Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

Hongjie Dong; Fa Peng; Yi Ru-Ya Zhang; Yuan Zhou

doi:10.1515/crelle-2024-0016

Article

Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

Hongjie Dong , Fa Peng , Yi Ru-Ya Zhang and Yuan Zhou

Published/Copyright: April 11, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 812

Abstract

We introduce a distributional Jacobian determinant det D ⁢ V β ⁢ ( D ⁢ v ) in dimension two for the nonlinear complex gradient V β ⁢ ( D ⁢ v ) = | D ⁢ v | β ⁢ ( v x 1 , − v x 2 ) for any β > − 1 , whenever v ∈ W loc 1 , 2 and β ⁢ | D ⁢ v | 1 + β ∈ W loc 1 , 2 . This is new when β ≠ 0 . Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant det D ⁢ V β ⁢ ( D ⁢ u ) is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications.

Applying this result with β = 0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 𝑢 is an ∞-harmonic function in the whole R 2 with
lim inf R → ∞ inf c ∈ R 1 R ⁢ ⨍ B ⁢ ( 0 , R ) | u ⁢ ( x ) − c | ⁢ d x < ∞ ,
then u = b + a ⋅ x for some b ∈ R and a ∈ R 2 .
Denoting by u p the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that det D ⁢ V β ⁢ ( D ⁢ u p ) → det D ⁢ V β ⁢ ( D ⁢ u ) as p → ∞ in the weak-⋆ sense in the space of Radon measure. Recall that V β ⁢ ( D ⁢ u p ) is always quasiregular mappings, but V β ⁢ ( D ⁢ u ) is not in general.

Funding source: Simons Foundation

Award Identifier / Grant number: 709545

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2055244

Funding source: China Postdoctoral Science Foundation

Award Identifier / Grant number: BX20220328

Funding source: Chinese Academy of Sciences

Award Identifier / Grant number: 11688101

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871088

Award Identifier / Grant number: 12025102

Funding statement: H. Dong was partially supported by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244. F. Peng was supported by China Postdoctoral Science Foundation funded project (No. BX20220328). Y. Zhang was supported by the Chinese Academy of Science and NSFC grant No. 11688101. Y. Zhou was supported by NSFC (No. 11871088 & No. 12025102) and by the Fundamental Research Funds for the Central Universities.

A Some sharpness in the plane

At the borderline case β = − 1 , we have the following result, which will be used later.

Lemma A.1

Let 1 < p < ∞ . If u p is a nonconstant 𝑝-harmonic function in a domain Ω ⊂ R 2 , then

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 = | D ⁢ log ⁡ | D ⁢ u p | | 2 + ( p − 2 ) ⁢ p ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 4 a.e.

In particular,

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 = | D ⁢ log ⁡ | D ⁢ u p | | 2 ⁢ a.e. if ⁢ p = 2 ;

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 ≥ | D ⁢ log ⁡ | D ⁢ u p | | 2 ⁢ a.e. if ⁢ p > 2 ;

| D ⁢ log ⁡ | D ⁢ u p | | 2 ≥ | D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 ≥ ( p − 1 ) 2 ⁢ | D ⁢ log ⁡ | D ⁢ u p | | 2 ⁢ a.e. if ⁢ 1 < p < 2 .

Proof

In Ω ∖ E u p , one has

| D ⁢ log ⁡ | D ⁢ u p | | 2 = | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 4 ≤ | D 2 ⁢ u p | 2 | D ⁢ u p | 2

and

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 = | | D ⁢ u p | − 1 ⁢ D 2 ⁢ u p − | D ⁢ u p | − 3 ⁢ D 2 ⁢ u p ⁢ D ⁢ u p ⊗ D ⁢ u p | 2 = | D 2 ⁢ u p | 2 | D ⁢ u p | 2 − | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 4 .

Recall that

D 2 ⁢ u p ⁢ D ⁢ u p 2 − Δ ⁢ u p ⁢ Δ ∞ ⁢ u p = 1 2 ⁢ [ | D 2 ⁢ u p | 2 − ( Δ ⁢ u p ) 2 ] ⁢ | D ⁢ u p | 2 in ⁢ Ω ∖ E u p .

Replacing Δ ⁢ u p with ( p − 2 ) ⁢ Δ ∞ ⁢ u p | D ⁢ u p | 2 , one has

1 2 ⁢ [ | D 2 ⁢ u p | 2 − ( p − 2 ) 2 ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 4 ] = | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 2 − ( p − 2 ) ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 4 ,

or equivalently,

| D 2 ⁢ u p | 2 = 2 ⁢ | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 2 + ( p − 2 ) ⁢ p ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 4 .

Thus,

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 = | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 4 + ( p − 2 ) ⁢ p ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 6 = 1 2 ⁢ | D 2 ⁢ u p | 2 | D ⁢ u p | 2 + ( p − 2 ) ⁢ p 2 ⁢ ( Δ ∞ ⁢ u p ) 2 | D ⁢ u p | 6 .

When p = 2 , then

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 = 1 2 ⁢ | D 2 ⁢ u p | 2 | D ⁢ u p | 2 = | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 4 in ⁢ Ω ∖ E u ;

When p > 2 , we have

| D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 ≥ 1 2 ⁢ | D 2 ⁢ u p | 2 | D ⁢ u p | 2 ≥ | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 2 ,

while when 1 < p < 2 , we have

1 2 ⁢ | D 2 ⁢ u p | 2 | D ⁢ u p | 2 ≥ | D ⁢ [ | D ⁢ u p | − 1 ⁢ D ⁢ u p ] | 2 ≥ ( p − 1 ) 2 2 ⁢ | D 2 ⁢ u p | 2 | D ⁢ u p | 2 ≥ ( p − 1 ) 2 2 ⁢ | D 2 ⁢ u p ⁢ D ⁢ u p | 2 | D ⁢ u p | 4 . ∎

For 1 < p < ∞ , we recall the extremal 𝑝-harmonic function constructed by [23, Section 7]. Here we keep notation the same as therein. Let

(A.1) H ⁢ ( ξ ) = ( ξ | ξ | + ε ⁢ | ξ | 3 ξ 3 ) ⁢ | ξ | 1 d for all ⁢ ξ ∈ C

with

1 d = 1 2 ⁢ ( − p + 16 ⁢ ( p − 1 ) + ( p − 2 ) 2 ) > 0 and ε = 1 − d 1 + 3 ⁢ d .

If p = 2 , then d = 1 and ε = 0 , and hence H ⁢ ( ξ ) = ξ . If p ≠ 2 , then d > 0 and ε ≠ 0 , and 𝐻 is a quasiconformal homeomorphism on the whole plane. According to [23, Theorem 2], H ⁢ ( ξ ) satisfies [23, (18) with n = 1 ], that is,

(A.2) H ξ ̄ = ( 1 2 − 1 p ) ⁢ [ ξ ξ ̄ ⁢ H ξ + ξ ̄ ξ ⁢ H ξ ̄ ] ,

where H ξ = 1 2 ⁢ ( H x − i ⁢ H y ) and H ξ ̄ = 1 2 ⁢ ( H x + i ⁢ H y ) for ξ = x + i ⁢ y .

Let f ⁢ ( z ) denote the inverse of H ⁢ ( ξ ) in ℂ so that f ⁢ ( H ⁢ ( ξ ) ) = ξ and H ⁢ ( f ⁢ ( z ) ) = z for all z , ξ ∈ C . From (A.2), one deduces

f z ̄ = ( 1 p − 1 2 ) ⁢ [ f f ̄ ⁢ f z ̄ + f ̄ f ⁢ f z ] ,

where f z = 1 2 ⁢ ( f x − i ⁢ f y ) and f z ̄ = 1 2 ⁢ ( f x + i ⁢ f y ) for z = x + i ⁢ y . This then defines a 𝑝-harmonic function 𝑤 in the whole plane so that its complex derivative is w z = f .

We have the following properties.

Lemma A.2

One has log ⁡ | D ⁢ w | = log ⁡ | f | ∉ W loc 1 , 2 ⁢ ( R 2 ) and | D ⁢ w | − 1 ⁢ D ⁢ w ∉ W loc 1 , 2 ⁢ ( R 2 ) .

Proof

By Lemma A.1, we only need to prove log ⁡ | f | ∉ W loc 1 , 2 ⁢ ( R 2 ) . We argue by contradiction. Assume that log ⁡ | f | ∈ W loc 1 , 2 ⁢ ( C ) . Note that f ⁢ ( t ⁢ z ) = t d ⁢ f ⁢ ( z ) for any t ≥ 0 . A direct calculation implies that

(A.3) D ⁢ | f | ⁢ ( z ) = t 1 − d ⁢ D ⁢ | f | ⁢ ( t ⁢ z ) , D ⁢ log ⁡ | f | ⁢ ( z ) = D ⁢ | f | ⁢ ( z ) | f ⁢ ( z ) | , z ∈ C ∖ { f − 1 ⁢ ( 0 ) } , t ≥ 0 ,

where f − 1 ⁢ ( 0 ) = { z ∈ C : f ⁢ ( z ) = 0 } . For each R > 0 , we know that

f − 1 ⁢ ( 0 ) ∩ { z ∈ C : | z | < R }

is discrete, and from f ⁢ ( t ⁢ z ) = t d ⁢ f ⁢ ( z ) , (A.3), we conclude that

∫ | z | < R | D ⁢ log ⁡ | f | ⁢ ( z ) | 2 ⁢ d z = ∫ | z | < R | D ⁢ | f | ⁢ ( z ) | 2 | f ⁢ ( z ) | 2 ⁢ d z = ∫ | ξ | < t ⁢ R | D ⁢ | f | ⁢ ( ξ t ) | 2 | f ⁢ ( ξ t ) | 2 ⁢ d ξ t = ∫ | ξ | < t ⁢ R ( 1 t ) 2 ⁢ d − 2 ⁢ | D ⁢ | f | ⁢ ( ξ ) | 2 ( 1 t ) 2 ⁢ d ⁢ | f ⁢ ( ξ ) | 2 ⁢ 1 t 2 ⁢ d ξ = ∫ | ξ | < t ⁢ R | D ⁢ log ⁡ | f | ⁢ ( ξ ) | 2 ⁢ d ξ for all ⁢ t > 0 .

Letting t → 0 , we conclude that D ⁢ log ⁡ | f | ⁢ ( z ) = 0 whenever | z | < R , and hence, by the arbitrariness of 𝑅, for all z ∈ C . Thus, | f | is a positive constant in the whole plane. This contradicts that f ⁢ ( t ⁢ z ) = t d ⁢ f ⁢ ( z ) for all t > 0 and z ∈ C , where we recall that d > 0 . ∎

Lemma A.3

One has

(A.4) sup C ∖ { 0 } | f z ̄ | | f z | = | p − 2 | p = K ⁢ ( p ) − 1 K ⁢ ( p ) + 1 with ⁢ K ⁢ ( p ) = max ⁡ { 1 p − 1 , p − 1 } .

In general, for β > − 1 , writing g = | f | β ⁢ f , one has

(A.5) sup C ∖ { 0 } | g z ̄ | | g z | = K ⁢ ( p , β ) − 1 K ⁢ ( p , β ) + 1 with ⁢ K ⁢ ( p , β ) = max ⁡ { p − 1 β + 1 , β + 1 p − 1 , β + 1 , 1 β + 1 } .

Proof

Since H ⁢ ( ξ ) is the inverse of f ⁢ ( z ) , (A.4) is equivalent to

sup C ∖ { 0 } | H ξ ̄ | | H ξ | = | p − 2 | p .

We already have

| H ξ ̄ | = 1 2 ⁢ | ξ ξ ̄ ⁢ H ξ + ξ ̄ ξ ⁢ H ̄ ξ | ≤ | p − 2 | p ⁢ | H ξ | in ⁢ C ∖ { 0 } .

Taking the derivative ∂ ξ on both sides of (A.1), one has

H ξ = 1 2 ⁢ | ξ | 1 d − 1 ⁢ [ ( 1 d + 1 ) + ( 1 d − 3 ) ⁢ ε ⁢ | ξ | 4 ξ 4 ] .

If ξ ∈ R , then H ξ ⁢ ( ξ ) ∈ R , and hence (A.1) gives H ξ ̄ = p − 2 p ⁢ H ξ as desired.

Next, for β > − 1 , write g = | f | β ⁢ f and G = g − 1 . By [9, Section 3], one has

g z ̄ = − 1 2 ⁢ ( p − 2 − β p + β + β β + 2 ) ⁢ g ̄ g ⁢ g z − 1 2 ⁢ ( p − 2 − β p + β − β β + 2 ) ⁢ g g ̄ ⁢ g ̄ z

and hence

(A.6) G ξ ̄ = 1 2 ⁢ ( p − 2 − β p + β + β β + 2 ) ⁢ ξ ̄ ξ ⁢ G ξ + 1 2 ⁢ ( p − 2 − β p + β − β β + 2 ) ⁢ ξ ξ ̄ ⁢ G ̄ ξ .

Thus,

sup C ∖ { 0 } | G ξ ̄ | | G ξ | ≤ max ⁡ { | p − 2 − β | p + β , | β | β + 2 } = K ⁢ ( p , β ) − 1 K ⁢ ( p , β ) + 1

with K ⁢ ( p , β ) as in (A.5). Moreover, note that

G ⁢ ( ξ ) = H ⁢ ( | ξ | − β β + 1 ⁢ ξ ) = ( ξ | ξ | + ε ⁢ | ξ | 3 ξ 3 ) ⁢ | ξ | − 1 ( β + 1 ) ⁢ d for all ⁢ ξ ∈ C ,

and hence

G ξ ⁢ ( ξ ) = 1 2 ⁢ | ξ | 1 ( β + 1 ) ⁢ d − 1 ⁢ [ ( 1 ( β + 1 ) ⁢ d + 1 ) + ( 1 ( β + 1 ) ⁢ d − 3 ) ⁢ ε ⁢ | ξ | 4 ξ 4 ] .

| p − 2 − β | p + β ≥ | β | β + 2 ,

for ξ ∈ R , we have G ξ ⁢ ( ξ ) ∈ R , and therefore, (A.6) gives

G ξ ̄ ⁢ ( ξ ) = p − 2 − β p + β ⁢ G ξ ⁢ ( ξ ) ,

as desired. If

| p − 2 − β | p + β < | β | β + 2 ,

for ξ ∈ R − i ⁢ R , we have ξ ̄ = i ⁢ ξ , G ξ ⁢ ( ξ ) ∈ R , and

ξ ξ ̄ ⁢ G ξ ⁢ ( ξ ) = − ξ ̄ ξ ⁢ G ̄ ξ ⁢ ( ξ ) = − i ⁢ G ξ ⁢ ( ξ ) ,

which together with (A.6) gives G ξ ̄ ⁢ ( ξ ) = β 2 + β ⁢ i ⁢ G ξ ⁢ ( ξ ) , as desired. ∎

Lemma A.3 gives the sharpness of constants in (1.7).

Remark A.4

By a standard calculation, (A.4) gives

ess ⁢ sup C ⁡ 2 ⁢ [ | f z | 2 + | f z ̄ | 2 ] | f z | 2 − | f z ̄ | 2 = ( p − 1 ) 2 + 1 p − 1 .

Since | D 2 ⁢ w | 2 = 2 ⁢ [ | f z | 2 + | f z ̄ | 2 ] and − det D 2 ⁢ w = | f z | 2 − | f z ̄ | 2 , we write this as

ess ⁢ sup C ⁡ | D 2 ⁢ w | 2 − det D 2 ⁢ w = ( p − 1 ) 2 + 1 p − 1 = ( p − 1 ) + 1 p − 1 .

Thus, the constant in (1.7) is sharp. Note that ( p − 1 ) + 1 p − 1 converges to ∞ as p → ∞ .

For β > − 1 , in a similar way, (A.5) gives

ess ⁢ sup ⁡ | D ⁢ [ | D ⁢ w | β ⁢ D ⁢ w ] | 2 − det D ⁢ [ | D ⁢ w | β ⁢ D ⁢ w ] = K ⁢ ( p , β ) 2 + 1 K ⁢ ( p , β ) = K ⁢ ( p , β ) + 1 K ⁢ ( p , β ) ,

and hence the constant in (1.7) is sharp. We also note that K ⁢ ( p , β ) + 1 K ⁢ ( p , β ) converges to ∞ as p → ∞ .

Acknowledgements

Y. Zhou would like to thank Professor Juan J. Manfredi and Professor Nageswari Shanmugalingam for their kind suggestions and comments on the best constants as in Lemma A.3 and Remark A.4.

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Received: 2023-01-20

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Published Online: 2024-04-11

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