A Comparison Principle for Some Types of Elliptic Equations
In this paper a comparison principle between a continuous viscosity supersolution and a continuous viscosity subsolution is presented. The operator of interest is a fully nonlinear uniformly elliptic one with a gradient term which could be noncontinuous and grow like some BMO functions, as shown in the last section.
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 , in the case of classical solutions, the comparison principle states the following: (i)let be, respectively, a sub and a supersolution of the equation, if on , then in , as proven in  for convex operator and in  for uniformly elliptic ones.
Some years later, Jensen in , using his known approximation functions, proved such a kind of principle between a viscosity subsolution and a viscosity supersolution, both in , for operators which grow linearly in the gradient term and could be uniformly elliptic and nonincreasing in the variable or degenerate elliptic and decreasing in . In the same time, in , Trudinger was able to compare solutions which are and .
Then Jensen et al.  extended these results considering a zero order term and sub- and supersolutions which are only . Soon after, Ishii in  and Jensen in  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 variable, while the second concerns uniformly elliptic operators which are Lipschitz in the gradient term and nonincreasing in .
Then Ishii and Lions in  obtain this kind of result between bounded viscosity sub- and supersolution for strictly elliptic operators which grow quadratically in the variable and are nonincreasing in . In the same article, these two authors weakened the structure conditions on and compared continuous bounded functions where at least one has to be locally Lipschitz; then, this result was sharped by Crandall in  (see also ).
Crandall et al., in their pioneering paper  were able to prove such a kind of results between viscosity solutions for degenerate elliptic equations, nonincreasing in , extending the results obtained before (see also [11, 12]). Then Koike and Takahashi in their work  compared -viscosity sub- and supersolutions, when at least one of them is -strong.
In the last years, Bardi and Mannucci in  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 ) and Sirakov, , has the same result for fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type with unbounded ingredients and the most quadratic growth in .
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.
2. Preliminaries and Auxiliary Results
First of all, it is useful to give some definitions. We say that is a paraboloid of opening when where is a positive constant, is a constant, and is a linear function. is a convex paraboloid if there is the + sign in (2.1), concave otherwise.
Given two functions and on an open set , touches from the above in when In this case, one could also say that touches from below.
Consider the following: From , we have the following.
Lemma 2.1. Assume that and that (in the sense of distribution) for all direction . If has an interior maximum then there exist two constants and such that
Then some lemmas from  are needed for the sequel.
Lemma 2.2. Let and assume that
Then there exists a function and a matrix valued measure such that (1), (2) is singular with respect to Lebesgue measure, (3)(
Lemma 2.3. Let and assume that
If has an interior maximum then there exists a constant such that for (as in the previous lemma) as
Lemma 2.4. Assume that and that (2.5) holds. If is the decomposition of Lemma 2.2, then for almost every we have
3. Sup and Inf Convex Envelope
The aim of this paper is to consider equations of the following form: where and are such that the following hold:(1) is a continuous function on ;(2) there exist two constants and such that for all and ;(3) for all and ;(4) there exists a positive function on such that for , all , where has to satisfy the following: if , has an interior maximum, and (in the sense of distribution), then there exists a constant such that for and 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 , the convex envelope of a function.
Definition 3.1. Let be a bounded domain of , a subset of such that , . We call, respectively, sup and inf convex envelope of as in the following objects:
Now it is possible to give some properties of the sup convex envelope, noting that similar ones hold for the inf convex envelope.
Proposition 3.2 (see ). Let be a bounded domain of , a subset such that , and . Then (1): , (2), (3), (4), (5), (6).
Theorem 3.3 (see ). Let be an open set such that , we have (1) uniformly in for ; (2)for all there exists a concave paraboloid of opening which touches from below in .
Then is from below in .
In particular is pointwise differentiable to the second order for almost every .
Before going further, it is useful to give the definition of viscosity solution.
Definition 3.4. A viscosity subsolution of is a function such that and . If has a local maximum in , then the following holds A viscosity supersolution of is a function such that and . If has a local minimum in , then the following holds A continuous function is a viscosity solution of if and only if is both a viscosity sub- and supersolution.
Remember the following.(i) is the set of upper semicontinuous function in such that . (ii) is the set of lower semicontinuous function in such that .
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.
Theorem 3.5. Let be bounded functions which are, respectively, viscosity subsolution and supersolution of . If is uniformly elliptic and nonincreasing then there exist two Lipschitz continuous and bounded functions e and an open set , with , such that is semiconvex, is semiconcave on which are, respectively, viscosity subsolution and supersolution of in .
4. Comparison Principle
Now it is possible to prove the comparison principle
Theorem 4.1 (Comparison Principle). Let . Assume that is a viscosity supersolution and is a viscosity subsolution of
Suppose that on . If and satisfy the structure condition, then
Proof. Suppose that the contrary holds true as
Define and .
By Theorem 3.5, and 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
Note that and satisfies
Take from Lemma 2.1, for some , we have
Now, from Lemma 2.2, the following hold so and give a decomposition of .
From Lemma 2.4, for almost every , there exist and and
Moreover, by the definition of , we have
Applying the definition of viscosity subsolution and supersolution, it is possible to write
By (4.4) of Lemma 2.2, we have for almost every
Since and satisfy the structural condition, for almost every , we obtain
Computing we can show that it is nonnegative. In fact, from Lemma 2.3 (fixed a constant ) we have where ; while from the structural condition where . Then, for , since , we have which contradicts (4.3). So in .
Remark 4.2. Note that in the last line it is essential that and are finite.
It is possible to give some examples for . Assume that and let and , as in Lemma 2.2.(1) Consider Since then where the last equality is given by Lemma 2.3.(2). We have since .(3). Arguing as in the previous example, we can obtain the result.(4). In fact, suppose then which is a contradiction to the previous example.
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