε| ≤ 1. ˆ uε ≥ ˆ Oε. In {ˆ uε > ˆ Oε}, ∆ˆ uε = 0. In {ˆ uε = ˆ Oε}, ∆ˆ uε = −1/ε = ∆ ˆ Oε. That is, ˆ uε solves the obstacle problem with ˆ Oε as the obstacle. Lemma: [ˆ uε]Lip ≤ L in B1/2.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Up to a subseqn of ε → 0, ˆ uε → ˆ u0 uniformly in B1/2 and in C 2

loc(B1/2\{x1 = 0}).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Up to a subseqn of ε → 0, ˆ uε → ˆ u0 uniformly in B1/2 and in C 2

loc(B1/2\{x1 = 0}).

ˆ u0 solves the thin obstacle problem      ∆ˆ u0 = 0 in B1/2 ∩ ({x1 = 0} ∪ {x1 = 0, ˆ u0 > 0}), ∆ˆ u0 ≤ 0 in B1/2, ˆ u0 ≥ 0 along B1/2 ∩ {x1 = 0}.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Up to a subseqn of ε → 0, ˆ uε → ˆ u0 uniformly in B1/2 and in C 2

loc(B1/2\{x1 = 0}).

ˆ u0 solves the thin obstacle problem      ∆ˆ u0 = 0 in B1/2 ∩ ({x1 = 0} ∪ {x1 = 0, ˆ u0 > 0}), ∆ˆ u0 ≤ 0 in B1/2, ˆ u0 ≥ 0 along B1/2 ∩ {x1 = 0}. Bad news: 1. ˆ u0 = Re(x2 + ix1)3/2; and 2. ˆ u0 = α(x1)+ + β(x1)−.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Up to a subseqn of ε → 0, ˆ uε → ˆ u0 uniformly in B1/2 and in C 2

loc(B1/2\{x1 = 0}).

ˆ u0 solves the thin obstacle problem      ∆ˆ u0 = 0 in B1/2 ∩ ({x1 = 0} ∪ {x1 = 0, ˆ u0 > 0}), ∆ˆ u0 ≤ 0 in B1/2, ˆ u0 ≥ 0 along B1/2 ∩ {x1 = 0}. Bad news: 1. ˆ u0 = Re(x2 + ix1)3/2; and 2. ˆ u0 = α(x1)+ + β(x1)−. Good news: Up to a rescaling again, these are the only non-C 2 solutions.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 1. ˆ u0 = Re(x2 + ix1)3/2. D2 ˆ u0 > 0 in B1/2 ∩ {x1 = 0}. = ⇒ D2 ˆ u0 > δ in B1/2 ∩ {|x1| > η}. = ⇒ D2 ˆ uε ≥ 1

2δ in B1/2 ∩ {|x1| > η} for small ε.

= ⇒ D2u ≥ 1

2δε in B1/2 ∩ {|x1| > η}. (u = 1 2x2 1 + εˆ

uε) Combining with D2u ≥ −Lε in B1/2, this implies D2u ≥ 0 in B1/2. A perturbation gives Deu ≥ 0 in B1/2 for all e ∈ S1 close to e2.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 1. ˆ u0 = Re(x2 + ix1)3/2. D2 ˆ u0 > 0 in B1/2 ∩ {x1 = 0}. = ⇒ D2 ˆ u0 > δ in B1/2 ∩ {|x1| > η}. = ⇒ D2 ˆ uε ≥ 1

2δ in B1/2 ∩ {|x1| > η} for small ε.

= ⇒ D2u ≥ 1

2δε in B1/2 ∩ {|x1| > η}. (u = 1 2x2 1 + εˆ

uε) Combining with D2u ≥ −Lε in B1/2, this implies D2u ≥ 0 in B1/2. A perturbation gives Deu ≥ 0 in B1/2 for all e ∈ S1 close to e2.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 2. ˆ u0 = α(x1)+ + β(x1)−. If α > 0, then u = 1

2x2 1 + εαx1 + εo(1) for x1 > 0.

= ⇒ u(0) > 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 2. ˆ u0 = α(x1)+ + β(x1)−. If α > 0, then u = 1

2x2 1 + εαx1 + εo(1) for x1 > 0.

= ⇒ u(0) > 0. Thus α, β ≤ 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 2. ˆ u0 = α(x1)+ + β(x1)−. If α > 0, then u = 1

2x2 1 + εαx1 + εo(1) for x1 > 0.

= ⇒ u(0) > 0. Thus α, β ≤ 0. If α < 0, then u = 0 on a small line segment.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Ruling out 2. ˆ u0 = α(x1)+ + β(x1)−. If α > 0, then u = 1

2x2 1 + εαx1 + εo(1) for x1 > 0.

= ⇒ u(0) > 0. Thus α, β ≤ 0. If α < 0, then u = 0 on a small line segment. Lemma: If D2u ≥ −c0ε and |u − 1

2x2 1| < ε in B1, then

Deeu ≥ 0 in B1/2 for all e ∈ S1 close to e1.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ u0 is C 2 at 0. Let q be its second order Taylor expansion. Then |u − (1 2x2

1 + εq)| ≤ ε(o(r2) + o(1)) in Br.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ u0 is C 2 at 0. Let q be its second order Taylor expansion. Then |u − (1 2x2

1 + εq)| ≤ ε(o(r2) + o(1)) in Br.

∆q = 0, D22q ≥ 0. = ⇒ Up to O(ε2)-error, p′ = 1

2x2 1 + εq satisfies ∆p′ = 1 and D2p′ ≥ 0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

ˆ u0 is C 2 at 0. Let q be its second order Taylor expansion. Then |u − (1 2x2

1 + εq)| ≤ ε(o(r2) + o(1)) in Br.

∆q = 0, D22q ≥ 0. = ⇒ Up to O(ε2)-error, p′ = 1

2x2 1 + εq satisfies ∆p′ = 1 and D2p′ ≥ 0.

Lemma Given δ > 0, there is r0 and ε0 such that if 0 ∈ Sing(u), and |u − 1

2x2 1| < ε and D2u ≥ −c0ε in B1 for some

ε < ε0, then there is a parabola solution p′ satisfying |u − p′| < δεr2

0 in Br0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Away from {x1 = 0}, |u − p′| < δǫr2

0 implies

D2u ≥ D2p′ − Cδǫ ≥ −Cδǫ in B 1

4 r0(1

2r0e1).

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Away from {x1 = 0}, |u − p′| < δǫr2

0 implies

D2u ≥ D2p′ − Cδǫ ≥ −Cδǫ in B 1

4 r0(1

2r0e1). Together with Deeu + c0ε ≥ 0, ∆(Deeu + c0ǫ) = 0 in {u > 0}, and Deeu ≥ 0 in {u = 0}, we have Deeu ≥ −c0(1 − β)ε in Br0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Lemma: Quadratic approximation Case (0.5) There are universal ε0, c0 > 0 , and β, r0 ∈ (0, 1) such that the following is true: Suppose |u − 1

2x2 1| ≤ ε and D2u ≥ −c0ε in B1 for ε < ε0 and

0 ∈ Sing(u). Then there is a parabola solution p′ such that |u − p′| ≤ (1 − β)εr2

0 and D2u ≥ −c0(1 − β)ε in Br0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Lemma: Quadratic approximation Case (0.5) There are universal ε0, c0 > 0 , and β, r0 ∈ (0, 1) such that the following is true: Suppose |u − 1

2x2 1| ≤ ε and D2u ≥ −c0ε in B1 for ε < ε0 and

0 ∈ Sing(u). Then there is a parabola solution p′ such that |u − p′| ≤ (1 − β)εr2

0 and D2u ≥ −c0(1 − β)ε in Br0.

Lemma: Quadratic approximation Case (1) Given κ > 0, there are ε0, c0 > 0 , and β, r0 ∈ (0, 1), depending on κ, such that the following is true: Suppose for some parabola solution p = 1

2Σajx2 j with

a1 ≥ a2 ≥ . . . ad and a2 ≤ κε, |u − p| ≤ ε and D2u ≥ −c0ε in B1 for ε < ε0 and 0 ∈ Sing(u). Then there is a parabola solution p′ such that |u − p′| ≤ (1 − β)εr2

0 and D2u ≥ −c0(1 − β)ε in Br0.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Lemma: Quadratic approximation Case (1) Given κ > 0, there are ε0, c0 > 0 , and β, r0 ∈ (0, 1), depending on κ, such that the following is true: Suppose for some parabola solution p = 1

2Σajx2 j with

a1 ≥ a2 ≥ . . . ad and a2 ≤ κε, |u − p| ≤ ε and D2u ≥ −c0ε in B1 for ε < ε0 and 0 ∈ Sing(u). Then there is a parabola solution p′ such that |u − p′| ≤ (1 − β)εr2

0 and D2u ≥ −c0(1 − β)ε in Br0.

Suppose we can iteratively apply this lemma, then we get a sequence of parabola solutions pk such that |u − pk| < (1 − β)kεr2k in Brk

0 , and

|D2pk − D2pk+1| < (1 − β)kε. In particular pk → p, which satisfies |u − p| ≤ Cr2+α in Br.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Lemma: Quadratic approximation Case (2) There are universal κ, ε0, c0 > 0, and β, r0 ∈ (0, 1) such that the following is true: Suppose |u − p| ≤ ε and D2u ≥ −c0ε in B1 for ε < ε0, a2 ≥ κε and 0 ∈ Sing(u). Then there is a parabola solution p′ such that |u − p′| ≤ ε′r2

0 and

D2u ≥ −c0ε′ in Br0. We have the following dichotomy for ε′:

1 ε′ ≤ (1 − β)ε; or 2 ε′ ≤ ε − εµ and (u − h)( 1

2r0e1) ≤ C(ε − ε′), where h is the

solution to

• F(D2h) = 1

in B1, h = u along ∂B1.

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thank you!

Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set