In this paper, we investigate the convexity property of viscosity solutions to a homogeneous normalized Finsler infinity Laplacian equation. Weak and strong forms for convexity property have been addressed.

1. Introduction

In this article, we study the characterization of Finsler infinity harmonic function, i.e., a viscosity solution of singular and degenerate elliptic equation

()

where

is a bounded domain and

()

in terms of maximum and minimum of u on the boundary of a Finsler ball B_F(x₀, r) ⊂ Ω as a function of r. Recently, the Finsler infinity Laplacian has the attention of researcher and thus, a lot of articles have been appeared, for instance [1–7]. Our problem was investigated in [8] when u ∈ C(Ω) is the infinity harmonic function, i.e., u ∈ C(Ω) is a viscosity solution of

()

The operator Δ_∞u = 〈D²u Du, Du〉 is a special case of the Finsler infinity Laplacian operator

()

when F(ξ) = |ξ|. Viscosity solution and its characterization of (3) have been exhaustively investigated. We may lead the reader directly to [9–14]. The operator

or Δ_F;∞u is a type of Aronsson operator

()

which was proposed in 1960s [15–17]. It was applied in recent papers [18, 19]. Here,

is a function, D_xG(Du(x), u(x), x) is the gradient of the map x⟶G(Du(x), u(x), x), and G_p is the gradient of G(p, s, x) with respect to p. If

becomes

, and if

becomes Δ_F;∞u(x).

We organize the paper as follows. In Section 2, we recall Finsler–Minkowski norm, and we state our main results. In Section 3, we discuss about viscosity solution of (1). Finally, we prove our main results in Section 4

Here, we have used the following notations.

•
O denotes zero vector in .
•
∂Ω denotes the boundary of Ω.
•
and r > 0.
•
and r > 0.
•
dist_F(x₀, ∂Ω) = min_x∈∂Ω F^∗(x − x₀), x₀ ∈ Ω.
•
Du denotes the gradient of u ∈ C¹(Ω).
•
D²u represents the Hessian matrix of u ∈ C²(Ω).
•
sgn(t) = 1 if t > 0, and sgn(t) = −1 if t < 0.
•
x^t represents the transpose of .
•
〈x, y〉 represents the inner product of and .
•
x ⊗ y represents the tensor product of and .

2. Main Result

In this section, we state the main results of the paper. For its purpose, we begin by exploring the Finsler–Minkowski norm and its properties. Let F be a non-negative function defined in

satisfying

(1)
.
(2)
F is absolutely homogeneous of degree one, i.e.,
()
(3)
.

The function F is known as the Finsler–Minkowski norm. It is obvious that F(O) = 0. Differentiating both sides of F(tξ) = |t|F(ξ) for

and

with respect to ξ_i, i = 1, 2, …, n, and t, we get, respectively,

()

and

()

Differentiating both sides of (8) with respect to ξ_i, i = 1, 2, …, n, we obtain

()

By direct computation, we have

()

Using (8)–(10), we have

()

This follows that

. Let us now consider

. When ξ and ζ are linearly dependent, it is clear that F(ξ + ζ) ≤ F(ξ) + F(ζ). Let us consider the case where ξ and ζ are linearly independent. The set S = {ξ + λ(ξ − ζ) : 0 ≤ λ ≤ 1} does not contain the origin. By Taylor’s theorem, there is a point Θ ∈ S such that

()

The last inequity follows from the following argument. Let

such that ζ = η + κΘ with 〈DF(Θ), η〉 = 0 and

. We observe that

()

From (12), the inequality F(ξ) ≤ 1/2(F(ξ + ζ) + F(ξ − ζ)) is obtained. Taking ξ + ζ = 2x and ξ − ζ = 2y results F(x + y) ≤ F(x) + F(y). We thus conclude

()

Consequently, F is a norm in the sense of functional analysis. Let

and |ω| = 1. Since

()

For any

and using the absolute homogeneity condition,

()

Using (8) and (16), we obtain

()

The dual norm F^∗(x) of the Finsler–Minkowski norm F(ξ) is given by

()

It is obvious that F^∗(O) = 0 and

and

. The supremum in (18) is indeed a maximum, i.e., it is attained at a particular point ξ with F(ξ) = 1. Thus, we may write it as

()

, then from (18), we have the inequality

()

Since F(O) = 0 and

, (20) can be written as

()

We now define a C¹ mapping x⟼F(x)DF(x) for each . It is not difficult to show this mapping is bijective and the Jacobean matrix D²(1/2F²(x)) is invertible. By the inverse function theorem, the inverse mapping F(x)DF(x)⟼x is . Since . We next prove the following lemma.

Lemma 1. Let . Then,

()

Proof 1. Let . There exists such that

()

Hence,

()

It follows that

()

Using (24) and (25) and homogeneity condition of F and F^∗, we obtain

()

and

()

Notice that F(F(ξ_x)DF^∗(x)) = F(ξ_x), and substituting (27) into (26), we get F^∗(x)DF(F(ξ_x)DF^∗(x)) = x. By applying the homogeneity condition of F and DF, we can find F(DF^∗(x)) = 1 and DF(DF^∗(x)) = x/F^∗(x).

Given , with the same techniques that have been applied in obtaining (26) and (27), there exists such that

()

and

()

We observe that

()

Thus, the mapping is invertible and

()

This shows the mapping is C¹. Since , we conclude that . Finally, by considering

()

where ξ = F(x)DF(x), we can show that D²F^∗(ξ)² > O. This shows F^∗ is a Finsler–Minkowski norm and thus satisfy all the properties that have been proved before. We state these properties as a remark as follows.

Remark 1. Let with x ≠ O. The following holds true.

(1)
〈DF^∗(x), y〉 ≤ F^∗(y). Equality holds if y = κx for κ ≥ 0.
(2)
.
(3)
〈D²F^∗(x), x〉 = O.
(4)
F^∗(y + z) ≤ F^∗(y) + F^∗(z).

Consequently, we have the following remark.

Remark 2. Let . Then,

()

Proof 2. Replacing y by −y in Remark 1 (1), we get

()

()

For x ≠ O, it is clear that DF^∗(x) ≠ O. Let such that w^tD²F^∗(DF^∗(x)) ≠ O. We see that

()

and

()

by Remark 1. Thus, we can find

such that 〈DF^∗(x), z〉 = 0. Take y = z/F^∗(z) and so F^∗(y) = 1. Consequently,

()

Let α = min{(1/F(ξ)) : |ξ| = 1} and β = max{(1/F(ξ)) : |ξ| = 1}. The Euclidean and Finsler–Minkowski norms in are related by

()

For a detail discussion on the Finsler–Minkowski norm, reader can refer [20]. We now define Finsler ball and its boundary by

()

and

()

respectively.

Let x₀ ∈ Ω and 0 < r < dist_F(x₀, ∂Ω). We denote

()

for u ∈ C(Ω). Our main results are the following.

Proposition 1. (Weak form)

(1)
Suppose that u ∈ C(Ω) is a viscosity solution of .Then, ρ(r) satisfies −ρ^″(r) < 0 in a viscosity sense.
(2)
Suppose that u ∈ C(Ω) is a viscosity solution of . Then, η(r) satisfies −η^″(r) > 0 in a viscosity sense.

Theorem 1. (Strong form)

(1)
A function u ∈ C(Ω) is a Finsler infinity subharmonic function if and only if ρ(r) is convex in (0, dist_F(x, ∂Ω)).
(2)
A function u ∈ C(Ω) is a Finsler infinity superharmonic function if and only if η(r) is concave in (0, dist_F(x, ∂Ω)).

3. Viscosity Solutions

In this section we discuss about viscosity solutions of (1) and their properties. We begin as follows. The lower and upper Finsler infinity Laplacian of a twice differentiable function ϕ at x^∗ ∈ Ω are, respectively, denoted by

and

. These are defined by

()

and

()

From (43) and (44), we conclude that

()

Definition 1.

(1)
A function is called a viscosity subsolution of (1) if for every function such that u − ϕ has local maximum at x^∗ ∈ Ω, we have
()
In this case, we write .
(2)
A function is called a viscosity supersolution of (1) if for every function such that u − ϕ has local minimum at x^∗ ∈ Ω, we have
()
In this case, we write .
(3)
A function is called a viscosity solution of (1) if u is both a viscosity subsolution and supersolution of (1).

A viscosity subsolution of (1) is called a Finsler infinity subharmonic function whereas a viscosity supersolution of (1) is called a Finsler infinity superharmonic function. A viscosity solution of (1) is called Finsler infinity harmonic functions.

Remark 3. If u is a viscosity subsolution (or supersolution) of (1), then −u is a viscosity supersolution (subsolution) of (1).

The following lemma is taken from [6].

Lemma 2. Let be a subdomain. For any and and C(x) = a + bF^∗(x − x₀).

(1)
A function u ∈ C(Ω) is a Finsler infinity subharmonic function in Ω iff implies .
(2)
A function u ∈ C(Ω) is a Finsler infinity superharmonic function in Ω iff implies .

4. Proofs

Proof of Proposition 1. We prove only (1) and (2) can be proved analogously. Suppose ϕ ∈ C²(0, dist_F(x₀, ∂Ω)) such that ρ − ϕ has local maximum at r₀ ∈ (0, dist_F(x₀, ∂Ω)). That is, there exists δ > 0 such that

()

Let such that . So, we have

()

This implies u − ϕ has local maximum at . By the hypothesis that u is the Finsler infinity subharmonic function, we have

()

For , direct computations show that

()

and

()

And hence,

()

by Lemma 1 and Remark 1. We now consider the following cases.

Case 1. .

Here, from (51), we have . We see that

()

Case 2. .

Using (51), we get . We observe that

()

Since , . Therefore, ρ(r) is a viscosity solution of −ρ^″(r) < 0.

Proof of Theorem 1. We prove that only (1) and (2) can be proved analogously. We first suppose u ∈ C(Ω) is the Finsler infinity subharmonic function. Let 0 < r₁ < r₂ < dist_F(x₀, ∂Ω). Define a cone function

()

where

()

We can easily see that

()

By Lemma 2 (i), we have

()

Thus, for any θ ∈ (0, 1) and F^∗(x − x₀) = θr₁ + (1 − θ)r₂, we have

()

This concludes ρ is convex. Conversely, suppose ρ is convex. For x₀ ∈ Ω and 0 < r < R < dist_F(x₀, ∂Ω),

()

Sending r⟶0, ρ(r)⟶u(x₀) and hence

()

Take

()

then equation (51) becomes

()

Clearly,

()

If we take

()

the result follows by Lemma 2.

Conflicts of Interest

The author declares no conflicts of interest.

Funding

This study did not receive any funding in any form.

Open Research

Data Availability Statement

No data were used to support the findings of this study.

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Convexity Property of Finsler Infinity Harmonic Functions

Abstract

1. Introduction

2. Main Result

3. Viscosity Solutions

4. Proofs

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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Convexity Property of Finsler Infinity Harmonic Functions

Abstract

1. Introduction

2. Main Result

3. Viscosity Solutions

4. Proofs

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

Related

Information