Nonuniformly elliptic problems Presentation June 2020 CITATIONS - PDF document

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/341792977 Nonuniformly elliptic problems Presentation · June 2020 CITATIONS READS 0 7 1 author: Giuseppe Mingione Università di Parma 167 PUBLICATIONS 7,029 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Nonlinear Calderón-Zygmund theory View project Double phase functionals View project All content following this page was uploaded by Giuseppe Mingione on 01 June 2020. The user has requested enhancement of the downloaded file.

Nonuniformly elliptic problems Giuseppe Mingione January 27, 19 Singular Problems Associated to Quasilinear Equations ShanghaiTech, June 1, 2020 Giuseppe Mingione Non-uniform ellipticity

The uniformly elliptic case We consider nonlinear equations with linear growth − div a ( Du ) = µ under assumptions � | a ( z ) | + | ∂ a ( z ) || z | ≤ L | z | p − 1 ν | z | p − 2 | ξ | 2 ≤ � ∂ a ( z ) ξ, ξ � A typical instance is − div ( | Du | p − 2 Du ) = µ Emphasis on Lipschitz estimates. We want to consider more general growth and ellipticity assumptions. Giuseppe Mingione Non-uniform ellipticity

Intrinsic rewriting Theorem If u solves − div a ( Du ) = µ , then d | µ | ( y ) � � | a ( Du ( x )) | � | x − y | n − 1 + − | a ( Du ) | dy B R ( x ) B R ( x ) holds Duzaar & Min. (JFA 2010) for 2 − 1 / n < p < 2 Kuusi & Min. (CRAS 2011 + ARMA 2013) for the case p ≥ 2 Nguyen & Phuc (Math. Ann. 19) for 3 n − 2 2 n − 1 < p ≤ 2 Giuseppe Mingione Non-uniform ellipticity

A global estimate Theorem If u solves − div a ( Du ) = µ , and decays properly at infinity, then d | µ | ( y ) � | Du ( x ) | p − 1 � | x − y | n − 1 R n Giuseppe Mingione Non-uniform ellipticity

A classical theorem of Stein Theorem (Stein, Ann. Math. 1981) Dv ∈ L ( n , 1) = ⇒ v is continuous Recall that � ∞ |{ x : | g ( x ) | > λ }| 1 / n d λ < ∞ g ∈ L ( n , 1) ⇐ ⇒ 0 An example of L ( n , 1) function is given by 1 β > 1 in the ball B 1 / 2 | x | log β (1 / | x | ) △ u = µ ∈ L ( n , 1) = ⇒ Du is continuous Giuseppe Mingione Non-uniform ellipticity

A nonlinear Stein theorem Now notice that d | µ | ( y ) � µ ∈ L ( n , 1) = ⇒ lim | x − y | n − 1 = 0 uniformly w.r.t. x R → 0 B R ( x ) From the results of Kuusi & Min. it also follows that if d | µ | ( y ) � lim | x − y | n − 1 = 0 uniformly = ⇒ Du is continuous . R → 0 B R ( x ) Giuseppe Mingione Non-uniform ellipticity

Linear and nonlinear Stein theorems Theorem (Stein, Ann. Math. 1981) If u solves the Poisson equation − div Du = △ u = µ ∈ L ( n , 1) then Du is continuous . Giuseppe Mingione Non-uniform ellipticity

Linear and nonlinear Stein theorems Theorem (Stein, Ann. Math. 1981) If u solves the Poisson equation − div Du = △ u = µ ∈ L ( n , 1) then Du is continuous . Theorem (Kuusi & Min. ARMA 2013) If u solves the p-Laplacean equation − div a ( Du ) = µ ∈ L ( n , 1) then Du is continuous . Giuseppe Mingione Non-uniform ellipticity

Linear and nonlinear Stein theorems Theorem (Kuusi & Min. ARMA 2013) If u solves the p-Laplacean type equation − div a ( x , Du ) = µ ∈ L ( n , 1) with � ω ( ̺ ) d ̺ < ∞ ̺ 0 then Du is continuous . Here it is | a ( x , z ) − a ( y , z ) | � ω ( | x − y | ) | z | p − 1 Giuseppe Mingione Non-uniform ellipticity

Emphasis on external ingredients µ and c ( · ) Theorem (Kuusi & Min. ARMA 2013 - Calc. Var. 2014) If u solves the p-Laplacean equation − div ( c ( x ) a ( Du )) = µ with 0 < c ( · ) is Dini continuous and µ ∈ L ( n , 1) then Du is continuous . Giuseppe Mingione Non-uniform ellipticity

A classic from Ladyzhenskaya & Ural’tseva (1970) Giuseppe Mingione Non-uniform ellipticity

A classic from Trudinger (1971) Giuseppe Mingione Non-uniform ellipticity

A classic from Leon Simon (1976) Giuseppe Mingione Non-uniform ellipticity

Verifying uniform ellipticity Minimizers of F ( z ) := | z | p � v �→ [ F ( Dv ) − fv ] dx for p Ω are solutions to − div ∂ F ( Du ) = f . In this case we have ( p − 1) | z | p − 2 Id ≤ ∂ 2 F ( z ) ≤ c | z | p − 2 Id therefore highest eigenvalue of ∂ 2 F ( z ) p lowest eigenvalue of ∂ 2 F ( z ) ≈ min { p − 1 , 1 } Giuseppe Mingione Non-uniform ellipticity

Non-uniformly elliptic problems I consider functionals � v �→ [ F ( Dv ) − fv ] dx , Ω so that the Euler-Lagrange reads as − div ∂ F ( Du ) = f and non-uniform ellipticity reads as highest eigenvalue of ∂ 2 F ( z ) | z |→∞ R ( z ) = lim lim lowest eigenvalue of ∂ 2 F ( z ) = ∞ . | z |→∞ Giuseppe Mingione Non-uniform ellipticity

Connection: functionals with non-standard growth of polynomial type (Marcellini) � W 1 , 1 ∋ v �→ Ω ⊂ R n F ( Dv ) dx Ω with | z | p � F ( z ) � | z | q + 1 and q > p > 1 Giuseppe Mingione Non-uniform ellipticity

A first example: almost polynomial This means � W 1 , 1 ∋ v �→ | Dv | p log(1 + | Dv | ) dx p ≥ 1 Ω in particular, we have the almost linear growth condition � W 1 , 1 ∋ v �→ | Dv | log(1 + | Dv | ) dx Ω Giuseppe Mingione Non-uniform ellipticity

Oscillating and polynomial This time it is � W 1 , 3 ∋ v �→ F ( | Dv | ) dx Ω with et 3 if t ≤ e   F ( t ) := t 4+sin(log log t ) if t > e  Giuseppe Mingione Non-uniform ellipticity

Anisotropic growth conditions In this case the model is n � W 1 , 1 ∋ v �→ | Dv | p + � | D i v | p i dx Ω i =1 with 1 ≤ p ≤ p 1 ≤ . . . ≤ p n Giuseppe Mingione Non-uniform ellipticity

Fast growth conditions This means we are considering functionals of the type � exp(exp( . . . exp( | Dv | p ) . . . )) dx , v �→ p ≥ 1 , Ω Duc & Eells (1991), Lieberman (1992), Marcellini (1996) Giuseppe Mingione Non-uniform ellipticity

A basic condition � W 1 , 1 ∋ v �→ Ω ⊂ R n F ( Dv ) dx Ω with | z | p � F ( z ) � | z | q + 1 and q > p > 1 then q p < 1 + o ( n ) is a sufficient (Marcellini) and necessary (Giaquinta and Marcellini) condition for regularity Giuseppe Mingione Non-uniform ellipticity

A model result by Marcellini We consider functionals of the type � F ( v ) := v : Ω → R F ( Dv ) dx Ω assuming that z �→ F ( z ) is C 2 and ν | z | p ≤ F ( z ) ≤ L (1 + | z | q ) � p − 2 2 | λ | 2 ≤ � ∂ 2 F ( z ) λ, λ � ≤ L ( | z | 2 + 1) q − 2 ν ( | z | 2 + 1) 2 | λ | 2 Giuseppe Mingione Non-uniform ellipticity

A model result by Marcellini Theorem (Marcellini JDE 1991) Under the above assumptions, if q p < 1 + 2 n then any local W 1 , p -minimizer is locally Lipschitz continuous. Moreover, we have 2 � � � ( n +2) p − nq � Du � L ∞ ( B R / 2 ) � − F ( Du ) dx + 1 B R for every ball B R ⋐ Ω Giuseppe Mingione Non-uniform ellipticity

Additional interesting results Bella & Sh¨ affner, Analysis & PDE, to appear q 2 ⇒ Du ∈ L ∞ p < 1 + n − 1 = loc Sh¨ affner, Arxiv 2020, to appear q 2 ⇒ Du ∈ L q p < 1 + n − 1 = loc (vectorial case) Hirsch & Sh¨ affner, Comm. Cont. Math., to appear. p − 1 1 1 ⇒ u ∈ L ∞ q < n − 1 = loc De Filippis & Kristensen & Koch, to appear 2 q ⇒ Du ∈ L ∞ p < 1 + n − 2 = loc by duality methods, under special assumptions Giuseppe Mingione Non-uniform ellipticity

Non-standard growth conditions Bounded minimisers give better bounds q < p + 1 the first example of this result I know is from a paper of Uraltseva & Urdaletova (1984). Later results by Choe (Nonlinear Anal. 1992) – Kristensen & co. (Ann. IHP 2011) – De Filippis & Min. (JGA 2020). Giuseppe Mingione Non-uniform ellipticity

Special structures Theorem (Bousquet & Brasco Rev. Mat. Iber. to appear) If u is a local minimizer of the functional n � | D k v | p k dx , � v �→ Ω k =1 where 2 ≤ p 1 ≤ . . . ≤ p n Then u ∈ L ∞ ⇒ Du ∈ L ∞ loc = loc . Giuseppe Mingione Non-uniform ellipticity

Special structures Theorem (Bousquet & Brasco Rev. Mat. Iber. to appear) If u is a local minimizer of the functional n � | D k v | p k dx , � v �→ Ω k =1 where 2 ≤ p 1 ≤ . . . ≤ p n Then u ∈ L ∞ ⇒ Du ∈ L ∞ loc = loc . No upper bound on p n / p 1 is needed. Giuseppe Mingione Non-uniform ellipticity

The variational setting We consider functionals � v �→ [ F ( Dv ) − fv ] dx , Ω Double control on the eigenvalues g 1 ( | z | ) Id � ∂ 2 F ( z ) � g 2 ( | z | ) Id Balance condition �� | z | � R ( z ) � g 2 ( | z | ) g 1 ( | z | ) � H g 1 ( s ) s ds 0 for a suitable increasing function H ( · ) which is of power type Giuseppe Mingione Non-uniform ellipticity

The non-uniformly elliptic case Theorem (Beck & Min. CPAM 2020) If u is a local minimizer and f ∈ L ( n , 1) , then Du ∈ L ∞ loc (Ω) . Moreover the estimate � � Du � L ∞ ( BR / 2) � γ 2 � � + � f � γ 1 − g 1 ( s ) s ds � F ( Du ) dx L ( n , 1)( B R ) + 1 0 B R holds for every ball. The result still holds in the vectorial case provided F ( Du ) ≡ ˜ F ( | Du | ) . Giuseppe Mingione Non-uniform ellipticity