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

Artikel

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

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

Veröffentlicht/Copyright: 11. April 2024

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2024 Heft 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.

References

[1] G. Aronsson, Minimization problems for the functional sup x ⁡ F ⁢ ( x , f ⁢ ( x ) , f ′ ⁢ ( x ) ) , Ark. Mat. 6 (1965), 33–53. 10.1007/BF02591326Suche in Google Scholar

[2] G. Aronsson, Minimization problems for the functional sup x ⁡ F ⁢ ( x , f ⁢ ( x ) , f ′ ⁢ ( x ) ) . II, Ark. Mat. 6 (1966), 409–431. 10.1007/BF02590964Suche in Google Scholar

[3] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561. 10.1007/BF02591928Suche in Google Scholar

[4] G. Aronsson, On the partial differential equation u x 2 ⁢ u x ⁢ x + 2 ⁢ u x ⁢ u y ⁢ u x ⁢ y + u y 2 ⁢ u y ⁢ y = 0 , Ark. Mat. 7 (1968), 395–425. 10.1007/BF02590989Suche in Google Scholar

[5] G. Aronsson, Minimization problems for the functional sup x ⁡ F ⁢ ( x , f ⁢ ( x ) , f ′ ⁢ ( x ) ) . III, Ark. Mat. 7 (1969), 509–512. 10.1007/BF02590888Suche in Google Scholar

[6] G. Aronsson, On certain singular solutions of the partial differential equation u x 2 ⁢ u x ⁢ x + 2 ⁢ u x ⁢ u y ⁢ u x ⁢ y + u y 2 ⁢ u y ⁢ y = 0 , Manuscripta Math. 47 (1984), no. 1–3, 133–151. 10.1007/BF01174590Suche in Google Scholar

[7] G. Aronsson, Representation of a 𝑝-harmonic function near a critical point in the plane, Manuscripta Math. 66 (1989), no. 1, 73–95. 10.1007/BF02568483Suche in Google Scholar

[8] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in R n , Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. 10.5186/aasfm.1983.0806Suche in Google Scholar

[9] B. Bojarski and T. Iwaniec, 𝑝-harmonic equation and quasiregular mappings, Partial differential equations (Warsaw 1984), Banach Center Publ. 19, PWN, Warsaw (1987), 25–38. 10.4064/-19-1-25-38Suche in Google Scholar

[10] A. Cianchi and V. G. Maz’ya, Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 569–599. 10.1007/s00205-018-1223-7Suche in Google Scholar

[11] M. G. Crandall and L. C. Evans, A remark on infinity harmonic functions, Proceedings of the USA–Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso 2000), Electron. J. Differ. Equ. Conf. 6, Southwest Texas State University, San Marcos (2001), 123–129. Suche in Google Scholar

[12] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123–139. 10.1007/s005260000065Suche in Google Scholar

[13] E. DiBenedetto, C 1 + α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. 10.1016/0362-546X(83)90061-5Suche in Google Scholar

[14] H. Dong, F. Peng, Y. R.-Y. Zhang and Y. Zhou, Hessian estimates for equations involving 𝑝-Laplacian via a fundamental inequality, Adv. Math. 370 (2020), Article ID 107212. 10.1016/j.aim.2020.107212Suche in Google Scholar

[15] L. D’Onofrio, F. Giannetti, T. Iwaniec, J. Manfredi and T. Radice, Divergence forms of the ∞-Laplacian, Publ. Mat. 50 (2006), no. 1, 229–248. 10.5565/PUBLMAT_50106_13Suche in Google Scholar

[16] L. C. Evans, A new proof of local C 1 , α regularity for solutions of certain degenerate elliptic p.d.e., J. Differential Equations 45 (1982), no. 3, 356–373. 10.1016/0022-0396(82)90033-XSuche in Google Scholar

[17] L. C. Evans, Estimates for smooth absolutely minimizing Lipschitz extensions, Electron. J. Differential Equations 1993 (1993), Paper No. 3. Suche in Google Scholar

[18] L. C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159–177. 10.1007/s00526-002-0164-ySuche in Google Scholar

[19] L. C. Evans and O. Savin, C 1 , α regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations 32 (2008), no. 3, 325–347. 10.1007/s00526-007-0143-4Suche in Google Scholar

[20] L. C. Evans and C. K. Smart, Adjoint methods for the infinity Laplacian partial differential equation, Arch. Ration. Mech. Anal. 201 (2011), no. 1, 87–113. 10.1007/s00205-011-0399-xSuche in Google Scholar

[21] L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations 42 (2011), no. 1–2, 289–299. 10.1007/s00526-010-0388-1Suche in Google Scholar

[22] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Math. ETH Zürich, Birkhäuser, Basel 1993. Suche in Google Scholar

[23] T. Iwaniec and J. J. Manfredi, Regularity of 𝑝-harmonic functions on the plane, Rev. Mat. Iberoam. 5 (1989), no. 1–2, 1–19. 10.4171/rmi/82Suche in Google Scholar

[24] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal. 123 (1993), no. 1, 51–74. 10.1007/BF00386368Suche in Google Scholar

[25] H. Koch, Y. R.-Y. Zhang and Y. Zhou, An asymptotic sharp Sobolev regularity for planar infinity harmonic functions, J. Math. Pures Appl. (9) 132 (2019), 457–482. 10.1016/j.matpur.2019.02.008Suche in Google Scholar

[26] J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), no. 6, 849–858. 10.1512/iumj.1983.32.32058Suche in Google Scholar

[27] E. Lindgren and P. Lindqvist, The gradient flow of infinity-harmonic potentials, Adv. Math. 378 (2021), Article ID 107526. 10.1016/j.aim.2020.107526Suche in Google Scholar

[28] P. Lindqvist and J. J. Manfredi, The Harnack inequality for ∞-harmonic functions, Electron. J. Differential Equations 1995 (1995), Paper No. 4. Suche in Google Scholar

[29] J. J. Manfredi, 𝑝-harmonic functions in the plane, Proc. Amer. Math. Soc. 103 (1988), no. 2, 473–479. 10.1090/S0002-9939-1988-0943069-2Suche in Google Scholar

[30] J. J. Manfredi and A. Weitsman, On the Fatou theorem for 𝑝-harmonic functions, Comm. Partial Differential Equations 13 (1988), no. 6, 651–668. 10.1080/03605308808820556Suche in Google Scholar

[31] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. 10.1090/S0894-0347-08-00606-1Suche in Google Scholar

[32] Y. Peres and S. Sheffield, Tug-of-war with noise: A game-theoretic view of the 𝑝-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. 10.1215/00127094-2008-048Suche in Google Scholar

[33] S. Sarsa, Note on an elementary inequality and its application to the regularity of 𝑝-harmonic functions, Ann. Fenn. Math. 47 (2022), no. 1, 139–153. 10.54330/afm.112699Suche in Google Scholar

[34] O. Savin, C 1 regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal. 176 (2005), no. 3, 351–361. 10.1007/s00205-005-0355-8Suche in Google Scholar

[35] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. 10.1016/0022-0396(84)90105-0Suche in Google Scholar

[36] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3–4, 219–240. 10.1007/BF02392316Suche in Google Scholar

[37] N. N. Uralceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222. Suche in Google Scholar

Received: 2023-01-20

Revised: 2024-03-01

Published Online: 2024-04-11

Published in Print: 2024-07-01

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/crelle-2024-0016