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¨

• gskolan, Sweden

August 2019

Yi Zhang A fundamental inequality

Denote by ∆ and ∆∞ the Laplacian and ∞-Laplacian, respectively, in Rn with n ≥ 2, i.e. ∆v = div(Dv) and ∆∞v = D2vDv · Dv ∀v ∈ C ∞. 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 Rn with n ≥ 2, i.e. ∆v = div(Dv) and ∆∞v = D2vDv · Dv ∀v ∈ C ∞. 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 F(u, Ω) = ess sup

Ω

|Du|2 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(x1, x2) = x 4/3

1

− x 4/3

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(x1, x2) = x 4/3

1

− x 4/3

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(x1, x2) = x 4/3

1

− x 4/3

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 up ∈ W 1, p(Ω) is called a p-harmonic function if it minimizes the Dirichlet p-energy

• Ω

|∇up|p dx ≤

• Ω

|∇v|p dx whenever up − v ∈ W 1, p (Ω). Equivalently, −∆pup := −div(|∇up|p−2∇up) = 0 in Ω in the weak sense or viscosity sense. It is clear that, by the additivity of Lp-integral, we automatically have that up’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 up ∈ W 1, p(Ω) is called a p-harmonic function if it minimizes the Dirichlet p-energy

• Ω

|∇up|p dx ≤

• Ω

|∇v|p dx whenever up − v ∈ W 1, p (Ω). Equivalently, −∆pup := −div(|∇up|p−2∇up) = 0 in Ω in the weak sense or viscosity sense. It is clear that, by the additivity of Lp-integral, we automatically have that up’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 pi → ∞ and u ∈ W 1, ∞(Ω) so that u = lim

pi→∞ upi

weakly in ∩q>1W 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 pi → ∞ and u ∈ W 1, ∞(Ω) so that u = lim

pi→∞ upi

weakly in ∩q>1W 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, ∆pu = (p − 2)|∇u|p−4 |∇u|2 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 ∆N

p u :=

1 p − 2∆u + |∇u|−2∆∞u the normalized p-Laplacian, which can be regarded as an ”interpolation” between Laplacian and (normalized) ∞-Laplacian.

Yi Zhang A fundamental inequality

We call ∆N

p u :=

1 p − 2∆u + |∇u|−2∆∞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 Rn. For any v ∈ C ∞(U ), we have

• |D2vDv|2 − ∆v∆∞v − 1

2[|D2v|2 − (∆v)2]|Dv|2

• ≤ n − 2

2 [|D2v|2|Dv|2 − |D2vDv|2] in U .

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 < γn,p, where γn,p := min{p +

n n−1, 3 + p−1 n−1}. For any weak/viscosity solution u to

∆pu = 0 in Ω, we have |Du|

p−γ 2 Du ∈ W 1,2

loc (Ω) and, for anyB = B(z, r) ⊂ 2B ⊂⊂ Ω

• B

|D[|Du|

p−γ 2 Du]|2 dx ≤ C(n, p, γ) 1

r 2

• 2B

|Du|p−γ+2 dx. 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 D2u, which was shown earlier by using the Cordes condition. Corollary 2 Let n ≥ 2 and p ∈ (1, 2) ∪ (2, 3 +

2 n−2). There exists δn,p ∈ (0, 1) such

that for any weak/viscosity solution u to ∆pu = 0 in Ω, we have D2u ∈ Lq

loc (Ω) for any q < 2 + δn,p and, for any

B = B(z, r) ⊂ 2B ⊂⊂ Ω,

• –
• B

|D2u|q dx 1/q ≤ C(n, p, q)1 r

• –
• 2B

|Du|2 dx 1/2

Yi Zhang A fundamental inequality

Similar results also hold in the parabolic case, and some of them were completely open problems. Write Qr(z, s) := (s − r 2, s) × B(z, r). Theorem 2 Let n ≥ 2 and p ∈ (1, 2) ∪ (2, 3 +

2 n−2). There exists δn,p ∈ (0, 1) such

that for any viscosity solution u = u(x, t) to ut − ∆N

p u = 0

in ΩT := Ω × (0, T), we have ut, D2u ∈ Lq

loc (Ω) for any q < 2 + δn,p, and for every

Qr = Qr(z, s)⊂ Q2r ⊂⊂ ΩT.

• –
• Qr

[|ut|q + |D2u|q] dx 1/q ≤ C(n, p, q)1 r

• –
• Q2r

|Du|2 dx 1/2 .

Yi Zhang A fundamental inequality

Theorem 3 Let n ≥ 1. For any weak/viscosity solution u = u(x, t) to ut − ∆pu = 0 in ΩT, the following results hold. (i) For p ∈ (1, 2) ∪ (2, ∞), we have ut ∈ L2

loc (ΩT) and, for any

Qr = Qr(z, s)⊂ Q2r ⊂⊂ ΩT,

• Qr

(ut)2 dx dt ≤ C r 2

• Q2r

|Du|p + |Du|2p−2 dx dt (ii) For p ∈ (1, 2) ∪ (2, 3), we have D2u ∈ L2

loc (ΩT) and, for any

Qr = Qr(z, s)⊂ Q2r ⊂⊂ ΩT,

• Qr

|D2u|2 dx dt ≤ C(n, p) 1 r 2

• Q2r

|Du|2 + |Du|4−p dx dt. 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

Unlike the p-Laplacian, lots of problems on infinity Laplace equations are open. One of the main problems for infinity Laplace equations is to show the continuity of it gradient. Here is a list of known results on the interior regularity.

Yi Zhang A fundamental inequality

Unlike the p-Laplacian, lots of problems on infinity Laplace equations are open. One of the main problems for infinity Laplace equations is to show the continuity of it gradient. Here is a list of known results on the interior regularity. ∆∞u = 0 in Rn: Crandall and Evans, 2001, linear approximation property; Savin, 2005, C 1

loc -regularity when n = 2;

Evans and Savin, 2008, C 1, α

loc -regularity when n = 2;

Evans and Smart, 2011, differentiable everywhere;

Yi Zhang A fundamental inequality

Unlike the p-Laplacian, lots of problems on infinity Laplace equations are open. One of the main problems for infinity Laplace equations is to show the continuity of it gradient. Here is a list of known results on the interior regularity. ∆∞u = 0 in Rn: Crandall and Evans, 2001, linear approximation property; Savin, 2005, C 1

loc -regularity when n = 2;

Evans and Savin, 2008, C 1, α

loc -regularity when n = 2;

Evans and Smart, 2011, differentiable everywhere; ∆∞u = f in Rn with |f | > 0: Lu and Wang, 2008, W 1, ∞

loc -regularity;

Lindgren, 2014, linear approximation property and differentiable everywhere when f is Lipschitz.

Yi Zhang A fundamental inequality

Unlike the p-Laplacian, lots of problems on infinity Laplace equations are open. One of the main problems for infinity Laplace equations is to show the continuity of it gradient. Here is a list of known results on the interior regularity. ∆∞u = 0 in Rn: Crandall and Evans, 2001, linear approximation property; Savin, 2005, C 1

loc -regularity when n = 2;

Evans and Savin, 2008, C 1, α

loc -regularity when n = 2;

Evans and Smart, 2011, differentiable everywhere; ∆∞u = f in Rn with |f | > 0: Lu and Wang, 2008, W 1, ∞

loc -regularity;

Lindgren, 2014, linear approximation property and differentiable everywhere when f is Lipschitz. The difficulty to show the regularity comes from the following fact

• bserved by Evans in 1993: For any smooth infinity harmonic

function in Rn, one has a uniform C 1, 1

loc -estimate in terms of uC 0, 1.

Therefore, one cannot approximate every infinity harmonic function via smooth ones.

Yi Zhang A fundamental inequality

Here is the main result of the joint work with H. Koch and Y. Zhou. Theorem 4 Let Ω ⊂ R2 be a domain and u be an ∞-harmonic function in Ω. For each α > 0, we have |Du|α ∈ W 1,2

loc (Ω) and

(|Du|α)iui = 0 almost everywhere in Ω. (2) Moreover, the distributional determinant − det D2udx is a Radon measure satisfying − det D2u ≥ |D|Du||2 where = holds when u ∈ C 2(Ω). Quantitative estimates are also given.

Yi Zhang A fundamental inequality

Remark 5

• 1. The result above is sharp when α → 0, i.e. for the Aronsson

function log |Dw| / ∈ W 1,2

loc (Ω)

• 2. Note that the function u(x) = |x| ∈ W 1,∞

loc (R2) satisfies

|Du|2 ∈ W 1,2

loc (R2) and (2), but is not ∞-harmonic.

• 3. In the plane for p-harmonic functions we have D2u ∈ Lγ

loc for

γ < 3, while for the Aronsson function D2w ∈ Lβ

loc with β < 3 2.

However, |D2wDw|

|Dw|

∈ Lγ

loc .

Yi Zhang A fundamental inequality

Theorem 6 Suppose Ω ⊂ R2 is a bounded domain and f ∈ BV loc (Ω) ∩ C 0(Ω) with |f | > 0 in Ω. Let u ∈ C 0(Ω) be a viscosity solution to −∆∞u = f in Ω. Then we have: (i) For α > 3/2, we have |Du|α ∈ W 1,2

loc (Ω), which is (asymptotic)

sharp when α → 3/2. (ii) For α ∈ (0, 3/2] and p ∈ [1, 3/(3 − α)), we have |Du|α ∈ W 1,p

loc (Ω), which is sharp when p → 3/(3 − α).

(iii) For ǫ > 0, we have |Du|−3+ǫ ∈ L1

loc (Ω), which is sharp when

ǫ → 0. (iv) For α > 0, we have −(|Du|α)iui = 2α|Du|α−2f almost everywhere in Ω. Some quantative bounds are given.

Yi Zhang A fundamental inequality

Remark 7

• 1. The function w(x1, x2) = −x 4/3

1

as a viscosity solution to −∆∞w = 43

34 in R2, clarifies the above sharpness:

|Dw|−3 / ∈ L1

loc (R2);|Dw|α /

∈ W 1,3/(3−α)

loc

(R2) whenever α ∈ (0, 3/2]; and for any p > 2, |Dw|α / ∈ W 1,p

loc (R2) whenever

α ∈ (3/2, 3 − 3/p).

• 2. For the solutions u to the infinity Laplace equations in both of

the cases above, we conjecture that, there exists ǫ > 0 so that |Du|2 ∈ W 1, 2+ǫ

loc

(Ω).

Yi Zhang A fundamental inequality

Here is a sketch proof of the first theorem on the infinity Laplacian.

1)

For ǫ ∈ (0, 1], consider uǫ ∈ C ∞(U ) ∩ C(U ) satisfies −∆∞uǫ − ǫ∆uǫ = 0 in Ω, uǫ = u on ∂Ω. It is known that uǫ → u locally uniformly as ǫ → 0.

Yi Zhang A fundamental inequality

Here is a sketch proof of the first theorem on the infinity Laplacian.

1)

For ǫ ∈ (0, 1], consider uǫ ∈ C ∞(U ) ∩ C(U ) satisfies −∆∞uǫ − ǫ∆uǫ = 0 in Ω, uǫ = u on ∂Ω. It is known that uǫ → u locally uniformly as ǫ → 0.

2)

Observe that, for any smooth function v in the plane, − det D2v = −1 2div(∆vDv − D2vDv) and (− det D2v)|Dv|2 = |D2vDv|2 − ∆v∆∞v.

Yi Zhang A fundamental inequality

Here is a sketch proof of the first theorem on the infinity Laplacian.

1)

For ǫ ∈ (0, 1], consider uǫ ∈ C ∞(U ) ∩ C(U ) satisfies −∆∞uǫ − ǫ∆uǫ = 0 in Ω, uǫ = u on ∂Ω. It is known that uǫ → u locally uniformly as ǫ → 0.

2)

Observe that, for any smooth function v in the plane, − det D2v = −1 2div(∆vDv − D2vDv) and (− det D2v)|Dv|2 = |D2vDv|2 − ∆v∆∞v.

3)

Integral by parts to obtain a uniform Caccioppoli-type inequality for 1

2|Duǫ|2 and D2uǫDuǫ.

Yi Zhang A fundamental inequality

Here is a sketch proof of the first theorem on the infinity Laplacian.

1)

For ǫ ∈ (0, 1], consider uǫ ∈ C ∞(U ) ∩ C(U ) satisfies −∆∞uǫ − ǫ∆uǫ = 0 in Ω, uǫ = u on ∂Ω. It is known that uǫ → u locally uniformly as ǫ → 0.

2)

Observe that, for any smooth function v in the plane, − det D2v = −1 2div(∆vDv − D2vDv) and (− det D2v)|Dv|2 = |D2vDv|2 − ∆v∆∞v.

3)

Integral by parts to obtain a uniform Caccioppoli-type inequality for 1

2|Duǫ|2 and D2uǫDuǫ.

4)

Integral by parts to obtain a uniform L2-flatness estimate (roughly speaking, |Duǫ|2 − Duǫ · DP| is small in the L2-average sense) and conclude uǫ → u strongly in W 1, p

loc for any 1 < p < ∞.

Yi Zhang A fundamental inequality

Here is a sketch proof of the first theorem on the infinity Laplacian.

1)

For ǫ ∈ (0, 1], consider uǫ ∈ C ∞(U ) ∩ C(U ) satisfies −∆∞uǫ − ǫ∆uǫ = 0 in Ω, uǫ = u on ∂Ω. It is known that uǫ → u locally uniformly as ǫ → 0.

2)

Observe that, for any smooth function v in the plane, − det D2v = −1 2div(∆vDv − D2vDv) and (− det D2v)|Dv|2 = |D2vDv|2 − ∆v∆∞v.

3)

Integral by parts to obtain a uniform Caccioppoli-type inequality for 1

2|Duǫ|2 and D2uǫDuǫ.

4)

Integral by parts to obtain a uniform L2-flatness estimate (roughly speaking, |Duǫ|2 − Duǫ · DP| is small in the L2-average sense) and conclude uǫ → u strongly in W 1, p

loc for any 1 < p < ∞.

5)

Let ǫ → 0.

Yi Zhang A fundamental inequality

Yi Zhang A fundamental inequality