A fundamental inequality for the p-Laplacian and the -Laplacian Yi - PowerPoint PPT Presentation

A fundamental inequality for the p-Laplacian and the ∞ -Laplacian Yi Ru-Ya Zhang ETH Z¨ urich Kungliga Tekniska h¨ ogskolan, Sweden August 2019 Yi Zhang A fundamental inequality

Denote by ∆ and ∆ ∞ the Laplacian and ∞ -Laplacian, respectively, in R n with n ≥ 2, i.e. ∆ ∞ v = D 2 vDv · Dv ∀ v ∈ C ∞ . ∆ v = div( Dv ) and Observe that ∆ ∞ is a highly degenerate nonlinear second elliptic partial differential operator whose coefficient matrix for the second derivative has always rank 1 everywhere. Yi Zhang A fundamental inequality

Denote by ∆ and ∆ ∞ the Laplacian and ∞ -Laplacian, respectively, in R n with n ≥ 2, i.e. ∆ ∞ v = D 2 vDv · Dv ∀ v ∈ C ∞ . ∆ v = div( Dv ) and Observe that ∆ ∞ is a highly degenerate nonlinear second elliptic partial differential operator whose coefficient matrix for the second derivative has always rank 1 everywhere. The equation ∆ ∞ u = 0 is introduced by Aronsson in 1960’s (assuming u ∈ C 2 (Ω)) as the Euler-Lagrange’s equation while absolutely minimizing the L ∞ -functional | Du | 2 F ( u , Ω) = ess sup Ω A function u ∈ W 1 , ∞ loc (Ω) is an absolute minimizer in Ω if for any V ⊂⊂ Ω we have F ( u , V ) ≤ F ( v , V ) provided that v ∈ W 1 , ∞ loc ( V ) and v = u on ∂ V . Yi Zhang A fundamental inequality

However, not every ∞ -harmonic is C 2 : Aronsson gives an example w ( x 1 , x 2 ) = x 4 / 3 − x 4 / 3 1 2 in the plane which is only C 1 , 1 / 3 . This function is usually called the Aronsson function. Yi Zhang A fundamental inequality

However, not every ∞ -harmonic is C 2 : Aronsson gives an example w ( x 1 , x 2 ) = x 4 / 3 − x 4 / 3 1 2 in the plane which is only C 1 , 1 / 3 . This function is usually called the Aronsson function. Jensen in 1993 identified the viscosity solutions of ∆ ∞ u = 0 with absolute minimizers of such L ∞ -functional and proved the uniqueness. Yi Zhang A fundamental inequality

However, not every ∞ -harmonic is C 2 : Aronsson gives an example w ( x 1 , x 2 ) = x 4 / 3 − x 4 / 3 1 2 in the plane which is only C 1 , 1 / 3 . This function is usually called the Aronsson function. Jensen in 1993 identified the viscosity solutions of ∆ ∞ u = 0 with absolute minimizers of such L ∞ -functional and proved the uniqueness. Later, Lu and Wang considered the inhomogeneous ∞ -Laplace equation − ∆ ∞ u = f , u = g ∈ C ( ∂ Ω) (1) in Ω in the viscosity sense, where f ∈ C (Ω) is always assumed. They proved the existence and uniqueness of such an equation under the assumption that f is bounded and | f | > 0. However, when f changes sign, they gave a counter-example to the uniqueness of (1). The uniqueness for the case where f ≥ 0 or f ≤ 0 is still open. Yi Zhang A fundamental inequality

On the other hand, it is well-known that there is a similar connection between the p -Laplacian and minimizers of the Dirichlet p -energy. Yi Zhang A fundamental inequality

On the other hand, it is well-known that there is a similar connection between the p -Laplacian and minimizers of the Dirichlet p -energy. Recall that u p ∈ W 1 , p (Ω) is called a p -harmonic function if it minimizes the Dirichlet p -energy � � |∇ u p | p dx ≤ |∇ v | p dx Ω Ω whenever u p − v ∈ W 1 , p (Ω). Equivalently, 0 − ∆ p u p := − div( |∇ u p | p − 2 ∇ u p ) = 0 in Ω in the weak sense or viscosity sense. It is clear that, by the additivity of L p -integral, we automatically have that u p ’s are also absolute minimizers. Yi Zhang A fundamental inequality

On the other hand, it is well-known that there is a similar connection between the p -Laplacian and minimizers of the Dirichlet p -energy. Recall that u p ∈ W 1 , p (Ω) is called a p -harmonic function if it minimizes the Dirichlet p -energy � � |∇ u p | p dx ≤ |∇ v | p dx Ω Ω whenever u p − v ∈ W 1 , p (Ω). Equivalently, 0 − ∆ p u p := − div( |∇ u p | p − 2 ∇ u p ) = 0 in Ω in the weak sense or viscosity sense. It is clear that, by the additivity of L p -integral, we automatically have that u p ’s are also absolute minimizers. The regularity of p -harmonic functions has been widely studied and is understood quite well. (Uraltseva, Lewis, Dibenedetto, Evans, Uhlenbeck, Iwaniec, Manfredi, Lindqvist, Fusco, Kinnunen...) Yi Zhang A fundamental inequality

By the standard energy estimate, there exists a subsequence p i → ∞ and u ∈ W 1 , ∞ (Ω) so that u = lim p i →∞ u p i weakly in ∩ q > 1 W 1 , q (Ω) and u absolutely minimizing the L ∞ -functional. Therefore, u is infinity harmonic. Yi Zhang A fundamental inequality

By the standard energy estimate, there exists a subsequence p i → ∞ and u ∈ W 1 , ∞ (Ω) so that u = lim p i →∞ u p i weakly in ∩ q > 1 W 1 , q (Ω) and u absolutely minimizing the L ∞ -functional. Therefore, u is infinity harmonic. Moreover, a formal calculation gives the following: For a p -harmonic function, � |∇ u | 2 � ∆ p u = ( p − 2) |∇ u | p − 4 p − 2 ∆ u + ∆ ∞ u = 0 . In particular, we have |∇ u | 2 p − 2 ∆ u + ∆ ∞ u = 0 . By letting p → ∞ , we obtain ∆ ∞ u = 0 . Yi Zhang A fundamental inequality

We call 1 ∆ N p − 2∆ u + |∇ u | − 2 ∆ ∞ u p u := the normalized p -Laplacian, which can be regarded as an ”interpolation” between Laplacian and (normalized) ∞ -Laplacian. Yi Zhang A fundamental inequality

We call 1 ∆ N p − 2∆ u + |∇ u | − 2 ∆ ∞ u p u := the normalized p -Laplacian, which can be regarded as an ”interpolation” between Laplacian and (normalized) ∞ -Laplacian. Indeed, there is a rigorous way to interpret this relation via probability model. Recall that harmonic functions is related to Brownian motion. Peres et al. introduced a notion of tug-of-war and use it to model the (normalized) infinity Laplacian | Du | − 2 ∆ ∞ and then the (normalized) p -Laplacian. Yi Zhang A fundamental inequality

In a recent paper by H. Dong, P. Fa, Zhang and Y. Zhou, we show the following inequality, whose the planar version was obtained by H. Koch, Zhang and Y. Zhou earlier. Yi Zhang A fundamental inequality

In a recent paper by H. Dong, P. Fa, Zhang and Y. Zhou, we show the following inequality, whose the planar version was obtained by H. Koch, Zhang and Y. Zhou earlier. Lemma 1 Let n ≥ 2 and U be a domain of R n . For any v ∈ C ∞ ( U ) , we have � � � | D 2 vDv | 2 − ∆ v ∆ ∞ v − 1 2[ | D 2 v | 2 − (∆ v ) 2 ] | Dv | 2 � � � � � ≤ n − 2 [ | D 2 v | 2 | Dv | 2 − | D 2 vDv | 2 ] in U . 2 Yi Zhang A fundamental inequality

Based on this inequality, we are able to show the following results for euqations involving p -Laplacian: Yi Zhang A fundamental inequality

Based on this inequality, we are able to show the following results for euqations involving p -Laplacian: Theorem 1 Let n ≥ 2 , p ∈ (1 , 2) ∪ (2 , ∞ ) and γ < γ n , p , where n − 1 , 3 + p − 1 n γ n , p := min { p + n − 1 } . For any weak/viscosity solution u to ∆ p u = 0 in Ω , p − γ 2 Du ∈ W 1 , 2 we have | Du | loc (Ω) and, for anyB = B ( z , r ) ⊂ 2 B ⊂⊂ Ω � 2 Du ] | 2 dx ≤ C ( n , p , γ ) 1 � | Du | p − γ +2 dx . p − γ | D [ | Du | r 2 B 2 B Theorem 1 improves the earlier result by Bojarski and Iwaniec, where the convexity and the monotonicity of the p -Laplacian were heavily used in their proof. Yi Zhang A fundamental inequality

As a byproduct, we reprove the following higher integrability of D 2 u , which was shown earlier by using the Cordes condition. Corollary 2 2 Let n ≥ 2 and p ∈ (1 , 2) ∪ (2 , 3 + n − 2 ) . There exists δ n , p ∈ (0 , 1) such that for any weak/viscosity solution u to ∆ p u = 0 in Ω , we have D 2 u ∈ L q loc (Ω) for any q < 2 + δ n , p and, for any B = B ( z , r ) ⊂ 2 B ⊂⊂ Ω , � 1 / q � 1 / 2 � � ≤ C ( n , p , q )1 � � | D 2 u | q dx | Du | 2 dx – – r 2 B B Yi Zhang A fundamental inequality

Similar results also hold in the parabolic case, and some of them were completely open problems. Write Q r ( z , s ) := ( s − r 2 , s ) × B ( z , r ). Theorem 2 2 Let n ≥ 2 and p ∈ (1 , 2) ∪ (2 , 3 + n − 2 ) . There exists δ n , p ∈ (0 , 1) such that for any viscosity solution u = u ( x , t ) to u t − ∆ N p u = 0 in Ω T := Ω × (0 , T ) , we have u t , D 2 u ∈ L q loc (Ω) for any q < 2 + δ n , p , and for every Q r = Q r ( z , s ) ⊂ Q 2 r ⊂⊂ Ω T . � 1 / q � 1 / 2 � � ≤ C ( n , p , q )1 � � [ | u t | q + | D 2 u | q ] dx | Du | 2 dx . – – r Q r Q 2 r Yi Zhang A fundamental inequality

Theorem 3 Let n ≥ 1 . For any weak/viscosity solution u = u ( x , t ) to u t − ∆ p u = 0 in Ω T , the following results hold. (i) For p ∈ (1 , 2) ∪ (2 , ∞ ) , we have u t ∈ L 2 loc (Ω T ) and, for any Q r = Q r ( z , s ) ⊂ Q 2 r ⊂⊂ Ω T , ( u t ) 2 dx dt ≤ C � � | Du | p + | Du | 2 p − 2 dx dt r 2 Q r Q 2 r (ii) For p ∈ (1 , 2) ∪ (2 , 3) , we have D 2 u ∈ L 2 loc (Ω T ) and, for any Q r = Q r ( z , s ) ⊂ Q 2 r ⊂⊂ Ω T , � | D 2 u | 2 dx dt ≤ C ( n , p ) 1 � | Du | 2 + | Du | 4 − p dx dt . r 2 Q r Q 2 r The range of p (including p = 2 from the classical result) here is sharp for the W 2 , 2 loc -regularity. Yi Zhang A fundamental inequality