A Comparison Principle for Some Types of Elliptic Equations

as proven in 1 for convex operator and in 2 for uniformly elliptic ones. Some years later, Jensen in 3 , using his known approximation functions, proved such a kind of principle between a viscosity subsolution and a viscosity supersolution, both in W1,∞ Ω , for operators which grow linearly in the gradient term and could be uniformly elliptic and nonincreasing in the t variable or degenerate elliptic and decreasing in t. In the same time, in 4 , Trudinger was able to compare solutions which are C Ω ∩ C0,1 Ω and


Introduction
The aim of this paper is to study some fully nonlinear uniformly elliptic equations, where the gradient term could be noncontinuous and growing like some BMO functions.Given an equation F X, p, t, x 0, in the case of classical solutions, the comparison principle states the following: i let u, v ∈ C 2 Ω be, respectively, a sub and a supersolution of the equation, if u ≤ v on ∂Ω, then u ≤ v in Ω, as proven in 1 for convex operator and in 2 for uniformly elliptic ones.Some years later, Jensen in 3 , using his known approximation functions, proved such a kind of principle between a viscosity subsolution and a viscosity supersolution, both in W 1,∞ Ω , for operators which grow linearly in the gradient term and could be uniformly elliptic and nonincreasing in the t variable or degenerate elliptic and decreasing in t.In the same time, in 4 , Trudinger was able to compare solutions which are C Ω ∩ C 0,1 Ω and C Ω ∩ C 0,α Ω .
Then Jensen et al. 5 extended these results considering a zero order term and sub-and supersolutions which are only BUC Ω .Soon after, Ishii in 6 and Jensen in 3 ISRN Mathematical Analysis independently proved a comparison principle for only continuous bounded functions, where in the first paper the author considers continuous degenerate operators of Isaacs type, which grow linearly in the p variable, while the second concerns uniformly elliptic operators which are Lipschitz in the gradient term and nonincreasing in t.
Then Ishii and Lions in 7 obtain this kind of result between bounded viscosity suband supersolution for strictly elliptic operators which grow quadratically in the p variable and are nonincreasing in t.In the same article, these two authors weakened the structure conditions on F and compared continuous bounded functions where at least one has to be locally Lipschitz; then, this result was sharped by Crandall in 8 see also 9 .
Crandall et al., in their pioneering paper 10 were able to prove such a kind of results between viscosity solutions for degenerate elliptic equations, nonincreasing in t, extending the results obtained before see also 11,12 .Then Koike and Takahashi in their work 13 compared L p -viscosity sub-and supersolutions, when at least one of them is L p -strong.
In the last years, Bardi and Mannucci in 14 prove a comparison principle for fully nonlinear degenerate elliptic equations that satisfy some conditions of partial nondegeneracy, with linear growth in the gradient term see also 15 and Sirakov,16 , has the same result for fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type with unbounded ingredients and the most quadratic growth in p.
Also, it is interesting to mention the series of papers by Birindelli and Demengel, 17-19 , where they investigate on singular fully nonlinear equations.
The paper is organized as follows: in the first section some auxiliary results are stated; the second one is characterized by an overview on inf and supconvex envelope; the proof of the main result is given in the third section; finally, in the last one, some examples which justify the interest on this kind of operators are listed.

Preliminaries and Auxiliary Results
First of all, it is useful to give some definitions.We say that P is a paraboloid of opening M when where M is a positive constant, l 0 is a constant, and l x is a linear function.P is a convex paraboloid if there is the sign in 2.1 , concave otherwise.Given two functions u and v on an open set A, v touches u from the above in x 0 ∈ A when In this case, one could also say that u touches v from below.
Consider the following: From 3 , we have the following.
Lemma 2.1.Assume that w ∈ C Ω ∩ W 1,∞ Ω and that D 2 λ w ≥ −K 0 (in the sense of distribution) for all direction λ.If w has an interior maximum then there exist two constants c 0 > 0 and δ 0 > 0 such that Then some lemmas from 3 are needed for the sequel.
Then there exists a function M ∈ L 1 Ω; S n and a matrix valued measure Γ ∈ M Ω; S n such that 2 Γ is singular with respect to Lebesgue measure, 3 Γ(S) is positive semidefinite for all Borel subsets S di Ω, If w has an interior maximum then there exists a constant δ 0 > 0 such that for D 2 w M Γ (as in the previous lemma) as

ISRN Mathematical Analysis
Finally, set Q x, y distance x, y , graph u , 2.9

Sup and Inf Convex Envelope
The aim of this paper is to consider equations of the following form: where F and H are such that the following hold: 2 there exist two constants c 0 and c 1 such that for all M ≥ N and p, q, t ∈ R n × R n × R; 3 F M, p, t ≤ F M, p, s for all t > s and M, p ∈ S n × R n ; 4 there exists a positive function g on R n × R n such that H p − H q ≤ γg p, q p − q 3.3 for γ > 0, all p, q ∈ R n , where g has to satisfy the following then there exists a constant δ 0 > 0 such that for 0 < δ < δ 0 and M, N are the functions defined in Lemma 2.2.
We say that the structure condition holds if and only if 2.1 -4.4 are fulfilled.Define, as in 20 , the convex envelope of a function.
Definition 3.1.Let Ω be a bounded domain of R n , A a subset of Ω such that A ⊂ Ω, u ∈ C Ω .We call, respectively, sup and inf convex envelope of u as in the following objects: x − y 2 .

3.5
Now it is possible to give some properties of the sup convex envelope, noting that similar ones hold for the inf convex envelope.
Theorem 3.3 see 20 .Let A be an open set such that A ⊂ Ω, we have 2 for all x 0 ∈ A there exists a concave paraboloid of opening 2/ε which touches u ε from below in x 0 ∈ A.
In particular u ε is pointwise differentiable to the second order for almost every x ∈ A.
Before going further, it is useful to give the definition of viscosity solution.
Definition 3.4.A viscosity subsolution of F D 2 u, Du, u, x H Du 0 is a function u ∈ USC Ω such that ∀x 0 ∈ Ω and ∀φ ∈ C 2 Ω .If u − φ has a local maximum in x 0 , then the following holds A continuous function u is a viscosity solution of F D 2 u, Du, u, x H Du 0 if and only if u is both a viscosity sub-and supersolution.

ISRN Mathematical Analysis
Remember the following.i USC Ω is the set of upper semicontinuous function u in Ω such that u < ∞.
ii LSC Ω is the set of lower semicontinuous function v in Ω such that v > −∞.Now, to complete this section, note that, as stated in the following theorem see 5, 20 , the convex envelope of a viscosity solution is a viscosity solution of the same equation.

Comparison Principle
Now it is possible to prove the comparison principle Proof.Suppose that the contrary holds true as u < v in Ω.

4.3
Define v v ε − ε and u u − ε ε.By Theorem 3.5, v and u are, respectively, viscosity sub-and supersolution of the previous equation.By the properties of sup and inf convex envelope we know that for all direction λ

4.9
Moreover, by the definition of ζ δ , we have Applying the definition of viscosity subsolution and supersolution, it is possible to write

Examples
It

Theorem 3 . 5 .
Let u, v ∈ C Ω be bounded functions which are, respectively, viscosity subsolution and supersolution of F M, p, t H p 0. If F is uniformly elliptic and nonincreasing then there exist two Lipschitz continuous and bounded functions u * e v * and an open set A 1 , with A 1 ⊂ A, such that u * is semiconvex, v * is semiconcave on A 1 which are, respectively, viscosity subsolution and supersolution of F M, p, t H p 0 in A 1 .

Theorem 4 . 1
Comparison Principle .Let u, v ∈ C Ω .Assume that u is a viscosity supersolution and v is a viscosity subsolution of F D 2 w x , Dw x , Suppose that u ≥ v on ∂Ω.If F and H satisfy the structure condition, then u ≥ v in Ω. 4.2
Cg D u x , D v x |D w x | for every x ∈ ζ δ \ E 2 , 4.16 where meas E 2 0.Then, for E E 1 ∪ E 2 , since meas E 0, we have Remark 4.2.Note that in the last line it is essential that D u L ∞ and D v L ∞ are finite.
− ≥ F M x , D v x , v x H D v x < F M − x , D u x , u x H D u x for almost every x ∈ ζ δ \ E ⊂ ζ δ ,4.17 which contradicts 4.3 .So u ≥ v in Ω.