FINSLER INFINITY SUPERHARMONIC FUNCTIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We investigate a simple proof on properties of a non-negative Finsler infinity superharmonic function such as positivity, Harnack inequality, Liouville property and Lipschitz continuity using Finsler distance function. We also present Hopf boundary point lemma for a Finsler infinity subharmonic function.


INTRODUCTION
Let Ω ⊂ R n be open and connected set. In this paper we have presented properties of nonnegative Finsler infinity superharmonic function in Ω; that is properties of a non-negative viscosity supersolution of The normalized Finsler infinity Laplacian operator ∆ N F;∞ is a nonlinear, singular and degenerate elliptic. It is defined by (1.2) ∆ N F;∞ u(x) = D 2 uDF(Du(x)), DF(Du(x)) , where F is a Finsler minkowski norm in R n . The Finsler minkowski norm F in R n is defined as follows: Let F : R n → R + 0 be a function satisfying the following properties.
A function F : R n → R + 0 that satisfies regularity, positive homogeneity and strong convexity is called a Finsler-Minkowski norm on R n . We can see that [2,11]). The proof of the following Lemma can be found in [2]. Lemma 1.1. Let F be a Finsler-Minkowski norm. The following properties hold.
Equality holds if and only if w = κξ for some κ ≥ 0.
(3) The map FDF : R n → R n is invertible and For the proof of Lemma (1.3) we refer the reader to [5,12].
Lemma (1.3) (4) and properties of F * gives the following remark.
The following notations have been used.
B(x, r) = Euclidean ball center at x and radius r > 0 ·, · = The usual inner product We organized this paper as follows. In section two we state main results of the paper. Section three is devoted the definition of viscosity solution and proof of Lemma (2.1). In section four we give the proofs of Theorems (2.2),(2.3),(2.4),(2.5) and (2.6), respectively.

STATEMENT OF MAIN RESULTS
Lemma 2.1. Let u be non-negative Finsler infinity superharmonic function in Ω. If u(p) > 0,

Theorem 2.2. (Positivity) Let u be non-negative Finsler infinity superharmonic function in Ω.
If u is positive somewhere in Ω, then u is positive everywhere in Ω.

Theorem 2.3. (Harnack inequality) Let u be non-negative Finsler infinity superharmonic func-
Let u be a Finsler infinity harmonic function in Ω such that u(y) = inf Ω u and u(x 0 ) > u(y). Then u satisfies

VISCOSITY SOLUTION
In this section we give the definition of viscosity solution to problem (1.1) [11]. The lower and upper Finsler infinity Laplacian of a twice differentiable function ϕ at x 0 ∈ Ω are respectively denoted by ∆ − F;∞ ϕ(x 0 ) and ∆ + F;∞ ϕ(x 0 ). Which are defined by (1) A function u ∈ USC(Ω, R) is called a viscosity subsolution of (1.1) if for every function ϕ ∈ C 2 (Ω, R) and point x 0 ∈ Ω such that u ≺ x 0 ϕ we have In this case we write −∆ N F;∞ ϕ(x 0 ) ≤ 0. (2) A function u ∈ USC(Ω, R) is called a viscosity supersolution of (1.1) if for every function ϕ ∈ C 2 (Ω, R) and point x 0 ∈ Ω such that u 0 ϕ we have In this case we write −∆ N F;∞ u(x 0 ) ≥ 0. (3) A function u ∈ C(Ω, R) is called a viscosity solution of (1.1) if u is both a viscosity subsolution and supersolution of (1.1).
A viscosity subsolution of (1.1) is called Finsler infinity subharmonic where as a viscosity supersolution of (1.1) is called Finsler infinity superharmonic.
Proof. For x = x 0 , we observe that and We note that D(d α (x)) = o for x = x 0 , and DF(DF for any x ∈ R n \{x 0 }. We know that D 2 F * (x) x, x = 0 for any x ∈ R n and hence by Remark (1.4) we obtain We have also d(p) = r − F * (0) = r and thus v(p) = 1.
Let w = u c − v, for a fixed c. Then Thus w has a negative minimum in B F (p, r). This minimum value occurs at p. We show this by contradiction. Suppose there is a point x c = p such that Thus w α (p) = u c (p) − 1 < 0 and on ∂ B F (p, r), w α (x) ≥ 0. We can choose α sufficiently close to 1 such that the point of minimum of w α , denoted by x c,α = p and w α (x c,α ) < w α (p) = u c (p) − 1 < 0. This indicates x c,α / ∈ ∂ B F (p, r).

Again now
has a negative minimum at x c,α = p. We notice that v α (x) is C 2 near x c,α and as u is Finsler infinity superharmonic, we have Which is a contradiction. Hence the minimum of w occurs at p. r)and for all c < k.
we obtain Since Ω is open, there is a ball B(x, δ ) contained in Ω for some δ > 0. We observe that |x − y| ≤ 1

Proof of Theorem
nothing is done. So we assume z = x. In this case, by Lemma (2.1) we have Thus x ∈ S, that is S is closed. We see that S is both open and closed. It follows that S = Ω.
Therefore; u is positive in Ω.
Proof of Theorem (2.3). If u = 0 in Ω, then (2.1) holds true. So we assume u(x) > 0 for some For all x ∈ B F p, αr β κ , Equation (4.1) becomes Taking infinimum of (4.2) over B F p, αr β κ we get Take x ∈ B F p, αr β κ , p ∈ B F (x, r κ ). Let R be a mid point point of the segment joining the points p and x. Let F * (x − P) = l. In B F (x, l), From (4.3), (4.4) and (4.5) we obtain Taking the supremum of (4.6) over B F p, αr β κ we get Proof of Remark (2.4). Take two distinct points x and z in R n . Consider the ball B F (z, r) with and d(z) = d(x) + F * (z − x) = r. Letting r → ∞ we get u(z) ≤ u(x). Interchanging the roles of x and z we get the reverse inequality. Therefore; u is constant in R n .