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 ( D 2 u ) = χ { u > 0 } in Ω ⊂ R d , 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 ( D 2 u ) = χ { u > 0 } in Ω ⊂ R d , u ≥ 0 in Ω. F is uniformly elliptic. F (0) = 0 . F is convex and C 1 . Eg. F ( D 2 u ) = ∆ u . Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

The fully nonlinear obstacle problem � F ( D 2 u ) = χ { u > 0 } in Ω ⊂ R d , u ≥ 0 in Ω. F is uniformly elliptic. F (0) = 0 . F is convex and C 1 . Eg. F ( D 2 u ) = ∆ 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 u r ( x ) = 1 r 2 u ( rx ) . Up to a subseqn of r → 0, u r → u 0 in C 1 ,α loc ( R d ) . 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 u r ( x ) = 1 r 2 u ( rx ) . Up to a subseqn of r → 0, u r → u 0 in C 1 ,α loc ( R d ) . Thm (Lee ’98) 2 max { x · e , 0 } 2 for an e ∈ S d − 1 and Either u 0 ( x ) = γ F ( γ e ⊗ e ) = 1 (half-space solutions); or u 0 ( x ) = 1 2 x · 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, D 2 u r ≥ − o (1) in B 1 . 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, D 2 u r ≥ − o (1) in B 1 . Assuming 0 ∈ Sing ( u ), 3. As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 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, D 2 u r ≥ − o (1) in B 1 . Assuming 0 ∈ Sing ( u ), 3. As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 0 . 4. As r → 0, |{ u =0 }∩ B r | → 0 . r d 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, D 2 u r ≥ − o (1) in B 1 . Assuming 0 ∈ Sing ( u ), 3. As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 0 . 4. As r → 0, |{ u =0 }∩ B r | → 0 . r d 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, u r ∼ 1 2 max { x 1 , 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, u r ∼ 1 2 max { x 1 , 0 } 2 . = ⇒ ∂ { u > 0 } ∩ B r ∼ { x 1 = 0 } ∩ B r . 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, u r ∼ 1 2 max { x 1 , 0 } 2 . = ⇒ ∂ { u > 0 } ∩ B r ∼ { x 1 = 0 } ∩ B r . 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, u r ∼ 1 2 max { x 1 , 0 } 2 . = ⇒ ∂ { u > 0 } ∩ B r ∼ { x 1 = 0 } ∩ B r . Sing ( u )? Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F ( D 2 u ) = ∆ u , monotonicity formulae ⇒ | u − p | ≤ σ ( r ) r 2 in B r for a modulos of continuity σ. = (Previously we had: As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 0 . ) Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F ( D 2 u ) = ∆ u , monotonicity formulae ⇒ | u − p | ≤ σ ( r ) r 2 in B r for a modulos of continuity σ. = (Previously we had: As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 0 . ) No monotonicity formulae for fully nonlinear F . Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

When F ( D 2 u ) = ∆ u , monotonicity formulae ⇒ | u − p | ≤ σ ( r ) r 2 in B r for a modulos of continuity σ. = (Previously we had: As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 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 ( D 2 u ) = ∆ u , monotonicity formulae ⇒ | u − p | ≤ σ ( r ) r 2 in B r for a modulos of continuity σ. = (Previously we had: As r → 0, we can find parabola solutions p r such that � u r − p r � L ∞ ( B 1 ) → 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 =0 Σ k ( u ) . Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Thm (Savin-Y. ’19) Sing ( u ) = ∪ d − 1 k =0 Σ k ( u ) . Σ d − 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 =0 Σ k ( u ) . Σ d − 1 ( u ) is locally covered by a C 1 ,α -hypersurface. For k ≤ d − 2, Σ k ( u ) is locally covered by a k -dim C 1 , log ε -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 ) r 2 in B r . 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 ) r 2 in B r . One step in a discretized version: | u − p | < ε in B 1 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 ) r 2 in B r . One step in a discretized version: | u − p | < ε in B 1 for some ε very small. = ⇒ We can find p ′ such that | u − p ′ | < ε ′ r 2 0 in B r 0 for some r 0 ∈ (0 , 1) and ε ′ < ε. Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume � in B 1 ⊂ R 2 , ∆ u = χ { u > 0 } u ≥ 0 in B 1 , 0 ∈ Sing ( u ), and | u − 1 2 x 2 1 | < ε in B 1 . Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume � in B 1 ⊂ R 2 , ∆ u = χ { u > 0 } u ≥ 0 in B 1 , 0 ∈ Sing ( u ), and | u − 1 2 x 2 1 | < ε in B 1 . Normalize u ε = 1 ε ( u − 1 O ε = 1 ε ( − 1 2 x 2 1 ) , ˆ 2 x 2 ˆ 1 ) . Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set

Assume � in B 1 ⊂ R 2 , ∆ u = χ { u > 0 } u ≥ 0 in B 1 , 0 ∈ Sing ( u ), and | u − 1 2 x 2 1 | < ε in B 1 . Normalize u ε = 1 ε ( u − 1 O ε = 1 ε ( − 1 2 x 2 1 ) , ˆ 2 x 2 ˆ 1 ) . u 0 is C 2 at 0. The strategy: 1. ˆ u ε → ˆ u 0 as ε → 0. 2. ˆ Hui Yu (Columbia) Joint with Ovidiu Savin (Columbia) Regularity of the singular set