Weighted and pointwise bounds in measure datum problems with - PowerPoint PPT Presentation

Weighted and pointwise bounds in measure datum problems with applications Nguyen Cong Phuc Louisiana State University, USA LSU ShanghaiTech University and Masaryk University Zoom Talk – June 1st, 2020 In celebration of Marie-Fran¸ coise Bidaut-V´ eron and Laurent V´ eron’s 70th birthday N. C. Phuc (LSU) May 31, 2020 1 / 33

Acknowledgments Quoc-Hung Nguyen, Shanghai Tech University Simons Foundation N. C. Phuc (LSU) May 31, 2020 2 / 33

Gradient estimates – Introduction Consider the equation R n . − ∆ u = f in Then, under mild assumptions on f and u , one has a pointwise representation ˆ u ( x ) = c ( n ) R n Γ( x − y ) f ( y ) dy , where | x − y | 2 − n � if n > 2 Γ( x − y ) = − log ( | x − y | ) n = 2 . if N. C. Phuc (LSU) May 31, 2020 3 / 33

Gradient estimates – Introduction This pointwise representation is often written as u ( x ) = I 2 f ( x ) , n > 2 , and by differentiating |∇ u ( x ) | ≤ c I 1 | f | ( x ) , n ≥ 2 . Here I α , α ∈ (0 , n ) is a fractional integral d µ ( y ) ˆ I α µ ( x ) = c ( n , α ) | x − y | n − α R n ˆ ∞ µ ( B t ( x )) dt = c t . t n − α 0 N. C. Phuc (LSU) May 31, 2020 4 / 33

Gradient estimates – Introduction Now recall the fractional maximal function M α , α ∈ (0 , n ) : µ ( B t ( x )) x ∈ R n . M α µ ( x ) := sup , t n − α t > 0 Obviously, one has in R n . M α µ ≤ c I α µ The converse holds in the following sense: Theorem (Muckenhoupt-Wheeden ’74) Let q > 0 and w be a weight in the A ∞ class. We have ˆ ˆ R n ( I α µ ) q wdx ≤ C ( q , n , [ w ] A ∞ ) R n ( M α µ ) q wdx . N. C. Phuc (LSU) May 31, 2020 5 / 33

Gradient estimates – Introduction We recall that w ∈ A ∞ if there exist C , ν > 0 such that � ν w ( E ) � | E | w ( B ) ≤ C , | B | for all balls B and all measurable set E ⊂ B . The pair ( C , ν ) is called the A ∞ constants of w and is denoted by [ w ] A ∞ . N. C. Phuc (LSU) May 31, 2020 6 / 33

Gradient estimates – Introduction • Thus for the solution u above one has ˆ ˆ R n |∇ u | q wdx ≤ C ( q , n , [ w ] A ∞ ) R n ( M 1 | f | ) q wdx for all weights w ∈ A ∞ . • This bound has the advantage that it could hold for more general linear uniformly elliptic operator with possibly discontinuous coefficients, whereas for the pointwise bound |∇ u | ≤ c I 1 | f | to hold one needs at least Dini continuous coefficients. • Easier to handle up to the boundary of a bounded domain. • Moreover, this bound is usually enough in many applications to nonlinear PDEs. N. C. Phuc (LSU) May 31, 2020 7 / 33

Main goals • To obtain Muckenhoupt-Wheeden type (weighted) bounds for gradients of solutions to quasilinear elliptic equations with measure data: � − ∆ p u = µ in Ω , (1) u = 0 on ∂ Ω . Here Ω is a bounded open subset of R n , n ≥ 2 , and µ is a finite signed Radon measure in Ω . • To obtain pointwise estimates for gradients of solutions to − ∆ p u = µ in Ω . Assumption on p : For pointwise gradient estimate, we will be dealing mainly with the case 3 n − 2 2 n − 1 < p < + ∞ . N. C. Phuc (LSU) May 31, 2020 8 / 33

Main goals • As an application, we obtain characterizations of existence and removable singularities for the quasilinear Riccati type (viscous Hamilton-Jacobi type) equation with measure data: � − ∆ p u |∇ u | q + µ = in Ω , (2) u = 0 on ∂ Ω . Remark: We can deal with all p > 1 and q ≥ 1 for this equation. N. C. Phuc (LSU) May 31, 2020 9 / 33

A remark on principal operator We can replace the p -Laplacian ∆ p u = div ( |∇ u | p − 2 ∇ u ) with a more general operator of the form L p ( u ) = div A ( x , ∇ u ) , where the nonlinearity A : R n × R n → R n satisfies certain ellipticity and regularity conditions. • For (integral) Muckenhoupt-Weeden type bounds, we need that A satisfies a VMO condition in the x -variable. • For pointwise gradient bounds, we need that A satisfies a H¨ older or Dini condition in the x -variable. N. C. Phuc (LSU) May 31, 2020 10 / 33

Assumptions on Ω For the global gradient estimates, we also require certain regularity on the ground domain Ω . For our purpose C 1 domains would be enough. A sharper condition on ∂ Ω is the so-called Reifenberg flatness condition. Namely, at each boundary point and at every scale, we ask that the boundary of Ω be trapped between two hyperplanes separated by a distance proportional to the scale. N. C. Phuc (LSU) May 31, 2020 11 / 33

Muckenhoupt-Wheeden type (weighted) bounds Theorem (P., Adv. Math. ’14; Nguyen-P., Math. Ann. ’19) Let µ ∈ M b (Ω) . Let 3 n − 2 2 n − 1 < p < ∞ and q > 0 . For any w ∈ A ∞ and any renormalized solution u to − ∆ p u = µ in Ω , u = 0 on ∂ Ω , we have ˆ ˆ q |∇ u | q w ( x ) dx ≤ C p − 1 w ( x ) dx . M 1 ( µ ) Ω Ω Here C depends only on n , p , q , [ w ] A ∞ , and Ω . • Nguyen-P. (submitted): A similar bound is obtained for the case 1 < p ≤ 3 n − 2 2 n − 1 , but with q > 2 − p and w ∈ A 2 − p . q • Local unweighted setting: Mingione, Math. Ann. ’10. This paper treats with measurable coefficients for q ≤ p + ǫ . N. C. Phuc (LSU) May 31, 2020 12 / 33

The key comparison estimate Let u ∈ W 1 , p loc (Ω) be a solution of − ∆ p u = µ . For B R = B R ( x 0 ) ⋐ Ω , we let w ∈ W 1 , p ( B R ) + u be the unique solution to the equation 0 � − ∆ p w = 0 in B R , = on ∂ B R . w u Lemma 2 n − 1 < γ 0 < ( p − 1) n Assume that 3 n − 2 2 n − 1 < p ≤ 2 − 1 n n . Then for any ≤ 1 , n − 1 � 1 1 � γ 0 ≤ C � | µ | ( B R ) � p − 1 |∇ u − ∇ w | γ 0 dx + R n − 1 B R � 2 − p + C | µ | ( B 2 R ) � γ 0 . |∇ u | γ 0 dx R n − 1 B R For p > 2 − 1 n , this was obtained by Mingione ’07, Mingione-Duzaar ’11, with γ 0 = 1 . The case 1 < p ≤ 3 n − 2 2 n − 1 is still open. N. C. Phuc (LSU) May 31, 2020 13 / 33

A difficulty arises when p gets small Note that the fundamental solution of the p -Laplace equation is given by p − n p − 1 , x ∈ R n . v ( x ) = c ( n , p ) | x | n ( p − 1) n − 1 , ∞ , and ∇ v ∈ L 1 loc if and only if p > 2 − 1 Thus ∇ v ∈ L n . When p ≤ 2 − 1 n , ∇ v �∈ L 1 loc , and this prevents us from using the Sobolev’s inequality in the ‘traditional’ argument. Note that 3 n − 2 2 n − 1 < 2 − 1 n . (However, |∇ ( v δ ) | = δ |∇ v | v δ − 1 ∈ L 1 loc for certain δ ∈ (0 , 1) .) N. C. Phuc (LSU) May 31, 2020 14 / 33

Pointwise gradient estimates by Wolff’s potentials • Recall the estimates for functions: Theorem (Kilpel¨ ainen-Mal´ y, Acta Math. ’94) Suppose that u ≥ 0 is a solution of − ∆ p u = µ in Ω . Then for any ball B 2 r ( x ) ⊂ Ω and any γ > 0 , we have ˆ r 1 p − 1 dt � µ ( B t ( x )) � u ( x ) ≥ c 1 t . t n − p 0 ˆ 2 r 1 p − 1 dt u γ � 1 � µ ( B t ( x )) � � γ . u ( x ) ≤ c 2 t + c 2 t n − p 0 B 2 r ( x ) N. C. Phuc (LSU) May 31, 2020 15 / 33

Pointwise gradient estimates by Wolff’s potentials • For derivatives: Theorem (Duzaar-Mingione JFA ’10, Kuusi-Mingione ARMA ’13) Suppose that u is a solution of − ∆ p u = µ in Ω , where p > 2 − 1 n . Then for any ball B 2 r ( x ) ⊂ Ω , we have � ˆ 2 r 1 � | µ | ( B t ( x )) dt p − 1 |∇ u ( x ) | ≤ C + C |∇ u | dy . t n − 1 t 0 B 2 r ( x ) ´ 2 r | µ | ( B t ( x )) dt • Note that t is a truncated first order (linear) Riesz’s t n − 1 0 potential of | µ | . • Historically, the nonlinear case with p = 2 was done in [Mingione JEMS ’11]; the case p > 2 was done in [Duzaar-Mingione AJM ’11]. N. C. Phuc (LSU) May 31, 2020 16 / 33

Pointwise gradient estimates by Wolff’s potentials Theorem (Nguyen-P. JFA ’20) Suppose that u is a solution of − ∆ p u = µ in Ω , where 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . Then for any ball B 2 r ( x ) ⊂ Ω , we have � 1 1 � ˆ 2 r � γ 0 dt � � | µ | ( B t ( x )) � γ 0 γ 0( p − 1) |∇ u | γ 0 |∇ u ( x ) | ≤ C + C . t n − 1 t 0 B 2 r ( x ) � � 2 n − 1 , ( p − 1) n n Here γ 0 is any number in . n − 1 • γ 0 < 1 . � γ 1 dt � γ 2 dt ´ 2 r ´ 4 r � � | µ | ( B t ( x )) | µ | ( B t ( x )) • t ≤ C t whenever γ 1 > γ 2 > 0 . 0 t n − 1 0 t n − 1 N. C. Phuc (LSU) May 31, 2020 17 / 33

Sharp quantitative C 1 ,α estimates Two important ingredients in the proof of the above theorem: • Comparision estimates obtained in a previous lemma. • Sharp quantitative C 1 ,α estimates: Let w be a W 1 , p loc solution to the homogeneous equation − ∆ p w = 0 in Ω . Then we have 1) Classical C 1 ,α bound: |∇ w ( x ) − ∇ w ( y ) | ≤ C |∇ w | dx | x − y | α B r ( x 0 ) for any x , y ∈ B r / 2 ( x 0 ) ⊂ B r ( x 0 ) ⊂ Ω . 2) Duzaar-Mingione’s C 1 ,α bound: |∇ w − ( ∇ w ) B ρ ( x 0 ) | B ρ ( x 0 ) � ρ � α ≤ C |∇ w − ( ∇ w ) B r ( x 0 ) | r B r ( x 0 ) for every B r ( x 0 ) ⊂ Ω and ρ < r . N. C. Phuc (LSU) May 31, 2020 18 / 33