On the weak Maximum Principle for fully nonlinear elliptic pde s in general unbounded domains Italo Capuzzo Dolcetta

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1 Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 7(2008), pp On the weak Maximum Principle for fully nonlinear elliptic pde s in general unbounded domains Italo Capuzzo Dolcetta To Ermanno, with great esteem Abstract. The aim of this Note is to review some recent research on viscosity solutions of fully nonlinear equations of the form F x, u(x),du(x),d 2 u(x) =0, x 2 where is an open set in R N and F is a nonlinear function of its entries which is elliptic with respect to the Hessian matrix D 2 u of the unknown function u and satisfies some suitable structure condition. The main issues touched here are the Alexandrov-Bakelman- Pucci estimate, the weak Maximum Principle for bounded solutions in general unbounded domains and qualitative Phragmen-Lindelöf type theorems.. Introduction The paper focuses on some global and local properties of continuous functions u satisfying fully nonlinear elliptic equations of the form (.) F x, u(x),du(x),d 2 u(x) =0 in the viscosity sense in an open set R N. The main topics discussed here are the validity of Alexandrov-Bakelman-Pucci estimates, the Weak Maximum Principle (wmp in short) and qualitative Phragmen-Lindelöf type theorems in cylindrical and conical domains. The results presented apply to a wide class of unbounded domains, perhaps of infinite measure, which may have a quite irregular boundary and generalize in several aspects a number of well-known results for smooth or strong solution of linear elliptic equation see, for example, [9], [3], [2], [6], [7], [2], [4]. The content of this note is mostly taken from the recent papers [6], [7], [8], [9]. We refer to these papers for the detailed proofs of the result presented here. Author s address: I. Capuzzo Dolcetta, Università degli Studi di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Piazzale Aldo Moro 2, I-0085 Roma, Italy; capuzzo@mat.uniroma.it. Keywords. Elliptic equations, viscosity solutions, maximum principle in general domains. AMS Subject Classification. 35B05, 35B45, 35B50.

2 82 Italo Capuzzo Dolcetta 2. Viscosity solutions We report for the convenience of the reader a few facts about viscosity solutions. An upper semicontinuous function u 2 USC( ) is a viscosity subsolution of equation (.) if the inequality F x 0, (x 0 ),D (x 0 ),D 2 (x 0 ) 0 holds at any point x 0 2 and for all quadratic polynomials touching from above the graph of u at x 0. Observe that u is a viscosity solution of u 0 if and only if u is subharmonic in the sense of potential theory: for any ball B and for any function h such that h =0inB the inequality u apple h implies u apple h in B. Viscosity supersolutions are defined in a symmetric fashion: a lower semicontinuous function u 2 LSC( ) is a viscosity supersolution of (.) if F x 0, (x 0 ),D (x 0 ),D 2 (x 0 ) apple 0 at any point x 0 2 and for all for all quadratic polynomials touching from below the graph of u at x 0. A viscosity solution of (.) is a function u 2 C( ) which is simultaneously a sub and a supersolution. Most of the theory of strong solutions for linear elliptic equations in non-divergence form (2.) Tr A(x) D 2 u + b(x) Du + c(x) u =0 has been carried on to viscosity solutions of (.) under the leading assumption of ellipticity of F : (2.2) Tr Y apple F (x, t, p, X + Y ) F (x, t, p, X) apple Tr Y for some constants 0 < apple and for all X, Y 2S N with Y 0, where S N and Tr denote, respectively, the space of real symmetric N N matrices endowed with the partial ordering induced by non-negative definiteness and the trace of such a matrix. We refer to [4] and [0] for existence, uniqueness and stability viscosity solutions of (.), to [3] for regularity theory. Fundamental model examples of elliptic operators F are given by the Pucci extremal operators P, and P +, defined for X 2 SN and given parameters 0 < apple by (2.3) P, (X) = inf Tr (AX), A2A P+, (X) = suptr (AX) A2A where A = A (, ) = A 2S N : I apple A apple I see [20], [3]. Other important examples are the Isaac s operators sup j2k inf Tr (A k,j X) k2k with A k,j 2A,k,j 2 K, arising in stochastic di erential game theory []. Two fundamental tools in deriving the Alexandrov-Bakelman-Pucci estimates for viscosity solutions of equation (.) are the weak Harnack inequality and its socalled boundary version for nonnegative supersolutions of Pucci type di erential inequalities.

3 On the weak maximum principle for fully nonlinear elliptic pde s in unbounded domains 83 Proposition 2. (the weak Harnack inequality). Let A be an open bounded domain of R N.Ifw 2 LSC(A) satisfies (2.4) w 0, P, (D 2 w) b(x) Dw appleg(x) with b, g 2 C(A) \ L (A), in the viscosity sense, then there exist positive numbers C, p depending on N,,, kbk L (B 4) such that Z /p (2.5) apple C inf w + kgk B LN (B B 4) 2 B w p where B B 2 B 4 A are concentric balls of radii, 2 and 4, respectively. Let B R, B R/ with 2 (0, ) be concentric balls such that For w 2 LSC(Ā), w w m of w A \ B R 6= ;, B R/ \A 6= ;. 0, consider the following lower semicontinuous extension w m (x) = where m =inf x2@a\br/ w(x). ( min(w(x); m) if x 2 A m if x 62 A Proposition 2.2 (the boundary weak Harnack inequality). Let A be an open bounded domain of R N. If w 2 LSC(A) satisfies (2.4) in the viscosity sense, with b, g 2 C(A) \ L (A), then (2.6) B R Z /p (w m ) p apple C B R where p and C depend on N,,,,Rkbk L (A). inf A\B R w + R kg + k LN (A\B R/ ) See [3] for the case b 0 and [6] for the (slightly) more general case b 6= A general class of unbounded domains in R N As mentioned in the Introduction, the aim of this Note is to present some results about the validity of the wmp for equation (.) in general domains. Let us consider then domains in R N satisfying the following measure/geometric condition wg: there exist constants, 2 (0, ) such that for all y 2 there is a ball B Ry of radius R y containing y such that B Ry \ y, B Ry where y, is the connected component of \ B Ry/ containing y. The above condition, proposed first in [5], [22], generalizes the notion of G domains previously introduced by X. Cabrè: any domain fulfilling condition wg with R y = O() as y! is in fact a G domain in the sense of [4]. Condition wg is therefore satisfied by any G domain, for example: bounded domains: in this case, R y diam ( ) domains with finite measure: in this case, R y C(N) /N

4 84 Italo Capuzzo Dolcetta unbounded cylinders with bounded cross-section =R k! R N k, k in this case R y diam (!) periodically perforated domains with period >2 =R N \ X k2z N ( k + B (0)) in this case R y the complement of a plane spiral with constant step k, represented in polar coordinates as =R 2 \ = k 2 in this case R y k. Note that condition G implies in particular sup ) < +; on the other hand wg domains can have points at arbitrarily large distance from the boundary. Typical examples of unbounded domains satisfying wg but not G are: non-degenerate cones of R N (and their unbounded subsets); for such a set wg holds with R y = O( y ) as y! parabolically shaped domains, defined for k> by the inequalities v u u x 0 t N X i= x 2 i <x/k N, x N > 0 condition wg holds in this case with R = O(x /k N ) the complement of the logarithmic spiral: = R 2 \ % =e, 0. Condition wg is satisfied with R y = O( y ) as y!. To conclude this section, let us point out explicitly that exterior domains such as R N \ B R are not wg. 4. The structure conditions on F We shall assume that F = F (x, t, p, X) is continuous with respect to all variables and (degenerate) elliptic, that is (4.) F (x, t, p, X) F (x, t, p, Y ) for all x 2, t 2 R, p 2 R N and X, Y 2S N with X Y O. Moreover, we assume that the following structure condition holds for all x 2, t 0, p 2 R N and X 2S N : (4.2) F (x, t, p, X) applep +, (X)+b(x) p for some non-negative function b 2 C( ) \ L ( ) and for all x 2, t 0, p 2 R N, X 2S N. Here, P +, is the Pucci maximal operator. Assumptions (4.) and (4.2) are satisfied by any uniformly elliptic F, see (2.2), such that t! F (x, t, p, X) non increasing, F(x, 0,p,O) apple b(x) p.

5 On the weak maximum principle for fully nonlinear elliptic pde s in unbounded domains 85 Note, however, that some nonlinear degenerate elliptic operators fulfill our assumptions. An example is!! NX NX F (X) = '(µ + i ) (µ i ). i= Here, µ ± i, i =,...N, are the positive and negative eigenvalues of the matrix X 2S N and ', :[0, +)! [0, +) are continuous and nondecreasing functions such that '(s) apple s apple (s). Observe, finally, that while X!P +, (X) is convex, the structure condition does not require convexity nor concavity of X! F (x, t, p, X). We will also consider the case of F having quadratic growth in the gradient variable. In order to treat this case we will employ the structure condition (4.3) F (x, t, p, X) applep +, (X)+b(x) p + b 2 p 2 for t 0, where b 2 is a positive constant. 5. The Weak Maximum Principle in unbounded domains We present first in this section some recent result concerning the validity of the wmp for upper semicontinuous viscosity solutions of the partial di erential inequality (5.) F x, u(x),du(x),d 2 u(x) 0,x2, in unbounded domains of type wg and for degenerate elliptic functions F satisfying the structure condition (4.2) or (4.3)). We will consider in the next subsections a few di erent quite general situations in which the validity of the wmp can be established: bounded above solutions, linear growth in Du bounded above solutions, quadratic growth in Du bounded above solutions in domains of small measure and/or for operators with a small zero-order term exponentially growing solutions in narrow domains Phragmèn-Lindelöf theorems in cylindrical and conical domains 5.. Bounded above solutions, linear growth in Du. Our first result is an Alexandrov-Bakelman-Pucci type estimate for solutions of (5.2) F x, u(x),du(x),d 2 u(x) f(x),x2. Theorem 5.. Let u 2 USC( ) with sup u<+ be a viscosity solution of (5.2) where f 2 C( ) \ L ( ). Assume that F is continuous and satisfies (4.) and (4.2). Assume, moreover, that satisfies wg for some R y such that i= (5.3) Rb := sup R y kbk L ( y, ) <. Then (5.4) sup u apple u + + C sup R y kf k L N ( y, ) for some positive constant C depending on N,,,, and Rb.

6 86 Italo Capuzzo Dolcetta As an immediate consequence of the above result, the wmp holds: if u 2 USC( ) is bounded above and F (x, u, Du, D 2 u) 0, x2, in the viscosity sense then u apple 0 implies u apple 0in. Remark 5.2. For F = F (X) and bounded, the estimate (5.4) has been established in [3]. When F does not depend on Du, thenb 0 and condition (5.3) is trivially satisfied in any wg domain. In general, however, some condition relating the size of the domain with the size of first order terms at infinity is required for the validity of the wmp in unbounded domains. Indeed, u(x) =u(x,x 2 )= with 0 < <, is bounded and satisfies e x e x 2 =0, u > 0, u + B(x) Du =0 in in the cone = x =(x,x 2 ) 2 R 2 : x >, x 2 > with B given by B(x) =B(x,x 2 )= +, x x x 2 +. x 2 Since satisfies wg with R y = O( y ) as y!and the structure condition (4.2) holds with b(x) = B(x), condition (5.3) fails in this example. Remark 5.3. For bounded b, in order to enforce (5.3) the requirement on amounts to sup R y apple R 0 < + meaning that should be in fact a G domain. For G domains, the result of our Theorem 5. can be regarded essentially as a nonlinear version of the Alexandrov- Bakelman-Pucci estimate for linear elliptic equations with bounded coe cients proved in [4]. In the case of parabolically shaped domains, defined for k> by the inequalities v u x 0 t N X x i <x /k N,x N > 0, i= for which wg holds with R = O(x /k N ), one can show that (5.3) holds provided b(x) =O(/x /k N ) as x!. Note that cylindrical and conical domains can be seen as limiting cases of above situation when, respectively, k! + and k!. The detailed proof of Theorem 5. can be found in [6]. The first step is to observe that w(x) =M u + (x) withm :=sup x2 u + (x) < + satisfies w 0, P (D 2 w) b(x) Dw applef (x) in in the viscosity sense. By the boundary weak Harnack inequality (2.6) Z! /p (5.5) (w B Ry m ) p apple Cy inf w + R y kf k L B y, \B N ( y, ) Ry Ry

7 On the weak maximum principle for fully nonlinear elliptic pde s in unbounded domains 87 for positive constants p and Cy depending on N,,, and R y, kbk L ( y, ). Using the wg condition it is not hard to show that the left-hand side of the above inequality can be estimated from below as follows (5.6) B Ry From (5.5), (5.6) we deduce that Z B Ry (w m ) p! /p m /p. m /p apple C y M u + (y)+r y kf k L N ( y, ) at any point y 2. Observing that m M sup z2@ u + (z), after some simple computations we are led to the pointwise estimate /p /p (5.7) u + (y) apple sup u + + sup u + + R y kf k L N ( y, C y Thanks to the assumption (5.3), the constant C y / /p can be bounded above by some 2 (0, ), independently on y. Taking the supremum on both sides of (5.7), we obtain (5.4) Bounded above solutions, quadratic growth in Du. Let us briefly describe how the results of the previous section can be extended to the case of an F having at most quadratic growth in the gradient variable. Observe at this purpose that if v 0 is a viscosity solution of (5.8) P, (D 2 v) b(x) Dv b 2 Dv 2 apple g(x) then the Hopf-Cole type transform C y w = h (v) where h is smooth, non-negative, increasing and convex, satisfies P, (D 2 w)+ h00 (w) h 0 (w) ; Dw 2 b(x) Dw b 2 h 0 (w) Dw 2 apple g(x) h 0 (w) in the viscosity sense. The proof of this fact requires some viscosity calculus together with the superadditivity and the ellipticity of P,. Solving the ordinary di erential equation h 00 (t) b 2 (h 0 ) 2 (t) =0 one finds b 2 t h(t) = log b2 which satisfies the required properties for t 2 [0, /b 2 ). Correspondingly, the function w = e b2v/ b2 is a solution of w 0, P, (D 2 w) b(x) Dw appleg(x) b2 w. Applying inequality (2.5) of Section 2 to w and observing that, for M =supv, e b2m/ b 2 M/ v apple w apple v

8 88 Italo Capuzzo Dolcetta we conclude that the weak Harnack inequality Z /p (5.9) apple C inf v + kgk B L B N (B 4) 2 B v p holds for solutions of (5.8). Observe that the constant C depends on b 2 M. The dependence on the upper bound M in the estimate seems to be unavoidable, see [2], [5]. A boundary version of inequality (5.9) can be easily obtained in the present setting much in the same way as in Section 2: B R Z /p (v m ) p apple C B R inf A\B R v + R kg + k L N (A\B R/ ) where p and C are positive constants depending on N, Rkbk L (A\B R/ ).,,, b 2 M and The Alexandrov-Bakelman-Pucci estimate and the wmp continue therefore to hold true in the quadratic case under consideration: Theorem 5.4. Let u 2 USC( ) with sup u<+ be a viscosity solution of (5.2) where f 2 C( ) \ L ( ). Assume that F is continuous and satisfies (4.) and (4.3). Assume, moreover, that satisfies wg for some R y such that Rb := sup R y kbk L ( y, ) <. Then sup u apple sup u + + C sup R y kf k L N ( y, for some positive constant C depending on N,,,, and Rb. As an immediate consequence of the above result, the wmp holds: if u 2 USC( ) is bounded above and F (x, u, Du, D 2 u) 0, x2, in the viscosity sense then u apple 0 implies u apple 0in Bounded above solutions for the perturbed operator F + c(x). The next result, see [2] for the linear case, establishes the validity of a qualitative version of the wmp for the perturbed operator F + c(x) under a condition relating the radii R y in condition wg with the size of function c +. Note that the case c apple 0is trivially included in Theorem 5.. Theorem 5.5. Let u 2 USC( ) with sup u<+ and u apple 0 be a viscosity solution of F (x, u, Du, D 2 u)+c(x) u f(x) where f 2 C( ) \ L ( ). Assume that F is continuous and satisfies (4.) and (4.2). Assume, moreover, that satisfies wg for some R y such that and that Rb := sup R y kbk L ( y, ) < sup Ry 2 kc + k L ( y, ) is su ciently small.

9 On the weak maximum principle for fully nonlinear elliptic pde s in unbounded domains 89 Then sup u apple C sup R y kf k LN ( y, ) for some positive constant C depending on N,,,, and Rb. Since F (x, u, Du, D 2 u) c (x) u c + (x) u + f(x) a direct application of Theorem 5. yields sup apple C 0 sup u apple C sup R y k( c + u) k L N ( y, ) + kf k L N ( y, ) R 2 y kc + k L ( y, ) sup u + (y)+c sup R y kf k L N ( y, ). If sup R 2 ykc + k L ( y, ) is su ciently small, we conclude that sup u apple sup u + for some < and the statement follows. Remark 5.6. Theorem 5.5 applies of course when either sup R y or kc + k L ( ) are small enough, e.g. if is finite and su ciently small. A more interesting case is when a strictly convex cone with su ciently small opening and c + = O(/ y 2 ) as y!. Indeed, in this case we can apply Theorem 5.5 taking R y apple y, for su ciently small, in condition wg and kc + k L ( y, ) = O(/ y 2 ) Exponentially growing solutions in narrow domains. The next result shows that a qualitative version of the wmp holds even for unbounded above solutions of (5.2) provided the unbounded domain satisfies an appropriate narrowness condition related to the admissible rate of growth at infinity of the solution. More precisely, consider the unbounded cylinder =R k! with k + h = N, h,k, where! is a bounded domain of R h. As pointed out in Section 3 this is typical example of G domain. Theorem 5.7. For F as in Theorem 5. and as above, suppose kbk L ( ) apple b and let u 2 USC( ),F x, u(x),du(x),d 2 u(x) 0, x 2, with u apple 0 u + (x) =o(e x ) as x!+. Then for any >0 there exists a positive number d = d(n,,,b, ) such that, if diam (!) <d, then u apple 0 in. Conversely, for any fixed d>0 there exists = (N,,,b,d) such that if diam (!) <d, then u apple 0 in. Qualitative results of this type for general linear uniformly elliptic operators can be found in [], for semilinear equations on cylinders. In the special case of subharmonic functions on the 2-dimensional strip = R (0,d) there is a precise quantitative relationship between the diameter d and the growth exponent, namely = /d, see [2]. The proof of Theorem 5.7 is based on the construction of a suitable sequence of smooth barrier functions k on finite cylinders C k = B k (0)!, k 2 N, such that P + D 2 k(x) + b D k (x) apple0 in C k, k 0 in C k, k u + C k \@

10 90 Italo Capuzzo Dolcetta and for each fixed x 2 The barriers have the form lim k! k (x) =0. k(x, y) =K k /(e R cos h ( d/2)) exp( x ) hy sin y j for suitable choices of the parameters. It is a familiar technique in the case of a linear operator to use the wmp in bounded domains C k, considering di erences u k, and then passing to the limit as k!. The di culty in implementing this procedure in the present nonlinear setting where u need not to be smooth, is overcome by the use of the structure condition (4.2), together with the superadditivity of the maximal Pucci operator, since standard calculus rules apply due to the fact that k is twice continuously di erentiable, see [6] for details. A similar result holds for viscosity subsolutions with polynomial growth u(x) = O( x ) in angular sectors = R k!, where! is a cone in R h and h + k = N, provided (4.2) holds true with b(x) =O(/ x ) as x!. In this case, in order to deduce the validity of the wmp the opening of the cone has to be su ciently small depending on the exponent and the various structural parameters. We refer to [8] for previous results in this direction Phragmen-Lindelöf type theorems in general domains. In Subsection 5.4 we proposed some Phragmèn-Lindelöf type results for viscosity solutions in cylinders and narrow cones. On this basis, one may expect that wmp should hold in more general wg domains of cylindrical type (that is, wg holds with R y apple R<+) or conical type (that is, wg holds with R y = O( y ) as y!) under a suitable exponential, respectively, polynomial growth of subsolutions at infinity. However, the explicit constructions of the barrier functions used in the proofs of the above mentioned results, see [6] for more details, relies heavily on the simple geometry of cylinders and cones and cannot be easily carried over to geometrically more complex general G or wg domains. An alternative way to obtain qualitative Phragmèn-Lindelöf type results in general cylindrical or conical wg domains relies instead on the validity of the Maximum Principle for bounded above viscosity solutions of j= P +, (D2 w(x)) + (x) Dw(x) + 2 (x)w + (x) 0 where the coe cient 2 is allowed to be positive but suitably small. + Indeed, by Theorem 5.5, if 2 (x) apple c (in the case of cylindrical domains) and + 2 (x) apple c / x 2 as x!(inthecase of conical domains), then the wmp holds provided c, a positive number depending on the structure of F and on the geometric parameters occurring in the G or wg conditions, is small enough. Two model results in this direction are the following: Theorem 5.8. Assume that is a wg domain of R N of cylindrical type and that F satisfies F (x, t, p, X) applep +, (X)+b(x) p + c(x)t with b(x) appleb 0, c(x) apple c,

11 On the weak maximum principle for fully nonlinear elliptic pde s in unbounded domains 9 c > 0 small enough. Then there exists >0, depending on F and, such that if u 2 USC( ) is a viscosity solution of F x, u(x),du(x),d 2 u(x) 0 in with u apple 0 and u(x) =O(e x ) as x!+, then u apple 0 in. Theorem 5.9. Assume that is a wg domain of conical type and that F satisfies F (x, t, p, X) applep +, (X)+b(x) p + c(x)t with b 0 c b(x) apple, c(x) apple ( + x 2 ) 2 + x 2, c > 0 small enough. Then there exists >0, depending on F and, such that if u 2 USC( ) is a viscosity solution of F x, u(x),du(x),d 2 u(x) 0 in with u apple 0 and u(x) =O( x ) as x!+, then u apple 0 in. A sketchy proof of Theorem 5.9 starts with the consideration of the smooth positive function (x) =(+ x 2 ) /2 where >0 is a parameter. If u(x) =O( x )then w(x) = u(x) (x) is bounded above and obviously w(x) apple 0 A straightforward calculation shows now that D apple 2( + x 2 ), D 2 apple 2N /2 + x 2. By some viscosity calculus and using the decay condition on b we deduce that with P +, (D2 w(x)) + (x) Dw(x) + 2 (x)w + (x) 0 (x) = CN + b 0 2( + x 2 ), 2(x) = (CN2 +b 0 )+c /2 + x 2 for some positive constant C. The zero order coe cient 2 in the above inequality can be made are arbitrarily small by choosing suitably small values of. From Theorem 5.5 it follows then that w and u are non positive on. The proof of Theorem 5.8 goes the same way, modulo the use of the function instead of in the above computations. (x) =e (+ x 2 ) /2 Remark 5.0. Theorems 5.8 and 5.9 above extend in particular the results of [6], [23] in the direction of more general unbounded domains as well as of viscosity solutions of (non necessarily uniformly) elliptic fully nonlinear di erential inequalities containing lower order terms. Finally, let us point out that, in view of the discussion in Subsection 5.2, the Phragmèn-Lindelöf theorems above continue to hold true for operators with quadratic growth in the gradient variable.

12 92 Italo Capuzzo Dolcetta References [] H. Berestycki, L. A. Ca arelli & L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. I. A celebration of John F. Nash, Duke Math. J., 8(2)(996), [2] H. Berestycki. L. Nirenberg & S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm.PureAppl. Math., 47()(994), [3] L.A. Ca arelli, X. Cabrè, Fully nonlinear elliptic equations, AmericanMathematicalSociety Colloquium Publications, 43, American Mathematical Society, Providence, RI, 995. [4] X. Cabrè, On the Alexandro -Bakelman-Pucci estimate and reversed Holder inequalities for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48(995), [5] V. Cafagna & A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains, C.R.Acad.Sci.ParisSer.I,334(5)(2002), [6] I. Capuzzo Dolcetta, F. Leoni & A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Comm.PartialDifferential Equations, 30(2005), [7] I. Capuzzo Dolcetta & A. Vitolo, A Qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations, J.Di erentialequations,243(2)(2006), [8] I. Capuzzo Dolcetta & A. Vitolo, Local and global estimates for viscosity solutions of fully nonlinear elliptic equations, DynamicsofContinuous,Discrete&ImpulsiveSystems. Series A: Mathematical Analysis, 4(S2)(2007), 6. [9] I. Capuzzo Dolcetta & A. Vitolo, On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains, LeMatematiche,LXII(II)(2007), [0] M.G. Crandall, H. Ishii & P.L. Lions, User s guide to viscosity solutions of second order partial di erential equations, Bull.Amer.Math.Soc.(N.S.),27()(992), 67. [] W.H. Fleming & H.M. Soner, Controlled Markov processes and viscosity solutions, Applications of Mathematics 25, Springer-Verlag, Berlin-New York, 993. [2] L.E. Fraenkel, Introduction to maximum principles and symmetry in elliptic problems, Cambridge University Press, [3] D. Gilbarg & N.S. Trudinger, Elliptic partial di erential equations of second order, IIed., Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin-New York, 983. [4] H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic pde s, Comm.PureAppl.Math.,42(989),4 45. [5] S. Koike & A. Swiech, Maximum principle and existence of L p viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms, NoDEA, (2004), [6] V.A. Kondrate v & E.M. Landis, Qualitative theory of second order linear partial di erential equations, PartialDi erentialequationsiii,egorov,yu.v.&shubinm.a.eds., Encyclopedia of Mathematical Sciences, 32(99), [7] E.M. Landis, Second order equations of elliptic and parabolic type, TranslationsofMathematical Monographs 7, Amer. Math. Soc., Providence, R.I., 998. [8] K. Miller, Extremal barriers on cones with Phragmèn-Lindelöf theorems and other applications, Ann.Mat.PuraAppl.(4),90(97), [9] M.H. Protter & H.F Weinberger, Maximum principles in di erential equations, II ed., Springer-Verlag, New York, 984. [20] C. Pucci, Operatori ellittici estremanti, Ann.Mat.PuraAppl.(4),72(966),4 70. [2] N.S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, NoDEA,3-4(988), [22] A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J.Di erentialequations,94()(2003), [23] A. Vitolo, On the Phragmèn-Lindelöf principle for second-order elliptic equations, J. Math. Anal. Appl., 300()(2004),