Equivalence between distributional and viscosity solutions for the double-phase equation

Yuzhou Fang; Chao Zhang

doi:10.1515/acv-2020-0059

Open Access Published by De Gruyter October 17, 2020

Equivalence between distributional and viscosity solutions for the double-phase equation

Yuzhou Fang and Chao Zhang

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2020-0059

Abstract

We investigate the different notions of solutions to the double-phase equation

- div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 ,

which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the 𝒜 H ⁢ ( ⋅ ) -harmonic functions of nonlinear potential theory and then show that 𝒜 H ⁢ ( ⋅ ) -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

Keywords: Equivalence; double-phase equation; viscosity solution; distributional solution; comparison principle

MSC 2010: 35J92; 35D40; 35C45; 46E30

1 Introduction

Let Ω be a bounded domain in ℝ n , n ≥ 2 . In this work, we are concerned with the relationship of distributional and viscosity solutions to the double-phase problem

(1.1) - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 in ⁢ Ω

with some appropriate hypotheses, where 1 < p ≤ q < ∞ and a ⁢ ( x ) ≥ 0 . It is well known that equation (1.1) emerges naturally as the Euler–Lagrange equation of the functional

𝒫 ( u , Ω ) : = ∫ Ω ( 1 p | D u | p + a ⁢ ( x ) q | D u | q ) d x ,

which was first introduced by Zhikov [43, 46]. The functional 𝒫 can provide a model for characterizing the features of strongly anisotropic materials. To be precise, considering two diverse materials with power hardening exponents p and q, the coefficient a ⁢ ( ⋅ ) determines the geometry of the composite composed of the two, relying on the fact that x belongs to the set { a ⁢ ( x ) = 0 } or not. Also the functional 𝒫 gives new examples concerning the Lavrentiev phenomenon in [44, 45].

Over the last years, these functionals with non-standard growth conditions

W 1 , 1 ⁢ ( Ω ) ∋ u ↦ ∫ Ω F ⁢ ( x , u , D ⁢ u ) ⁢ d ⁢ x , ν ⁢ | z | p ≤ F ⁢ ( x , u , z ) ≤ L ⁢ ( | z | q + 1 )

have been a surge of interest. For the case of an autonomous energy density of the type F ⁢ ( x , u , D ⁢ u ) ≡ F ⁢ ( D ⁢ u ) , the regularity theory of minima of such functionals is by now well-understood from the seminal papers of Marcellini [36, 37, 38]. The study of nonautonomous functionals, especially for the double-phase problem (1.1), has been continued in a series of nice papers by Mingione et al. For example, the C 1 , α -regularity for local minimizers of 𝒫 was derived by Colombo and Mingione [13]. It is shown that, under the assumptions that

(1.2) 0 ≤ a ⁢ ( ⋅ ) ∈ C 0 , α ⁢ ( Ω ) , α ∈ ( 0 , 1 ] and q p < 1 + α n ,

the minimizers of 𝒫 belong to the class C loc 1 , β ⁢ ( Ω ) for some β ∈ ( 0 , 1 ) . Baroni, Colombo and Mingione [4] completed the regularity results for the borderline case q p = 1 + α n . Furthermore, a Harnack inequality for minimizers of 𝒫 was established by the same authors in [2]. If the minimizers of 𝒫 are bounded, the assumption imposed on p , q in [12] was relaxed to q ≤ p + α such that the C loc 1 , β ⁢ ( Ω ) regularity holds. For the inhomogeneous double-phase equation

- div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = - div ⁡ ( | F | p - 2 ⁢ F + a ⁢ ( x ) ⁢ | F | q - 2 ⁢ F ) in ⁢ Ω ,

Colombo and Mingione [14] proved the Calderón–Zygmund type estimate

( | F | p + a ⁢ ( x ) ⁢ | F | q ) ∈ L loc γ ⁢ ( Ω ) ⟹ ( | D ⁢ u | p + a ⁢ ( x ) ⁢ | D ⁢ u | q ) ∈ L loc γ ⁢ ( Ω ) , γ > 1

under assumption (1.2), and the estimate above was improved by De Fillippis and Mingione [18] for the borderline case q p = 1 + α n . Byun, Cho and Oh [8] obtained the Calderón–Zygmund estimates regarding a class of irregular obstacle problems with non-uniformly elliptic operator in divergence form of ( p , q ) -growth. For more results, see for instance [6, 3, 21, 9] and the references therein.

From the mentioned works above, we can see that rather abundant research results have been derived for double-phase problems. However, there are few results concerning the viscosity solutions for such kind of equations, except papers [16, 17] in the fully nonlinear setting and [20] in the nonlocal fractional framework concerned with the regularity theory for viscosity solutions. To this end, our interest in this work focuses on the different notions of solutions to equation (1.1). We define naturally the distributional solutions to (1.1) based on integration by parts, owing to the divergence form of this equation. At the same time, if the coefficient a ⁢ ( x ) is of class C 1 ⁢ ( Ω ) , then the notion of viscosity solutions is also applicable, which is defined according to pointwise touching test functions. Our aim is to show that the distributional solutions coincide with the viscosity solutions of (1.1). In fact, the equivalence of different notions of solutions is a much-investigated topic, which was first studied by Ishii [28] in the linear case. When it comes to the quasilinear case, employing the full uniqueness machinery of the theory of viscosity solutions, Juutinen, Lindqvist and Manfredi [30] verified the equivalence between distributional and viscosity solutions for the p-Laplace equation - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u ) = 0 in Ω, which is the special version of double-phase equation (1.1) in the case that a ⁢ ( x ) ≡ 0 . Moreover, the equivalence of two different solutions of p ⁢ ( x ) -Laplace equation - div ⁡ ( | D ⁢ u | p ⁢ ( x ) - 2 ⁢ D ⁢ u ) = 0 in Ω was obtained by Juutinen, Lukkari and Parviainen [31]. On the other hand, it is worth mentioning that a shorter proof for the equivalence of distributional and viscosity solutions for the p-Laplace equation was recently given in [29] by virtue of a technical regularization procedure via infimal convolutions. Exploiting the technique developed in [29], the equivalence of two different solutions of various equations has been obtained, for which one can refer to [39, 42] and references therein. For more results about viscosity solutions of equations of various nature featuring certain structures belonging to the Musielak–Orlicz class, see [7, 27, 32] and references therein.

In this paper, we revisit the methods developed by Juutinen, Lindqvist and Manfredi [30] relying on the uniqueness of solutions to show that the distributional and viscosity solutions of (1.1) coincide. The novelties of this paper are as follows. First, inspired by the known theory for p-harmonic functions and p ⁢ ( x ) -harmonic functions (see [22, 23, 24, 25, 33, 34]) which plays an important role in nonlinear potential theory, we introduce the notion of 𝒜 H ⁢ ( ⋅ ) -harmonic function for the double-phase equation and then show that distributional solutions coincide with 𝒜 H ⁢ ( ⋅ ) -harmonic functions. Second, the known results on double-phase problems are mainly from the variational point of view. As far as we know, there are not many studies in terms of the viscosity solutions. We derive the equivalence of two notions of solutions of double-phase problem (1.1) through justifying the distributional and viscosity solutions are the same as 𝒜 H ⁢ ( ⋅ ) -harmonic functions, respectively. Although we follow the ideas in [30] when investigating the relationship between 𝒜 H ⁢ ( ⋅ ) -harmonic functions and viscosity solutions, the double-phase problem is not translation invariant and possesses two diverse growth terms, which generates more delicate difficulties than equations of p-Laplace type. Our proof relies heavily on the well-established theory of distributional solutions to equation (1.1), such as the existence, uniqueness and regularity properties.

This paper is organized as follows. In Section 2, we first state some basic properties of function spaces and notions of solutions. Then we give some auxiliary results that will be used later. Section 3 is devoted to showing that distributional supersolutions and 𝒜 H ⁢ ( ⋅ ) -superharmonic functions are the same, and the equivalence between 𝒜 H ⁢ ( ⋅ ) -superharmonic functions and viscosity supersolutions is proved in Section 4. In Section 5, we verify the comparison principle for viscosity solutions, which is the indispensable element of the proof.

2 Preliminaries

In this section, we summarize some basic properties of the Musielak–Orlicz–Sobolev space W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) . In addition, we give the definition of 𝒜 H ⁢ ( ⋅ ) -harmonic functions, different notions of solutions to equation (1.1) together with some auxiliary results.

2.1 Function spaces

In the rest of this paper, unless otherwise stated, we assume that

(2.1) 0 ≤ a ⁢ ( ⋅ ) ∈ C 0 , α ⁢ ( Ω ) , α ∈ ( 0 , 1 ] and q p ≤ 1 + α n .

For all x ∈ Ω and ξ ∈ ℝ n , we shall use the notation H ( x , ξ ) : = | ξ | p + a ( x ) | ξ | q . With abuse of notation, we shall also denote H ⁢ ( x , ξ ) when ξ ∈ ℝ . Observe that H : Ω × [ 0 , + ∞ ) → [ 0 , + ∞ ) is a Musielak–Orlicz function that satisfies ( Δ 2 ) and ( ∇ 2 ) conditions (see [5, 40, 41]).

The Musielak–Orlicz space L H ⁢ ( ⋅ ) ⁢ ( Ω ) is defined as

L H ⁢ ( ⋅ ) ( Ω ) : = { u : Ω → ℝ measurable : ϱ H ( u ) < ∞ } ,

endowed with the norm

∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) : = inf { λ > 0 : ϱ H ( u λ ) ≤ 1 } ,

where

ϱ H ( u ) : = ∫ Ω H ( x , u ) d x = ∫ Ω | u | p + a ( x ) | u | q d x

is called ϱ H -modular.

The space L H ⁢ ( ⋅ ) ⁢ ( Ω ) is a separable, uniformly convex Banach space. From the definitions of ϱ H -modular and norm, we can find

(2.2) min ⁡ { ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) p , ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q } ≤ ϱ H ⁢ ( u ) ≤ max ⁡ { ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) p , ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q } .

It follows from (2.2) that

∥ u n - u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) → 0 ⇔ ϱ H ⁢ ( u n - u ) → 0 ,

which indicates the equivalence of convergence in ϱ H -modular and in norm.

The Musielak–Orlicz–Sobolev space W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) is the set of those functions u ∈ L H ⁢ ( ⋅ ) ⁢ ( Ω ) satisfying their distributional gradients D ⁢ u ∈ L H ⁢ ( ⋅ ) ⁢ ( Ω ) . We endow the space W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) with the norm

∥ u ∥ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) : = ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) .

The space W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) is a separable and reflexible Banach space. The local space W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) is composed of those functions belonging to W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ′ ) for any subdomain Ω ′ compactly involved in Ω. Finally, we denote by W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) the closure of C 0 ∞ ⁢ ( Ω ) in W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) . Indeed, conditions (2.1) ensure that the set C 0 ∞ ⁢ ( Ω ) is dense in W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) (see [21, 1]).

The following embedding theorem can be found in [11].

Lemma 2.1.

Let p * = n ⁢ p n - p if n < p and p * = ∞ otherwise.

L H ⁢ ( ⋅ ) ⁢ ( Ω ) ↪ L r ⁢ ( Ω ) and W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) ↪ W 0 1 , r ⁢ ( Ω ) for all r ∈ [ 1 , p ] .
If n ≠ p , then W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) ↪ L r ⁢ ( Ω ) for all r ∈ [ 1 , p * ] ; if n = p , then W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) ↪ L r ⁢ ( Ω ) for all r ∈ [ 1 , ∞ ) .
If p ≤ n , then W 0 1 , H ⁢ ( ⋅ ) ( Ω ) ↪ ↪ L r ( Ω ) for all r ∈ [ 1 , p * ) ; if p > n , then W 0 1 , H ⁢ ( ⋅ ) ( Ω ) ↪ ↪ L ∞ ( Ω ) .

Now we give a Poincaré type inequality coming from [11, Proposition 2.18].

Lemma 2.2.

If u ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , then there is a constant C > 0 , depending on n , p , diam ⁡ ( Ω ) (independent of u), such that ∥ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) ≤ C ⁢ ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) .

For more details about the space W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , we refer the readers to [10, 11, 35].

2.2 Notions of solutions

Define A ( x , ξ ) : = | ξ | p - 2 ξ + a ( x ) | ξ | q - 2 ξ for all x ∈ Ω and ξ ∈ ℝ n . Now we state the diverse type of solutions to (1.1) and the definition of 𝒜 H ⁢ ( ⋅ ) -superharmonic ( 𝒜 H ⁢ ( ⋅ ) -subharmonic) functions.

Definition 2.3 (Distributional solution).

A function u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) is called a distributional supersolution to (1.1) if ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ η 〉 ⁢ d ⁢ x ≥ 0 for every nonnegative function η ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) . The inequality is converse for distributional subsolution. We say that u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) is a distributional solution of (1.1) if and only if u is both super- and subsolution, that is ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ η 〉 ⁢ d ⁢ x = 0 for each η ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) .

Definition 2.4 ( A H ⁢ ( ⋅ ) -harmonic function).

We say that u : Ω → ( - ∞ , ∞ ] is an 𝒜 H ⁢ ( ⋅ ) -superharmonic function in Ω if

u is lower semicontinuous in Ω,
u is finite a.e. in Ω,
for any subdomain D ⊂⊂ Ω , the comparison principle holds: when h ∈ C ⁢ ( D ¯ ) is a distributional solution to (1.1), and u ≥ h on ∂ ⁡ D , then u ≥ h in D.

If - u is 𝒜 H ⁢ ( ⋅ ) -superharmonic, then u : Ω → [ - ∞ , ∞ ) is called 𝒜 H ⁢ ( ⋅ ) -subharmonic function. 𝒜 H ⁢ ( ⋅ ) -harmonic function means that it is both 𝒜 H ⁢ ( ⋅ ) -superharmonic and 𝒜 H ⁢ ( ⋅ ) -subharmonic.

Definition 2.5 (Viscosity solution).

A lower semicontinuous function u : Ω → ( - ∞ , ∞ ] is a viscosity supersolution of (1.1) in Ω if u is finite a.e. in Ω and for each φ ∈ C 2 ⁢ ( Ω ) such that

(2.3) { φ ⁢ ( x 0 ) = u ⁢ ( x 0 ) , x 0 ∈ Ω , φ ⁢ ( x ) < u ⁢ ( x ) , x ≠ x 0 , D ⁢ φ ⁢ ( x 0 ) ≠ 0 ,

there holds - div ⁡ A ⁢ ( x 0 , D ⁢ φ ⁢ ( x 0 ) ) ≥ 0 . A function u is a viscosity subsolution when - u is a viscosity supersolution of (1.1). A function u is called viscosity solution to (1.1) if and only if it is viscosity super- and subsolution.

Remark 2.6.

With condition (2.3), we say that the test function φ touches u from below at point x 0 ∈ Ω . In the case that 2 ≤ p ≤ q < ∞ , the equation is pointwise well-defined, so the requirement D ⁢ φ ⁢ ( x 0 ) ≠ 0 in (2.3) can be eliminated. At this time, - div ⁡ A ⁢ ( x 0 , D ⁢ φ ⁢ ( x 0 ) ) = 0 .

2.3 Auxiliary results

Now we first provide a comparison principle for distributional solutions. Set f + : = max { f , 0 } .

Proposition 2.7 (Comparison principle).

Let u , v ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) be such that ( u - v ) + ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) . If

∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ φ 〉 ⁢ d ⁢ x ≤ ∫ Ω 〈 A ⁢ ( x , D ⁢ v ) , D ⁢ φ 〉 ⁢ d ⁢ x

for each nonnegative φ ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , then u ≤ v a.e. in Ω.

Proof.

Take φ : = ( u - v ) + as a test function to obtain ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ v ) , D ⁢ ( u - v ) + 〉 ⁢ d ⁢ x ≤ 0 . After straightforward computation, it follows that u ≤ v a.e. in Ω. ∎

Proposition 2.8 (Caccioppoli type inequality).

Let u be a distributional solution to equation (1.1). Then, for any ζ ∈ C 0 ∞ ⁢ ( Ω ) with 0 ≤ ζ ≤ 1 , there holds

∫ Ω ζ q ⁢ H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ ∫ Ω H ⁢ ( x , u ⋅ D ⁢ ζ ) ⁢ d ⁢ x .

Proof.

Choosing η : = ζ q u as a test function in the weak form of equation (1.1) yields 0 = ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ η 〉 ⁢ d ⁢ x . We further get by Young’s inequality with ε that

∫ Ω ζ q ⁢ H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x ≤ ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , q ⁢ ζ q - 1 ⁢ u ⋅ D ⁢ ζ 〉 ⁢ d ⁢ x ≤ ε ⁢ ∫ Ω ζ ( q - 1 ) ⁢ p p - 1 ⁢ | D ⁢ u | p ⁢ d ⁢ x + C ⁢ ( ε ) ⁢ ∫ Ω q p ⁢ | u | p ⁢ | D ⁢ ζ | p ⁢ d ⁢ x + ε ⁢ ∫ Ω a ⁢ ( x ) ⁢ ζ q ⁢ | D ⁢ u | q ⁢ d ⁢ x + C ⁢ ( ε ) ⁢ ∫ Ω q q ⁢ a ⁢ ( x ) ⁢ | u | q ⁢ | D ⁢ ζ | q ⁢ d ⁢ x ≤ ε ⁢ ∫ Ω ζ q ⁢ H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + q q ⁢ C ⁢ ( ε ) ⁢ ∫ Ω H ⁢ ( x , u ⋅ D ⁢ ζ ) ⁢ d ⁢ x ,

where, in the last inequality, we used the fact that ( q - 1 ) ⁢ p p - 1 ≥ q and 0 ≤ ζ ≤ 1 . Therefore, taking a suitable value of ε, we arrive at

∫ Ω ζ q ⁢ H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ ∫ Ω H ⁢ ( x , u ⋅ D ⁢ ζ ) ⁢ d ⁢ x . ∎

We conclude this section by characterizing the viscosity properties of distributional solutions to (2.4) that approximates equation (1.1).

Lemma 2.9.

Let a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) and u ε ∈ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) be a distributional solution to

(2.4) - div ⁡ A ⁢ ( x , D ⁢ u ) = ε .

Assume that φ ∈ C 2 ⁢ ( Ω ) satisfies u ε ⁢ ( x 0 ) = φ ⁢ ( x 0 ) , u ε ⁢ ( x ) > φ ⁢ ( x ) for all x ≠ x 0 . Furthermore, if D ⁢ φ ⁢ ( x 0 ) ≠ 0 or x 0 is an isolated critical point of φ, then we arrive at

lim sup x → x 0 x ≠ x 0 ⁡ ( - div ⁡ A ⁢ ( x , D ⁢ φ ⁢ ( x ) ) ) ≥ ε .

Proof.

We argue by contradiction. If the claim does not hold, then there exists a constant r > 0 such that D ⁢ φ ⁢ ( x ) ≠ 0 and - div ⁡ A ⁢ ( x , D ⁢ φ ⁢ ( x ) ) < ε for 0 < | x - x 0 | < r . Set 0 < ρ < r . For all nonnegative η ∈ C 0 ∞ ⁢ ( B ⁢ ( x 0 , r ) ) , we get

- ∫ ρ < | x - x 0 | < r η ⁢ div ⁡ A ⁢ ( x , D ⁢ φ ) ⁢ d ⁢ x = ∫ ρ < | x - x 0 | < r A ⁢ ( x , D ⁢ φ ) ⋅ D ⁢ η ⁢ d ⁢ x + ∫ | x - x 0 | = ρ η ⁢ A ⁢ ( x , D ⁢ φ ) ⋅ x - x 0 ρ ⁢ d ⁢ S .

We now evaluate

| ∫ | x - x 0 | = ρ η ⁢ A ⁢ ( x , D ⁢ φ ) ⋅ x - x 0 ρ ⁢ d ⁢ S | ≤ ∥ η ∥ L ∞ ⁢ ∫ | x - x 0 | = ρ | D ⁢ φ | p - 1 + a ⁢ ( x ) ⁢ | D ⁢ φ | q - 1 ⁢ d ⁢ S ≤ C ⁢ ∥ η ∥ L ∞ ⁢ ( 1 + ∥ D ⁢ φ ∥ L ∞ q - 1 ) ⁢ ρ n - 1 → 0 when ⁢ ρ → 0 .

By the counter proposition,

- ∫ ρ < | x - x 0 | < r η ⁢ div ⁡ A ⁢ ( x , D ⁢ φ ) ⁢ d ⁢ x < ε ⁢ ∫ ρ < | x - x 0 | < r η ⁢ d ⁢ x ≤ ∫ B ⁢ ( x 0 , r ) ε ⁢ η ⁢ d ⁢ x .

Hence, by sending ρ → 0 , we know that

∫ B ⁢ ( x 0 , r ) A ⁢ ( x , D ⁢ φ ) ⋅ D ⁢ η ⁢ d ⁢ x ≤ ∫ B ⁢ ( x 0 , r ) ε ⁢ η ⁢ d ⁢ x ,

which means that φ is a distributional subsolution.

Let m : = inf x ∈ ∂ ⁡ B ⁢ ( x 0 , r ) ( u ε - φ ) > 0 . Then φ ~ : = φ + m is a distributional subsolution as well. Via φ ~ ≤ u ε on ∂ ⁡ B ⁢ ( x 0 , r ) and the comparison principle for distributional solutions, we have φ ~ ≤ u ε in B ⁢ ( x 0 , r ) , but φ ~ ⁢ ( x 0 ) > u ε ⁢ ( x 0 ) . That is a contradiction. ∎

3 Distributional solutions coincide with 𝒜 H ⁢ ( ⋅ ) -harmonic functions

In this section, we study the equivalence of distributional supersolutions and 𝒜 H ⁢ ( ⋅ ) -superharmonic functions, by symmetry deriving the same relation between subsolutions and 𝒜 H ⁢ ( ⋅ ) -subharmonic functions. Eventually, the equivalence can be extended to the distributional solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions.

We begin with stating that a distributional supersolution is 𝒜 H ⁢ ( ⋅ ) -superharmonic after a redefinition in a set of measure zero. The proof is similar to that of p ⁢ ( x ) -superharmonic functions in [23, Theorem 6.1]. To be self-contained, we repeat the proof here.

Theorem 3.1.

A distributional supersolution to (1.1), u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , is lower semicontinuous (after a possible change in a set of measure zero). We could define

(3.1) u ⁢ ( x ) = ess ⁢ lim inf y → x ⁡ u ⁢ ( y )

pointwise in Ω. Thus u is A H ⁢ ( ⋅ ) -superharmonic.

Proof.

Observe that u is lower semicontinuous and finite almost everywhere. In order to show that u obeys the comparison principle in Definition 2.4, we suppose that D ⊂⊂ Ω is a subdomain, and h ∈ C ⁢ ( D ¯ ) is a distributional solution in D such that u ≥ h on ∂ ⁡ D . By the lower semicontinuity, we choose ε > 0 and a subdomain D ′ ⊂⊂ D such that u + ε > h in D ∖ D ′ . Because the set { u - h ≤ - ε } is closed due to semicontinuity, the function min ⁡ { u + ε - h , 0 } is compactly supported in D ′ . Via the comparison principle (Proposition 2.7), u + ε ≥ h a.e. in D ′ . Thus u + ε ≥ h everywhere in D by virtue of (3.1), and sending ε → 0 finishes the proof. ∎

Next, we are ready to show that 𝒜 H ⁢ ( ⋅ ) -superharmonic functions are distributional supersolutions. To this end, consider an obstacle problem of the type

(3.2) ℱ ψ ( Ω ) : = { u ∈ W 1 , H ⁢ ( ⋅ ) ( Ω ) : u ≥ ψ a.e. in Ω and u - ψ ∈ W 0 1 , H ⁢ ( ⋅ ) ( Ω ) }

with ψ ∈ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) being an obstacle from below. Also ψ is the boundary value. If a function u ∈ ℱ ψ ⁢ ( Ω ) satisfies

∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ ( v - u ) 〉 ⁢ d ⁢ x ≥ 0 for any ⁢ v ∈ ℱ ψ ⁢ ( Ω ) ,

then u is called a solution of (3.2). We can easily find that a solution to the obstacle problem (3.2) is a distributional supersolution to equation (1.1).

We present the following results about existence and regularity properties of problem (3.2). We refer the readers to [10] for more details about the obstacle problem.

Lemma 3.2.

Under assumption (2.1), the following two conclusions hold.

For the obstacle problem ( 3.2 ), there is a unique solution u ψ ∈ ℱ ψ ⁢ ( Ω ) . Moreover, u ψ is a distributional supersolution of ( 1.1 ) in Ω.
If ψ ∈ C ⁢ ( Ω ) ∩ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , then u ψ is continuous in Ω and, in the distributional sense, solves equation ( 1.1 ) in the open set { x ∈ Ω : u ψ ⁢ ( x ) > ψ ⁢ ( x ) } .

Now we utilize this lemma to derive the following approximation result on the 𝒜 H ⁢ ( ⋅ ) -superharmonic functions.

Lemma 3.3.

If u is an A H ⁢ ( ⋅ ) -superharmonic function in Ω, there is an increasing sequence of continuous supersolutions { u j } in domain D satisfying u = lim j → ∞ ⁡ u j pointwise in D. Here D ⊂⊂ Ω is an arbitrary subdomain.

Proof.

By virtue of the lower semicontinuity of u, we can obtain a sequence of smooth functions ψ j ∈ C ∞ ⁢ ( Ω ) such that

ψ 1 ⁢ ( x ) ≤ ψ 2 ⁢ ( x ) ≤ ⋯ ≤ ψ n ⁢ ( x ) ≤ ⋯ and u ⁢ ( x ) = lim j → ∞ ⁡ ψ j ⁢ ( x )

everywhere in Ω. Fix a regular subdomain D ⊂⊂ Ω , and denote by u j : = u ψ j the solution of (3.2) in D with ψ j as an obstacle. Hence u j ∈ ℱ ψ j ⁢ ( D ) and u j ≥ ψ j in D. We claim that

u 1 ⁢ ( x ) ≤ u 2 ⁢ ( x ) ≤ ⋯ ≤ u n ⁢ ( x ) ≤ ⋯ and ψ j ⁢ ( x ) ≤ u j ⁢ ( x ) ≤ u ⁢ ( x )

at every point x ∈ D . In order to show u j ≤ u , we first observe this is true but possibly in the open set A j : = { x ∈ D : u j ( x ) > ψ j ( x ) } . Through Lemma 3.2, u j is a distributional solution in A j . Because ψ j and u j are continuous in A j ¯ (the closure of A j ), we get u j = ψ j on ∂ ⁡ A j . By means of u j ≤ u on ∂ ⁡ A j and the comparison principle obeyed by u in Definition 2.4, it follows that u j ≤ u in A j . Thus u j ≤ u in D. Analogously, we can justify u j ≤ u j + 1 , j = 1 , 2 , 3 , … , since u j + 1 obeys the comparison principle for distributional solutions by Lemma 3.2. Consequently, we deduce u = lim j → ∞ ⁡ ψ j ≤ lim j → ∞ ⁡ u j ≤ u everywhere in D. ∎

Lemma 3.4.

Let u be an A H ⁢ ( ⋅ ) -superharmonic function. If u is locally bounded from above in Ω, we can infer that u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) and u j satisfies lim j → ∞ ⁡ ∫ D H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x = 0 , where D ⊂⊂ Ω is a subdomain and u j is defined as in Lemma 3.3.

Proof.

Choose a regular domain D 1 such that D ⊂⊂ D 1 ⊂⊂ Ω . If u is locally bounded from above, then it is bounded in D 1 . Now we fix a smooth function ξ ⁢ ( x ) ∈ C 0 ∞ ⁢ ( Ω ) such that 0 ≤ ξ ⁢ ( x ) ≤ 1 in Ω, ξ ⁢ ( x ) ≡ 1 in D and ξ ⁢ ( x ) ≡ 0 in Ω ∖ D 1 . Since u j ⁢ ( x ) ↗ u ⁢ ( x ) as j → ∞ in D 1 ( u j is defined as in Lemma 3.3)), η : = ( sup D 1 u - u j ) ξ q is nonnegative in Ω. Thus, because of u j ( j = 1 , 2 , 3 , … ) being distributional supersolution, we could choose the function η ⁢ ( x ) to test the weak form of equation (1.1), arriving at

0 ≤ ∫ Ω 〈 A ⁢ ( x , D ⁢ u j ) , D ⁢ η 〉 ⁢ d ⁢ x = ∫ Ω 〈 A ⁢ ( x , D ⁢ u j ) , q ⁢ ξ q - 1 ⁢ ( sup D 1 ⁡ u - u j ) ⁢ D ⁢ ξ - ξ q ⁢ D ⁢ u j 〉 ⁢ d ⁢ x .

Similarly to the proof of Proposition 2.8, we further have

∫ Ω ξ q ⁢ H ⁢ ( x , D ⁢ u j ) ⁢ d ⁢ x ≤ ∫ Ω q ⁢ ξ q - 1 ⁢ 〈 A ⁢ ( x , D ⁢ u j ) , ( sup D 1 ⁡ u - u j ) ⁢ D ⁢ ξ 〉 ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ ∫ D 1 [ | ( sup D 1 ⁡ u - u j ) ⁢ D ⁢ ξ | p + a ⁢ ( x ) ⁢ | ( sup D 1 ⁡ u - u j ) ⁢ D ⁢ ξ | q ] ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ M q ⁢ ∫ D 1 [ | D ⁢ ξ | p + a ⁢ ( x ) ⁢ | D ⁢ ξ | q ] ⁢ d ⁢ x

with M : = sup D 1 u - inf D 1 ψ 1 + 1 , where, in the last inequality, we need to observe u j ≥ ψ j ≥ ψ 1 in D 1 , with ψ j being as in Lemma 3.3. Consequently, we get

(3.3) ∫ D H ⁢ ( x , D ⁢ u j ) ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ M q ⁢ ∫ D 1 H ⁢ ( x , D ⁢ ζ ) ⁢ d ⁢ x = : L , j = 1 , 2 , 3 , … ,

which means that D ⁢ u j ( j = 1 , 2 , 3 , … ) is uniformly bounded in L H ⁢ ( ⋅ ) ⁢ ( D ) . Thereby, we know that D ⁢ u j ⇀ D ⁢ u weakly in L H ⁢ ( ⋅ ) ⁢ ( D ) up to a subsequence, and u ∈ W 1 , H ⁢ ( ⋅ ) ⁢ ( D ) and ∫ D H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x ≤ L . We also deduce that u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) .

We now justify

lim j → ∞ ⁡ ∫ D H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x = 0 .

It suffices to show

lim j → ∞ ⁡ ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x = 0

whenever B r is a ball contained in D. In addition, we suppose B 2 ⁢ r ⊂⊂ D is a concentric ball. Let ζ ∈ C 0 ∞ ⁢ ( B 2 ⁢ r ) , 0 ≤ ζ ≤ 1 and ζ ≡ 1 in B r . We use η j : = ζ ( u - u j ) as a test function to get ∫ B 2 ⁢ r 〈 A ⁢ ( x , D ⁢ u j ) , D ⁢ η j 〉 ⁢ d ⁢ x ≥ 0 . Then we estimate

(3.4) I j : = ∫ B 2 ⁢ r 〈 A ( x , D u ) - A ( x , D u j ) , D ( ζ ( u - u j ) ) 〉 d x ≤ ∫ B 2 ⁢ r 〈 A ( x , D u ) , D ( ζ ( u - u j ) ) 〉 d x = ∫ B 2 ⁢ r ( u - u j ) ⁢ 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ ζ 〉 ⁢ d ⁢ x + ∫ B 2 ⁢ r ζ ⁢ 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ ( u - u j ) 〉 ⁢ d ⁢ x → 0 as ⁢ j → ∞ .

In fact, from u j converging to u monotonely and D ⁢ u j ⇀ D ⁢ u weakly in L H ⁢ ( ⋅ ) ⁢ ( D ) , we can obtain the limit.

On the other hand,

(3.5) I j = ∫ B 2 ⁢ r ζ ⁢ 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ ( u - u j ) 〉 ⁢ d ⁢ x + ∫ B 2 ⁢ r ( u - u j ) ⁢ 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ ζ 〉 ⁢ d ⁢ x = : I j , 1 + I j , 2 .

First, for I j , 2 , by the Hölder inequality, we arrive at

I j , 2 ≤ ∫ B 2 ⁢ r | D ⁢ ζ | ⁢ | u - u j | ⁢ ( | D ⁢ u | p - 1 + a ⁢ ( x ) ⁢ | D ⁢ u | q - 1 + | D ⁢ u j | p - 1 + a ⁢ ( x ) ⁢ | D ⁢ u j | q - 1 ) ⁢ d ⁢ x ≤ ∥ D ζ ∥ L ∞ ⁢ ( B 2 ⁢ r ) [ ( ∫ B 2 ⁢ r | u - u j | p d x ) 1 p ( ∫ B 2 ⁢ r | D u | p d x ) p - 1 p + ( ∫ B 2 ⁢ r a ( x ) | u - u j | q d x ) 1 q ( ∫ B 2 ⁢ r a ( x ) | D u | q d x ) q - 1 q + ( ∫ B 2 ⁢ r | u - u j | p d x ) 1 p ( ∫ B 2 ⁢ r | D u j | p d x ) p - 1 p + ( ∫ B 2 ⁢ r a ( x ) | u - u j | q d x ) 1 q ( ∫ B 2 ⁢ r a ( x ) | D u j | q d x ) q - 1 q ] ≤ ∥ D ζ ∥ L ∞ ⁢ ( B 2 ⁢ r ) ( 1 + ∫ B 2 ⁢ r H ( x , D u ) d x + ∫ B 2 ⁢ r H ( x , D u j ) d x ) max t ∈ { p , q } ( ∫ B 2 ⁢ r H ( x , u - u j ) d x ) 1 t .

Here we note that the exponents 1 p , 1 q , p - 1 p and q - 1 q are less than 1. Utilizing the Lebesgue dominated convergence theorem and (3.3) yield that

(3.6) I j , 2 → 0 as ⁢ j → ∞ .

Since I j , 1 ≥ 0 , combing (3.4), (3.5) and (3.6), we derive

lim j → ∞ ⁡ ∫ B 2 ⁢ r ζ ⁢ 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ ( u - u j ) 〉 ⁢ d ⁢ x = 0 .

Furthermore,

lim j → ∞ ⁡ ∫ B r 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ ( u - u j ) 〉 ⁢ d ⁢ x = 0 .

In what follows, we divide the proof into three cases.

Case 1. 2 ≤ p ≤ q < ∞ . It is easy to arrive at

∫ B r 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ ( u - u j ) 〉 ⁢ d ⁢ x ≥ C ⁢ ∫ B r | D ⁢ u - D ⁢ u j | p + a ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | q ⁢ d ⁢ x = C ⁢ ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x ≥ 0 .

Thus, by sending j → ∞ , we get ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x → 0 .

Case 2. 1 < p ≤ q < 2 . For each ε ∈ ( 0 , 1 ] , we have

∫ B r b ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | t ⁢ d ⁢ x ≤ C ⁢ ( t ) ⁢ ε t - 2 t ⁢ ∫ B r b ⁢ ( x ) ⁢ 〈 | D ⁢ u | t - 2 ⁢ D ⁢ u - | D ⁢ u j | t - 2 ⁢ D ⁢ u j , D ⁢ u - D ⁢ u j 〉 ⁢ d ⁢ x + ε ⁢ ∫ B r b ⁢ ( x ) ⁢ | D ⁢ u | t ⁢ d ⁢ x ,

where t ∈ { p , q } and b ⁢ ( x ) ∈ { 1 , a ⁢ ( x ) } . Therefore,

∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x ≤ C ⁢ ( p , q ) ⁢ ε p - 2 p ⁢ ∫ B r 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ u - D ⁢ u j 〉 ⁢ d ⁢ x + ε ⁢ ∫ B r H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x .

As j → ∞ , the above inequality becomes

lim j → ∞ ⁡ ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x ≤ ε ⁢ ∫ B r H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x .

Finally, since ε > 0 is arbitrary, we infer

lim j → ∞ ⁡ ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x = 0 .

Case 3. 1 < p < 2 ≤ q < ∞ . Merging case 1 and case 2, we can see

∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x ≤ C ⁢ ( p ) ⁢ ε p - 2 p ⁢ ∫ B r 〈 | D ⁢ u | p - 2 ⁢ D ⁢ u - | D ⁢ u j | p - 2 ⁢ D ⁢ u j , D ⁢ u - D ⁢ u j 〉 ⁢ d ⁢ x + ε ⁢ ∫ B r | D ⁢ u | p ⁢ d ⁢ x

+ C ⁢ ∫ B r a ⁢ ( x ) ⁢ 〈 | D ⁢ u | q - 2 ⁢ D ⁢ u - | D ⁢ u j | q - 2 ⁢ D ⁢ u j , D ⁢ u - D ⁢ u j 〉 ⁢ d ⁢ x

≤ C ⁢ ε p - 2 p ⁢ ∫ B r 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ u - D ⁢ u j 〉 ⁢ d ⁢ x + ε ⁢ ∫ B r | D ⁢ u | p ⁢ d ⁢ x .

Similarly, we deduce lim j → ∞ ⁡ ∫ B r H ⁢ ( x , D ⁢ u - D ⁢ u j ) ⁢ d ⁢ x = 0 . In summary, we reach the conclusion. ∎

Through the aforementioned approximation theorem, we could readily establish the result that bounded 𝒜 H ⁢ ( ⋅ ) -superharmonic functions are distributional supersolutions, which is stated as follows.

Theorem 3.5.

Assume that u is A H ⁢ ( ⋅ ) -superharmonic and locally bounded in Ω. Then u ∈ W loc 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) and u is a distributional supersolution to equation (1.1), that is, ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ η 〉 ⁢ d ⁢ x ≥ 0 for any nonnegative η ∈ C 0 ∞ ⁢ ( Ω ) .

Proof.

We need to verify

∫ Ω 〈 A ⁢ ( x , D ⁢ u ) , D ⁢ η 〉 ⁢ d ⁢ x = lim j → ∞ ⁡ ∫ Ω 〈 A ⁢ ( x , D ⁢ u j ) , D ⁢ η 〉 ⁢ d ⁢ x ≥ 0 ,

where u j is as in Lemma 3.3. It is well known that u j is distributional supersolution. Now we pass to the limit. We shall employ the elementary vector inequalities

(3.7) | | ξ 1 | t - 2 ⁢ ξ 1 - | ξ 2 | t - 2 ⁢ ξ 2 | ≤ { ( t - 1 ) ⁢ | ξ 1 - ξ 2 | ⁢ ( | ξ 1 | t - 2 + | ξ 2 | t - 2 ) if ⁢ t ≥ 2 , 2 2 - t ⁢ | ξ 1 - ξ 2 | t - 1 if ⁢ 1 < t < 2 ,

where ξ 1 , ξ 2 ∈ ℝ n . We split the proof into three cases.

Case 1. 2 ≤ p ≤ q < ∞ . Using the Hölder inequality and (3.7), we have

∫ Ω 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ η 〉 ⁢ d ⁢ x ≤ ∫ Ω | D ⁢ η | ⁢ | ( | D ⁢ u | p - 2 ⁢ D ⁢ u - | D ⁢ u j | p - 2 ⁢ D ⁢ u j ) + a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 ⁢ D ⁢ u - | D ⁢ u j | q - 2 ⁢ D ⁢ u j ) | ⁢ d ⁢ x ≤ C ⁢ ∫ Ω | D ⁢ η | ⁢ | D ⁢ u - D ⁢ u j | ⁢ [ ( | D ⁢ u | p - 2 + | D ⁢ u j | p - 2 ) + a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 + | D ⁢ u j | q - 2 ) ] ⁢ d ⁢ x ≤ C ⁢ ( ∫ Ω | D ⁢ η | p ⁢ d ⁢ x ) 1 p ⁢ ( ∫ Ω | D ⁢ u - D ⁢ u j | p ⁢ d ⁢ x ) 1 p ⁢ ( ∫ Ω ( | D ⁢ u | p - 2 + | D ⁢ u j | p - 2 ) p p - 2 ⁢ d ⁢ x ) p - 2 p + C ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ η | q ⁢ d ⁢ x ) 1 q ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | q ⁢ d ⁢ x ) 1 q ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 + | D ⁢ u j | q - 2 ) q q - 2 ⁢ d ⁢ x ) q - 2 q ≤ C ⁢ ( 1 + ∫ Ω H ⁢ ( x , D ⁢ η ) ⁢ d ⁢ x ) ⁢ ( 1 + ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + ∫ Ω H ⁢ ( x , D ⁢ u j ) ⁢ d ⁢ x ) ⋅ max t ∈ { p , q } ( ∫ Ω H ( x , D u - D u j ) d x ) 1 t → 0 as j → ∞ ,

where the limit is inferred by Lemma 3.4 and (3.3).

Case 2. 1 < p ≤ q < 2 . By the Hölder inequality and (3.7), then

∫ Ω | D ⁢ η | ⁢ | ( | D ⁢ u | p - 2 ⁢ D ⁢ u - | D ⁢ u j | p - 2 ⁢ D ⁢ u j ) + a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 ⁢ D ⁢ u - | D ⁢ u j | q - 2 ⁢ D ⁢ u j ) | ⁢ d ⁢ x ≤ C ⁢ ∫ Ω | D ⁢ η | ⁢ ( | D ⁢ u - D ⁢ u j | p - 1 + a ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | q - 1 ) ⁢ d ⁢ x ≤ C ⁢ ( ∫ Ω | D ⁢ η | p ⁢ d ⁢ x ) 1 p ⁢ ( ∫ Ω | D ⁢ u - D ⁢ u j | p ⁢ d ⁢ x ) p - 1 p + C ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ η | q ⁢ d ⁢ x ) 1 q ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | q ⁢ d ⁢ x ) q - 1 q ≤ C ( 1 + ∫ Ω H ( x , D η ) d x ) ⋅ max t ∈ { p , q } ( ∫ Ω H ( x , D u - D u j ) d x ) t - 1 t → 0 as j → ∞

by Lemma 3.4.

Case 3. 1 < p < 2 ≤ q < ∞ . Merging case 1 and case 2 leads to

∫ Ω | D ⁢ η | ⁢ | ( | D ⁢ u | p - 2 ⁢ D ⁢ u - | D ⁢ u j | p - 2 ⁢ D ⁢ u j ) + a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 ⁢ D ⁢ u - | D ⁢ u j | q - 2 ⁢ D ⁢ u j ) | ⁢ d ⁢ x ≤ C ⁢ ( ∫ Ω | D ⁢ η | p ⁢ d ⁢ x ) 1 p ⁢ ( ∫ Ω | D ⁢ u - D ⁢ u j | p ⁢ d ⁢ x ) p - 1 p + C ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ η | q ⁢ d ⁢ x ) 1 q ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ | D ⁢ u - D ⁢ u j | q ⁢ d ⁢ x ) 1 q ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ ( | D ⁢ u | q - 2 + | D ⁢ u j | q - 2 ) q q - 2 ⁢ d ⁢ x ) q - 2 q ≤ C ⁢ ( 1 + ∫ Ω H ⁢ ( x , D ⁢ η ) ⁢ d ⁢ x ) ⁢ ( 1 + ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + ∫ Ω H ⁢ ( x , D ⁢ u j ) ⁢ d ⁢ x ) ⋅ max s ∈ { p - 1 p , 1 q } ( ∫ Ω H ( x , D u - D u j ) d x ) s → 0 as j → ∞ .

Therefore, we have

lim j → ∞ ⁡ ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u j ) , D ⁢ η 〉 ⁢ d ⁢ x = 0 .

Now we finish the proof. ∎

The combination of Theorems 3.1 and 3.5 leads directly to the following conclusion serving as a bridge in the proof of equivalence of viscosity and distributional solutions. It is worth mentioning that the result on the equivalence between distributional solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions is of independent interest. Now let us state the main result of this section.

Corollary 3.6.

With condition (2.1), a (locally) bounded distributional solution is the same as a (locally) bounded A H ⁢ ( ⋅ ) -harmonic function in Ω.

4 Equivalence between viscosity solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions

To begin with, we shall verify the claim that 𝒜 H ⁢ ( ⋅ ) -superharmonic functions are viscosity supersolutions, which can be obtained by the comparison principle for distributional subsolutions and supersolutions (see Proposition 2.7).

Theorem 4.1.

Under the assumption that a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) , the A H ⁢ ( ⋅ ) -superharmonic functions are the viscosity supersolutions to (1.1).

Proof.

Let u be 𝒜 H ⁢ ( ⋅ ) -superharmonic in the domain Ω. If the conclusion does not hold, then we suppose that there exists a test function φ ∈ C 2 ⁢ ( Ω ) such that, for x 0 ∈ Ω ,

{ u ⁢ ( x 0 ) = φ ⁢ ( x 0 ) , D ⁢ φ ⁢ ( x 0 ) ≠ 0 , u ⁢ ( x ) > φ ⁢ ( x ) for ⁢ x ≠ x 0 ,

but it satisfies - div ⁡ A ⁢ ( x 0 , D ⁢ φ ⁢ ( x 0 ) ) < 0 . Owing to continuity, for x ∈ B ⁢ ( x 0 , δ ) with some small δ > 0 , D ⁢ φ ⁢ ( x ) ≠ 0 and - div ⁡ A ⁢ ( x , D ⁢ φ ⁢ ( x ) ) < 0 .

Set

m : = 1 2 min x ∈ ∂ ⁡ B ⁢ ( x 0 , δ ) { u ( x ) - φ ( x ) } > 0 and φ ~ : = φ + m .

Then we can see that φ ~ is a distributional subsolution and φ ~ ≤ u on ∂ ⁡ B ⁢ ( x 0 , δ ) . It follows from the comparison principle that φ ⁢ ( x ) + m ≤ u ⁢ ( x ) in B ⁢ ( x 0 , δ ) , which contradicts φ ⁢ ( x 0 ) = u ⁢ ( x 0 ) . ∎

Next, we present an essential approximation lemma, which states that the distributional solution to (1.1) could be approximated by the solution of (2.4).

Lemma 4.2.

Let u ∈ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) be a distributional solution of (1.1) and u ε a distributional solution to problem (2.4) with the Dirichlet boundary value u - u ε ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) with ε > 0 . Then u ε → u locally uniformly in Ω.

Proof.

We split the proof into two steps.

Step 1. We first obtain the boundedness of | D ⁢ u - D ⁢ u ε | in L H ⁢ ( ⋅ ) ⁢ ( Ω ) . We utilize u - u ε to test the weak formulation for u ε , which yields that

∫ Ω 〈 A ⁢ ( x , D ⁢ u ε ) , D ⁢ ( u - u ε ) 〉 ⁢ d ⁢ x = ε ⁢ ∫ Ω ( u - u ε ) ⁢ d ⁢ x .

Furthermore, making use of Hölder inequality and Young’s inequality, we have

∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x = ∫ Ω 〈 A ⁢ ( x , D ⁢ u ε ) , D ⁢ u 〉 ⁢ d ⁢ x - ε ⁢ ∫ Ω ( u - u ε ) ⁢ d ⁢ x

≤ σ ⁢ ∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x + C ⁢ ( σ ) ⁢ ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + δ ⁢ ∫ Ω | u - u ε | p ⁢ d ⁢ x + C ⁢ ( δ ) ⁢ | Ω | ⁢ ε p p - 1 .

Now we solely estimate the integral ∫ Ω | u - u ε | p ⁢ d ⁢ x . Since W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) ↪ W 0 1 , p ⁢ ( Ω ) (see Lemma 2.1) and u - u ε ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) , we can apply the Poincaré inequality to u - u ε in the Sobolev space W 0 1 , p ⁢ ( Ω ) , yielding that

∫ Ω | u - u ε | p ⁢ d ⁢ x ≤ C ⁢ ( n , p , Ω ) ⁢ ∫ Ω | D ⁢ u - D ⁢ u ε | p ⁢ d ⁢ x ≤ C ⁢ ( n , p , Ω ) ⁢ ( ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + ∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x ) .

Merging these two previous inequalities yields

∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x ≤ σ ⁢ ∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x + C ⁢ ( σ ) ⁢ ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + δ ⁢ C ⁢ ( n , p , Ω ) ⁢ ( ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + ∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x ) + C ⁢ ( δ ) ⁢ | Ω | ⁢ ε p p - 1 .

Selecting σ = 1 4 and δ = 1 4 ⁢ C ⁢ ( n , p , Ω ) , this inequality turns into

∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x ≤ C ⁢ ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x + C ⁢ ε p p - 1 .

Hence, noting ε p p - 1 ≤ 1 and using (2.2), it follows that

∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x ≤ C ⁢ ( 1 + ∫ Ω H ⁢ ( x , D ⁢ u ) ⁢ d ⁢ x ) ≤ C ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) .

On the other hand, by (2.2),

∥ D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) p ≤ 1 + ∫ Ω H ⁢ ( x , D ⁢ u ε ) ⁢ d ⁢ x .

Then

(4.1) ∥ D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) ≤ C ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) 1 p ≤ C ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) .

Consequently,

(4.2) ∥ D ⁢ u - D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) ≤ C ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) .

Step 2. We show that ∥ D ⁢ u - D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) → 0 as ε tends to 0. Then we can further conclude u ε → u in W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) .

We take u - u ε to test the weak forms of (1.1) and (2.4) and then subtract these two equations, which leads to

(4.3) ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u ε ) , D ⁢ ( u - u ε ) 〉 ⁢ d ⁢ x = - ε ⁢ ∫ Ω ( u - u ε ) ⁢ d ⁢ x .

In view of (4.2) and (2.2), as in step 1, the right-hand side in the last display can be evaluated,

ε ⁢ | ∫ Ω ( u - u ε ) ⁢ d ⁢ x | ≤ ε ⁢ | Ω | p p - 1 ⁢ ( ∫ Ω | u - u ε | p ⁢ d ⁢ x ) 1 p ≤ ε ⁢ C ⁢ ( n , p , Ω ) ⁢ ( ∫ Ω | D ⁢ u - D ⁢ u ε | p ⁢ d ⁢ x ) 1 p ≤ ε ⁢ C ⁢ ( n , p , Ω ) ⁢ ( ∫ Ω H ⁢ ( x , D ⁢ u - D ⁢ u ε ) ⁢ d ⁢ x ) 1 p ≤ ε ⁢ C ⁢ ( n , p , Ω ) ⁢ ( 1 + ∥ D ⁢ u - D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) 1 p ≤ ε ⁢ C ⁢ ( n , p , Ω ) ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) .

Next we focus on the term of the left-hand side in (4.3).

Case 1. If 2 ≤ p ≤ q < ∞ , by the elementary vector inequality, we derive

C ⁢ ∫ Ω H ⁢ ( x , D ⁢ u - D ⁢ u ε ) ⁢ d ⁢ x ≤ ∫ Ω 〈 A ⁢ ( x , D ⁢ u ) - A ⁢ ( x , D ⁢ u ε ) , D ⁢ ( u - u ε ) 〉 ⁢ d ⁢ x ≤ ε ⁢ C ⁢ ( 1 + ∥ D ⁢ u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) → 0 by letting ⁢ ε → 0 .

Case 2. If 1 < p ≤ q < 2 , we get

∫ Ω H ⁢ ( x , D ⁢ u - D ⁢ u ε ) ⁢ d ⁢ x = ∫ Ω { ( | D u | + | D u ε | ) p ⁢ ( 2 - p ) 2 ( | D u | + | D u ε | ) p ⁢ ( p - 2 ) 2 | D u - D u ε | p + a ( x ) ( | D u | + | D u ε | ) q ⁢ ( 2 - q ) 2 ( | D u | + | D u ε | ) q ⁢ ( q - 2 ) 2 | D u - D u ε | q } d x ≤ ( ∫ Ω ( | D ⁢ u | + | D ⁢ u ε | ) p ⁢ d ⁢ x ) 2 - p 2 ⁢ ( ∫ Ω ( | D ⁢ u | + | D ⁢ u ε | ) p - 2 ⁢ | D ⁢ u - D ⁢ u ε | 2 ⁢ d ⁢ x ) p 2 + ( ∫ Ω a ⁢ ( x ) ⁢ ( | D ⁢ u | + | D ⁢ u ε | ) q ⁢ d ⁢ x ) 2 - q 2 ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ ( | D ⁢ u | + | D ⁢ u ε | ) q - 2 ⁢ | D ⁢ u - D ⁢ u ε | 2 ⁢ d ⁢ x ) q 2 ≤ [ 1 + ( ∫ Ω ( | D ⁢ u | + | D ⁢ u ε | ) p + a ⁢ ( x ) ⁢ ( | D ⁢ u | + | D ⁢ u ε | ) q ⁢ d ⁢ x ) 2 - p 2 ] ⋅ max t ∈ { p , q } ( ∫ Ω ( | D u | + | D u ε | ) p - 2 | D u - D u ε | 2 + a ( x ) ( | D u | + | D u ε | ) q - 2 | D u - D u ε | 2 d x ) t 2 ≤ C [ 1 + ( ∫ Ω H ( x , D u ) + H ( x , D u ε ) d x ) 2 - p 2 ] max t ∈ { p , q } ( ∫ Ω 〈 A ( x , D u ε ) - A ( x , D u ) , D u ε - D u 〉 d x ) t 2 ≤ C ( 1 + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q + ∥ D u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q ) max t ∈ { p , q } ( ∫ Ω 〈 A ( x , D u ε ) - A ( x , D u ) , D u ε - D u 〉 d x ) t 2 ≤ C ( 1 + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) max t ∈ { p , q } [ ε C ( 1 + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) ] t 2 → 0 as ε → 0 ,

where, in the penultimate inequality, we employed (2.2) and the fact that 2 - p 2 < 1 , and the last inequality follows from (4.1).

Case 3. 1 < p < 2 ≤ q < ∞ . Combining case 1 and case 2, we obtain

∫ Ω H ⁢ ( x , D ⁢ u - D ⁢ u ε ) ⁢ d ⁢ x ≤ ( ∫ Ω ( | D ⁢ u | + | D ⁢ u ε | ) p ⁢ d ⁢ x ) 2 - p 2 ⁢ ( ∫ Ω ( | D ⁢ u | + | D ⁢ u ε | ) p - 2 ⁢ | D ⁢ u - D ⁢ u ε | 2 ⁢ d ⁢ x ) p 2 + ∫ Ω a ⁢ ( x ) ⁢ 〈 | D ⁢ u | q - 2 ⁢ D ⁢ u - | D ⁢ u ε | q - 2 ⁢ D ⁢ u ε , D ⁢ u - D ⁢ u ε 〉 ⁢ d ⁢ x ≤ [ 1 + ( ∫ Ω ( | D u | + | D u ε | ) p d x ) 2 - p 2 ] max t ∈ { p , 2 } ( ∫ Ω 〈 A ( x , D u ε ) - A ( x , D u ) , D u ε - D u 〉 d x ) t 2 ≤ C ( 1 + ∫ Ω ( H ( x , D u ) + H ( x , D u ε ) ) d x ) max t ∈ { p , 2 } ( ∫ Ω 〈 A ( x , D u ε ) - A ( x , D u ) , D u ε - D u 〉 d x ) t 2 ≤ C ( 1 + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) max t ∈ { p , 2 } [ ε C ( 1 + ∥ D u ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) q 2 ) ] t 2 → 0 when ε → 0 .

Consequently, we deduce ∥ D ⁢ u - D ⁢ u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) → 0 as ε → 0 . By Lemma 2.2, when ε → 0 , ∥ u - u ε ∥ L H ⁢ ( ⋅ ) ⁢ ( Ω ) → 0 . Therefore, we obtain

(4.4) u ε → u in ⁢ W 1 , H ⁢ ( ⋅ ) ⁢ ( Ω ) .

Let ε 1 ≤ ε 2 . Subtracting the corresponding equations, we get

∫ Ω 〈 A ⁢ ( x , D ⁢ u ε 2 ) - A ⁢ ( x , D ⁢ u ε 1 ) , D ⁢ η 〉 ⁢ d ⁢ x = ( ε 2 - ε 1 ) ⁢ ∫ Ω η ⁢ d ⁢ x ≥ 0

for any nonnegative η ∈ C 0 ∞ ⁢ ( Ω ) . From Proposition 2.7, u ε 2 ≥ u ε 1 almost everywhere. This together with (4.4) indicates that u ε → u a.e. in Ω. It follows from the uniform C loc α -estimates for u ε in ε that u ε → u locally uniformly in Ω. ∎

Before giving the main result of this section, we have to provide the following comparison principle for viscosity solutions which plays a crucial role in the equivalence of viscosity solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions. The proof of this lemma is rather long and delicate, postponed to the next section.

Lemma 4.3.

Assume that u is a viscosity subsolution to equation (1.1) and that v is a distributional solution, with local Lipschitz continuity, to equation (2.4). Under the assumption that 0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) , if u ≤ v on ∂ ⁡ Ω , we then conclude u ≤ v in Ω. Analogously, the claim about viscosity supersolution u ~ and locally Lipschitz continuous distributional solution v ~ to - div ⁡ A ⁢ ( x , D ⁢ v ~ ) = - ε holds true as well. If u ~ ≥ v ~ on ∂ ⁡ Ω , then u ~ ≥ v ~ in Ω.

Theorem 4.4.

When the conditions that 0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) and q p ≤ 1 + 1 n are in force, we can infer that the viscosity supersolutions to equation (1.1) are A H ⁢ ( ⋅ ) -superharmonic functions.

Proof.

Let u be a viscosity supersolution. To verify u is an 𝒜 H ⁢ ( ⋅ ) -superharmonic function, through Definition 2.4, it is enough to prove that u satisfies the comparison principle for distributional solutions to (1.1). To this aim, we suppose h ∈ C ⁢ ( D ¯ ) is such a distributional solution of (1.1) that u ≥ h on ∂ ⁡ D , where D ⊂⊂ Ω is a subdomain. According to the lower semicontinuity of u, we can find that for each δ > 0 and a smooth domain D ′ ⊂⊂ D such that h ≤ u + δ in D ∖ D ′ .

Now we are going to show h ≤ u + δ in D ′ . If this is true, then we arrive at h ≤ u + δ in D, and further, by letting δ → 0 , we reach the conclusion that h ≤ u in D.

Consider the Dirichlet problem

{ - div ⁡ A ⁢ ( x , D ⁢ h ε ) = - ε in ⁢ D ′ , h ε - h ∈ W 0 1 , H ⁢ ( ⋅ ) ⁢ ( D ′ ) .

Denote the distributional solution by h ε . Then h ε is locally Lipschitz continuous in D ′ (see [4, 13, 19]). Moreover, owing to the smoothness of D ′ , u + δ ≥ h ε on ∂ ⁡ D ′ . Via Lemma 4.3 and noting that u + δ is also a viscosity supersolution of (1.1), we apply Lemma 4.2 to infer u + δ ≥ h in D ′ . We now complete the proof. ∎

In summary, through merging Theorems 4.1 and 4.4, we obtain the equivalence of the viscosity solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions.

Corollary 4.5.

Let 0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) and q p ≤ 1 + 1 n . Viscosity solutions to equation (1.1) coincide with A H ⁢ ( ⋅ ) -harmonic functions.

Remark 4.6.

In this section, we only prove the equivalence between the viscosity supersolutions and 𝒜 H ⁢ ( ⋅ ) -superharmonic functions. Indeed, according to the Definitions 2.4 and 2.5, the case of equivalence between viscosity subsolutions and 𝒜 H ⁢ ( ⋅ ) -subharmonic functions is similar.

As has been stated above, by means of Corollaries 3.6 and 4.5, we shall gain the desired relationship between viscosity solutions and distributional solutions of equation (1.1). We end this section by stating the key result of this article as follows.

Theorem 4.7.

Under the hypotheses that

0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) 𝑎𝑛𝑑 q p ≤ 1 + 1 n ,

in a given domain, the (locally) bounded viscosity solutions and the (locally) bounded distributional solutions of equation (1.1) are the same.

Remark 4.8.

We would like to point out that the condition imposed on a ⁢ ( x ) is very natural because, to guarantee that the notion of viscosity solution is admissible, we should differentiate

| D ⁢ φ ⁢ ( x ) | p - 2 ⁢ D ⁢ φ ⁢ ( x ) + a ⁢ ( x ) ⁢ | D ⁢ φ ⁢ ( x ) | q - 2 ⁢ D ⁢ φ ⁢ ( x )

in the classical sense. Here φ ⁢ ( x ) ∈ C 2 is the test function.

5 The comparison principle

In this section, we present the proof of Lemma 4.3 that is the key ingredient to prove Theorem 4.4. We rewrite Lemma 4.3 as follows for the readability and completeness.

Proposition 5.1.

Let 0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) . Assume that u is a viscosity subsolution to - div ⁡ A ⁢ ( x , D ⁢ u ) = 0 in Ω and that v is a locally Lipschitz continuous distributional solution of - div ⁡ A ⁢ ( x , D ⁢ v ) = ε ( ε > 0 ) in Ω. If u ≤ v on ∂ ⁡ Ω , then we could infer u ≤ v in Ω.

Proof.

Let us prove it by contradiction. If the claim is not true, then we have 0 < sup Ω ⁡ ( u - v ) = : u ⁢ ( x 0 ) - v ⁢ ( x 0 ) for some x 0 ∈ Ω .

Consider the function

Ψ j ( x , y ) : = u ( x ) - v ( y ) - Φ j ( x , y ) ,

where Φ j ⁢ ( x , y ) = j s ⁢ | x - y | s with s > max ⁡ { 2 , p p - 1 , q q - 1 } . Let ( x j , y j ) ∈ Ω ¯ × Ω ¯ be the maximum point of Ψ j ⁢ ( x , y ) , i.e. Ψ j ⁢ ( x j , y j ) = max Ω ¯ × Ω ¯ ⁡ Ψ j ⁢ ( x , y ) . We can verify that ( x j , y j ) ∈ Ω × Ω for j large enough and ( x j , y j ) → ( x 0 , x 0 ) by sending j → ∞ (see [15, Lemma 7.2]).

Now we first show x j ≠ y j . By the definition of ( x j , y j ) , we get

u ⁢ ( x j ) - v ⁢ ( y ) - Φ j ⁢ ( x j , y ) ≤ u ⁢ ( x j ) - v ⁢ ( y j ) - Φ j ⁢ ( x j , y j ) for all ⁢ y ∈ Ω .

Then

v ⁢ ( y ) ≥ - Φ j ⁢ ( x j , y ) + Φ j ⁢ ( x j , y j ) + v ⁢ ( y j ) = : ϕ j ⁢ ( y ) .

That is to say, ϕ j ⁢ ( y ) touches v ⁢ ( y ) at y j from below. Thus, by Lemma 2.9,

(5.1) lim sup y → y j y ≠ y j ⁡ ( - div ⁡ A ⁢ ( y , D ⁢ ϕ j ⁢ ( y ) ) ) ≥ ε .

By expanding div ⁡ A ⁢ ( y , D ⁢ ϕ j ⁢ ( y ) ) , we obtain

Here we denote the trace of matrix M by tr ⁡ M . Furthermore,

D ⁢ ϕ j ⁢ ( y ) = - D ⁢ Φ j ⁢ ( x j , y ) = j ⁢ | x j - y | s - 2 ⁢ ( x j - y ) ,

D 2 ⁢ ϕ j ⁢ ( y ) = - D 2 ⁢ Φ j ⁢ ( x j , y ) = - j ⁢ | x j - y | s - 2 ⁢ I - j ⁢ ( s - 2 ) ⁢ | x j - y | s - 4 ⁢ ( x j - y ) ⊗ ( x j - y ) ,

where ξ ⊗ ξ denotes the matrix with entries ξ i ⁢ ξ j for ξ ∈ ℝ n . Therefore, we have

- div ⁡ A ⁢ ( y , D ⁢ ϕ j ⁢ ( y ) ) = j p - 1 ⁢ [ n + s - 2 + ( p - 2 ) ⁢ ( s - 1 ) ] ⁢ | x j - y | ( p - 2 ) ⁢ ( s - 1 ) + ( s - 2 ) + a ⁢ ( y ) ⁢ j q - 1 ⁢ [ n + s - 2 + ( q - 2 ) ⁢ ( s - 1 ) ] ⁢ | x j - y | ( q - 2 ) ⁢ ( s - 1 ) + ( s - 2 ) - j q - 1 ⁢ | x j - y | ( q - 2 ) ⁢ ( s - 1 ) + ( s - 2 ) ⁢ ( x j - y ) ⋅ D ⁢ a ⁢ ( y ) .

Note that, due to s > max ⁡ { 2 , p p - 1 , q q - 1 } ,

( p - 2 ) ⁢ ( s - 1 ) + ( s - 2 ) = s ⁢ ( p - 1 ) - p > 0 ,

( q - 2 ) ⁢ ( s - 1 ) + ( s - 2 ) = s ⁢ ( q - 1 ) - q > 0 .

Thus, if x j = y j , then lim y → y j - div ⁡ A ⁢ ( y , D ⁢ ϕ j ⁢ ( y ) ) = 0 , which contradicts (5.1).

Via the theorem of sums in [15], for each μ > 0 , there are symmetric n × n matrices X : = X ( μ ) and Y : = Y ( μ ) such that

( D x ⁢ Φ j ⁢ ( x j , y j ) , X ) ∈ J ¯ 2 , + ⁢ u ⁢ ( x j ) , ( - D y ⁢ Φ j ⁢ ( x j , y j ) , Y ) ∈ J ¯ 2 , - ⁢ v ⁢ ( y j )

and

( X - Y ) ≤ D 2 ⁢ Φ j ⁢ ( x j , y j ) + 1 μ ⁢ ( D 2 ⁢ Φ j ⁢ ( x j , y j ) ) 2 ,

where

D 2 ⁢ Φ j ⁢ ( x j , y j ) = ( D x ⁢ x ⁢ Φ j ⁢ ( x j , y j ) D x ⁢ y ⁢ Φ j ⁢ ( x j , y j ) D y ⁢ x ⁢ Φ j ⁢ ( x j , y j ) D y ⁢ y ⁢ Φ j ⁢ ( x j , y j ) ) .

By direct calculation,

D 2 ⁢ Φ j ⁢ ( x j , y j ) = : ( B - B - B B ) ,

where B = j ⁢ | z j | s - 2 ⁢ I + ( s - 2 ) ⁢ j ⁢ | z j | s - 4 ⁢ z j ⊗ z j with z j = x j - y j . After manipulation,

(5.2) ( X - Y ) ≤ ( B - B - B B ) + 2 μ ⁢ ( B 2 - B 2 - B 2 B 2 )

with B 2 = j 2 ⁢ | z j | 2 ⁢ s - 4 ⁢ I + j 2 ⁢ ( s - 2 ) ⁢ s ⁢ | z j | 2 ⁢ s - 6 ⁢ z j ⊗ z j . It is straightforward to derive that X ≤ Y , i.e. 〈 ( X - Y ) ⁢ ξ , ξ 〉 ≤ 0 for any ξ ∈ ℝ n . In the rest of the proof, we take a special value of μ, i.e., μ = j .

We now give some notation to be used below. Set

M ( x , ξ ) : = a ( x ) | ξ | q - 2 ( I + ( q - 2 ) ξ | ξ | ⊗ ξ | ξ | ) .

And define

F 1 ( ξ , X ) : = - | ξ | p - 2 ( tr X + ( p - 2 ) 〈 X ξ | ξ | , ξ | ξ | 〉 ) ,

F 2 ( x , ξ , X ) : = - tr ( M ( x , ξ ) X ) ,

F 3 ( x , ξ ) : = - | ξ | q - 2 ξ ⋅ D a ( x ) ,

for x ∈ Ω , ξ ∈ ℝ n and X being a symmetric n × n matrix. Then we can easily check that

- div ⁡ ( | D ⁢ φ | p - 2 ⁢ D ⁢ φ + a ⁢ ( x ) ⁢ | D ⁢ φ | p - 2 ⁢ D ⁢ φ ) = F 1 ⁢ ( D ⁢ φ , D 2 ⁢ φ ) + F 2 ⁢ ( x , D ⁢ φ , D 2 ⁢ φ ) + F 3 ⁢ ( x , D ⁢ φ ) = : F ⁢ ( x , D ⁢ φ , D 2 ⁢ φ ) .

Let

η j : = D x Φ j ( x j , y j ) = - D y Φ j ( x j , y j ) = j | x j - y j | s - 2 ( x j - y j ) .

Notice that η j ≠ 0 , which is of great importance. It is well known that u is a viscosity subsolution of (1.1), and v is a viscosity supersolution of (2.4), so we can arrive at F ⁢ ( x j , η j , X j ) ≤ 0 and F ⁢ ( y j , η j , Y j ) ≥ ε , where X j : = X ( j ) and Y j : = Y ( j ) . Subtracting these two inequalities yields

(5.3) ε ≤ F ⁢ ( y j , η j , Y j ) - F ⁢ ( x j , η j , X j ) = F 1 ⁢ ( η j , Y j ) - F 1 ⁢ ( η j , X j ) + F 2 ⁢ ( y j , η j , Y j ) - F 2 ⁢ ( x j , η j , X j ) + F 3 ⁢ ( y j , η j ) - F 3 ⁢ ( x j , η j ) .

We proceed by estimating the three terms on the right-hand side in (5.3), respectively. First, we observe F 1 ⁢ ( ξ , X ) is monotone decreasing with respect to the X-variable, so by X j ≤ Y j , we get

(5.4) F 1 ⁢ ( η j , Y j ) - F 1 ⁢ ( η j , X j ) ≤ 0 .

Next, we evaluate

(5.5) F 3 ⁢ ( y j , η j ) - F 3 ⁢ ( x j , η j ) = | η j | q - 2 ⁢ η j ⋅ ( D ⁢ a ⁢ ( x j ) - D ⁢ a ⁢ ( y j ) ) ≤ | η j | q - 1 ⁢ | D ⁢ a ⁢ ( x j ) - D ⁢ a ⁢ ( y j ) | → 0 as ⁢ j → ∞ .

Actually, in the last display, we have utilized the facts that | η j | ≤ C , a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) and x j , y j → x 0 . Finally, we deal with the second term F 2 ⁢ ( y j , η j , Y j ) - F 2 ⁢ ( x j , η j , X j ) . First observe that, from the matrix inequality (5.2), it follows that, when μ = j ,

(5.6) X j ⁢ ξ ⋅ ξ - Y j ⁢ η ⋅ η ≤ j ⁢ [ ( s - 1 ) ⁢ | z j | s - 2 + 2 ⁢ ( s - 1 ) 2 ⁢ | z j | 2 ⁢ ( s - 2 ) ] ⁢ | ξ - η | 2 .

Secondly, We can readily find the matrix M ⁢ ( x , ξ ) > 0 as a ⁢ ( x ) > 0 so that it possesses a matrix square root denoted by M 1 2 ⁢ ( x , ξ ) . We denote the l-th column of M 1 2 ⁢ ( x , ξ ) as M l 1 2 ⁢ ( x , ξ ) . Now we estimate

(5.7) F 2 ⁢ ( y j , η j , Y j ) - F 2 ⁢ ( x j , η j , X j ) = tr ⁡ ( M ⁢ ( x j , η j ) ⁢ X j ) - tr ⁡ ( M ⁢ ( y j , η j ) ⁢ Y j ) = tr ⁡ ( M 1 2 ⁢ ( x j , η j ) ⁢ M 1 2 ⁢ ( x j , η j ) ⁢ X j ) - tr ⁡ ( M 1 2 ⁢ ( y j , η j ) ⁢ M 1 2 ⁢ ( y j , η j ) ⁢ Y j ) = ∑ l = 1 n 〈 M l 1 2 ⁢ ( x j , η j ) , X j ⁢ M l 1 2 ⁢ ( x j , η j ) 〉 - ∑ l = 1 n 〈 M l 1 2 ⁢ ( y j , η j ) , Y j ⁢ M l 1 2 ⁢ ( y j , η j ) 〉 ≤ C ⁢ j ⁢ | x j - y j | s - 2 ⁢ ∥ M 1 2 ⁢ ( x j , η j ) - M 1 2 ⁢ ( y j , η j ) ∥ 2 2 ≤ C ⁢ j ⁢ | x j - y j | s - 2 ( λ min ⁢ ( M 1 2 ⁢ ( x j , η j ) ) + λ min ⁢ ( M 1 2 ⁢ ( y j , η j ) ) ) 2 ⁢ ∥ M ⁢ ( x j , η j ) - M ⁢ ( y j , η j ) ∥ 2 2 ,

where the penultimate inequality is obtained by (5.6) and the last inequality is derived from the local Lipschitz continuity of M ↦ M 1 2 (see [26, page 410]). Here λ min ⁢ ( M ) denotes the smallest eigenvalue of a symmetric n × n matrix M.

On the other hand,

∥ M ⁢ ( x j , η j ) - M ⁢ ( y j , η j ) ∥ 2 = ∥ a ⁢ ( x j ) ⁢ | η j | q - 2 ⁢ ( I + ( q - 2 ) ⁢ η j | η j | ⊗ η j | η j | ) - a ⁢ ( y j ) ⁢ | η j | q - 2 ⁢ ( I + ( q - 2 ) ⁢ η j | η j | ⊗ η j | η j | ) ∥ 2 ≤ n ⁢ | η j | q - 2 ⁢ | a ⁢ ( x j ) - a ⁢ ( y j ) | + | q - 2 | ⁢ | η j | q - 2 ⁢ | a ⁢ ( x j ) - a ⁢ ( y j ) | .

Additionally, we can derive

λ min ⁢ ( M 1 2 ⁢ ( x j , η j ) ) = ( λ min ⁢ ( M ⁢ ( x j , η j ) ) ) 1 2 ≥ min ⁡ { 1 , q - 1 } ⁢ | η j | q - 2 2 ⁢ a ⁢ ( x j ) .

Furthermore,

λ min ⁢ ( M 1 2 ⁢ ( x j , η j ) ) + λ min ⁢ ( M 1 2 ⁢ ( y j , η j ) ) ≥ min ⁡ { 1 , q - 1 } ⁢ | η j | q - 2 2 ⁢ ( a ⁢ ( x j ) + a ⁢ ( y j ) ) .

Hence, employing the fact that 0 < a ⁢ ( x ) ∈ C 1 ⁢ ( Ω ) and these previous estimates, inequality (5.7) becomes

(5.8) F 2 ⁢ ( y j , η j , Y j ) - F 2 ⁢ ( x j , η j , X j ) ≤ C ⁢ j ⁢ | x j - y j | s - 2 min ⁡ { 1 , q - 1 } ⁢ | η j | q - 2 ⁢ ( a ⁢ ( x j ) + a ⁢ ( y j ) ) 2 ⁢ C ⁢ ( n , q ) ⁢ | η j | 2 ⁢ ( q - 2 ) ⁢ | a ⁢ ( x j ) - a ⁢ ( y j ) | 2 = C 1 ⁢ | η j | q - 2 ⁢ j ⁢ | x j - y j | s - 2 ⁢ | a ⁢ ( x j ) - a ⁢ ( y j ) | 2 ( a ⁢ ( x j ) + a ⁢ ( y j ) ) 2 = C 1 ⁢ ( a ⁢ ( x j ) + a ⁢ ( y j ) ) - 2 ⁢ | η j | q - 1 ⁢ j ⁢ | x j - y j | s - 2 j ⁢ | x j - y j | s - 1 ⁢ | D ⁢ a ⁢ ( ζ j ) | 2 ⁢ | x j - y j | 2 ≤ C 2 ⁢ ( a ⁢ ( x j ) + a ⁢ ( y j ) ) - 2 ⁢ | η j | q - 1 ⁢ | x j - y j | → 0 as ⁢ j → ∞ .

Here the reason why we imposed a ⁢ ( x ) > 0 is that if a ⁢ ( x 0 ) = 0 , we cannot justify

( a ⁢ ( x j ) + a ⁢ ( y j ) ) - 2 ⁢ | x j - y j | → 0 as ⁢ j → ∞ .

Finally, merging (5.3), (5.4), (5.5) and (5.8), we deduce

ε ≤ F ⁢ ( y j , η j , Y j ) - F ⁢ ( x j , η j , X j ) ≤ 0

by letting j → ∞ . It is a contradiction. We now complete the proof. ∎

Communicated by Juha Kinnunen

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11671111

Award Identifier / Grant number: 12071098

Funding statement: This work was supported by the National Natural Science Foundation of China (No. 11671111, 12071098).

Acknowledgements

The authors wish to thank the anonymous reviewers for many valuable comments and suggestions to improve the manuscript.

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Received: 2020-06-09

Revised: 2020-09-16

Accepted: 2020-09-17

Published Online: 2020-10-17

Published in Print: 2022-10-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Equivalence between distributional and viscosity solutions for the double-phase equation

Abstract

1 Introduction

2 Preliminaries

2.1 Function spaces

Lemma 2.1.

Lemma 2.2.

2.2 Notions of solutions

Definition 2.3 (Distributional solution).

Definition 2.4 ( A H ⁢ ( ⋅ ) -harmonic function).

Definition 2.5 (Viscosity solution).

Remark 2.6.

2.3 Auxiliary results

Proposition 2.7 (Comparison principle).

Proof.

Proposition 2.8 (Caccioppoli type inequality).

Proof.

Lemma 2.9.

Proof.

3 Distributional solutions coincide with 𝒜 H ⁢ ( ⋅ ) -harmonic functions

Theorem 3.1.

Proof.

Lemma 3.2.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Theorem 3.5.

Proof.

Corollary 3.6.

4 Equivalence between viscosity solutions and 𝒜 H ⁢ ( ⋅ ) -harmonic functions

Theorem 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Theorem 4.4.

Proof.

Corollary 4.5.

Remark 4.6.

Theorem 4.7.

Remark 4.8.

5 The comparison principle

Proposition 5.1.

Proof.

Acknowledgements

References

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