Previous developments Maximum principles for supersolutions of the double-divergence equation; − → Littman (59) A potential theory, with analogous properties to the case of a ij ( · ) ∂ 2 x i x j ; − → 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 a ij ( · ) ∂ 2 x i x j ; − → Herv´ e (62) Improved maximum principles and preliminary approximation schemes; − → Littman (63) 6
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ) 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L d − 1 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L d − 1 ; − → Fabes-Stroock, Duke Mathematical Journal (84) 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L 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 older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L 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¨ ockner, Comm. Partial Differential Equations (01) 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L 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¨ ockner, Comm. Partial Differential Equations (01) − → Bogachev-Krylov-R¨ ockner (15) 7
Previous developments older spaces provided a ij ∈ C β Regularity theory in H¨ loc ( B 1 ); − → Sj¨ ogren (73) Properties of the associated Green’s function; gains of integrability of the type L 1 = d ⇒ L 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¨ ockner, Comm. Partial Differential Equations (01) − → Bogachev-Krylov-R¨ ockner (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 a ( x ) ℓ ( x ) a ( x ) φ xx ( x ) d x = 0 − 1 for every φ ∈ C 2 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 a ( x ) ℓ ( x ) a ( x ) φ xx ( x ) d x = 0 − 1 for every φ ∈ C 2 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 a ( x ) ℓ ( x ) a ( x ) φ xx ( x ) d x = 0 − 1 for every φ ∈ C 2 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 x 0 ∈ { u = 0 } 9
Our program Regularity theory along zero level-sets; focus on x 0 ∈ { u = 0 } ; − → nonphysical free boundary 9
Our program Regularity theory along zero level-sets; focus on x 0 ∈ { u = 0 } ; − → nonphysical free boundary Regularity transmission by approximation methods 9
Our program Regularity theory along zero level-sets; focus on x 0 ∈ { 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 x 0 ∈ { 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 x 0 ∈ { 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 x 0 ∈ { 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 a ij D 2 u � � Tr = 0 in B 1 , for which a richer regularity theory is available 10
Our program Import information from the well-understood non-divergence prob- lem a ij D 2 u � � Tr = 0 in B 1 , for which a richer regularity theory is available The regularity of the coefficients is an upper bound for the regularity of the solutions ‘in the large’. Therefore, we look for regularity improvements at x 0 ∈ { u = 0 } . 10
H¨ older continuity Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 11
H¨ older continuity Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 11
H¨ older continuity Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix 11
H¨ older continuity Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix. Then, u ∈ C 1 − loc ( B 1 ∩ ∂ { u > 0 } ) 11
H¨ older continuity Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix. Then, u ∈ C 1 − loc ( B 1 ∩ ∂ { u > 0 } ) and for every α ∈ (0 , 1) there exists C α > 0 such that | u ( x ) − u ( x 0 ) | ≤ C α r α , sup B r ( x 0 ) for every 0 < r ≪ 1 / 2 and x 0 ∈ ∂ { 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; a ij ∈ C ε u ∈ C ε loc ( B 1 ) = ⇒ loc ( B 1 ) 12
A few remarks Gains of regularity are independent of the data; a ij ∈ C ε u ∈ C ε loc ( B 1 ) = ⇒ loc ( B 1 ); however, at the free boundary u is of class C 1 − 12
A few remarks Gains of regularity are independent of the data; a ij ∈ C ε u ∈ C ε loc ( B 1 ) = ⇒ loc ( B 1 ); however, at the free boundary u is of class C 1 − ; Our results extend to equations involving lower-order terms ∂ 2 a ij ( x ) u ( x ) b i ( x ) u ( x ) � � � � + ∂ x i + c ( x ) u ( x ) = 0 in B 1 x i x j 12
A few remarks Gains of regularity are independent of the data; a ij ∈ C ε u ∈ C ε loc ( B 1 ) = ⇒ loc ( B 1 ); however, at the free boundary u is of class C 1 − ; Our results extend to equations involving lower-order terms ∂ 2 a ij ( x ) u ( x ) b i ( x ) u ( x ) � � � � + ∂ x i + c ( x ) u ( x ) = 0 in B 1 , x i x j provided b i , c : B 1 → 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 x 0 ∈ ∂ { 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 x 0 ∈ ∂ { 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 x 0 ∈ ∂ { 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 x 0 ∈ ∂ { 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 x 0 ∈ ∂ { 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 ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 14
Zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε sup B 9 / 10 14
Zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 one can find h ∈ C 1 , 1 loc ( B 1 ) satisfying � u − h � L ∞ ( B 9 / 10 ) < δ 14
Zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 one can find h ∈ C 1 , 1 loc ( B 1 ) satisfying � u − h � L ∞ ( B 9 / 10 ) < δ. Moreover, h ( x 0 ) = 0 for every x 0 ∈ ∂ { u > 0 } . 14
First-order oscillation control Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 15
First-order oscillation control Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every α ∈ (0 , 1) there exists ε > 0 and ρ ∈ (0 , 1 / 2) such that, if | a ij ( x ) − a ij | < ε sup B 9 / 10 15
First-order oscillation control Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every α ∈ (0 , 1) there exists ε > 0 and ρ ∈ (0 , 1 / 2) such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 then | u ( x ) | ≤ ρ α sup B ρ ( x 0 ) 15
First-order oscillation control Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ). For every α ∈ (0 , 1) there exists ε > 0 and ρ ∈ (0 , 1 / 2) such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 then | u ( x ) | ≤ ρ α , sup B ρ ( x 0 ) for every x 0 ∈ ∂ { u > 0 } . 15
Oscillation control at discrete scales Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 16
Oscillation control at discrete scales Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies | a ij ( x ) − a ij | < ε sup B 9 / 10 16
Oscillation control at discrete scales Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies | a ij ( x ) − a ij | < ε. sup B 9 / 10 Then | u ( x ) | ≤ ρ n α sup B ρ n ( x 0 ) 16
Oscillation control at discrete scales Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ C β loc ( B 1 ) satisfies | a ij ( x ) − a ij | < ε. sup B 9 / 10 Then | u ( x ) | ≤ ρ n α , sup B ρ n ( x 0 ) for every x 0 ∈ ∂ { u > 0 } and every n ∈ N . 16
H¨ older 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 a ij ∈ W 2 , p loc ( B 1 ) for every i , j = 1 , ..., d . 17
H¨ older 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 a ij ∈ W 2 , p loc ( B 1 ) for every i , j = 1 , ..., d . Consider a suitable zero level-set in this context: { u = | Du | = 0 } . 17
H¨ older continuity of the gradient Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 18
H¨ older continuity of the gradient Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 18
H¨ older continuity of the gradient Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix 18
H¨ older continuity of the gradient Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix. Then, u ∈ C 1 , 1 − loc ( B 1 ∩ ∂ { u > 0 } ) 18
H¨ older continuity of the gradient Theorem (Leit˜ ao, P., Santos, 2019) Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ) satisfies � a ij ( · ) − a ij � L ∞ ( B 1 ) ≪ 1 / 2 , where ( a ij ) d i , j =1 is a constant matrix. Then, u ∈ C 1 , 1 − loc ( B 1 ∩ ∂ { u > 0 } ) and there exists C α > 0 such that | Du ( x ) − Du ( x 0 ) | ≤ C α r α , sup B r ( x 0 ) for every 0 < r ≪ 1 / 2, and x 0 ∈ ∂ { u > 0 } ∩ ∂ {| Du | > 0 } and α ∈ (0 , 1). 18
Key: First-order zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation 19
Key: First-order zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε sup B 9 / 10 19
Key: First-order zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 one can find h ∈ C 1 , 1 loc ( B 1 ) satisfying � u − h � L ∞ ( B 9 / 10 ) < δ 19
Key: First-order zero level-set approximation lemma Proposition Let u ∈ L 1 loc ( B 1 ) be a weak solution to the double-divergence equation. Suppose a ij ∈ W 2 , p loc ( B 1 ). For every δ > 0 there exists ε > 0 such that, if | a ij ( x ) − a ij | < ε, sup B 9 / 10 one can find h ∈ C 1 , 1 loc ( B 1 ) satisfying � u − h � L ∞ ( B 9 / 10 ) < δ. Moreover, h ( x 0 ) = 0 and Dh ( x 0 ) = 0 for every x 0 ∈ ∂ { 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
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