On the viscosity approach to a class of fully nonlinear elliptic equations
aa r X i v : . [ m a t h . A P ] F e b ON THE VISCOSITY APPROACH TO A CLASS OF FULLYNONLINEAR ELLIPTIC EQUATIONS
HOANG-SON DO AND QUANG DIEU NGUYEN
Abstract.
In this paper, we study some properties of viscosity sub/super-solutions ofa class of fully nonlinear elliptic equations relative to the eigenvalues of the complexHessian. We show that every viscosity subsolution is approximated by a decreasingsequence of smooth subsolutions. When the equations satisfy some conditions on thelimit at infinity, we verify that the comparison principle holds, and as a sequence, weobtain a result about the existence of solution of the Dirichlet problem. Using thecomparison principle, we show that, under suitable conditions, a Perron-Bremermannenvelope can be approximated by a decreasing sequence of viscosity solutions.
Contents
1. Introduction 12. Preliminaries 33. Approximation of subsolutions 54. Comparison principle and applications 85. Maximal viscosity subsolutions 10References 121.
Introduction
Let Γ R n be an open, convex, symmetric cone with vertex at the origin such thatΓ n ⊆ Γ ⊆ Γ , where Γ k = { x ∈ R n : σ ( x ) > , ..., σ k ( x ) > } , for every 1 ≤ k ≤ n . Here σ k ( x ) is the k -th elementary symmetric sum of the coefficientsof x , i.e., σ k ( x ) = P ≤ j <...
Assume that ψ ( z, r ) does not depend on the last variable r ∈ R . If u ∈ U SC (Ω) is a viscosity subsolution of (3) then u is Γ -subharmonic and F ( H ( u ∗ χ ǫ )) ≥ ψ ∗ χ ǫ in Ω ǫ (in the classical sense). Here χ ǫ is the standard modifier, ∗ is the convolutionoperator and Ω ǫ = { z ∈ Ω : d ( z, ∂ Ω) > ǫ } . When f satisfies some conditions on the limit at infinity, we use Theorem 1.1 to showthat every viscosity subsolution of (3) can be approximated by a decreasing sequenceof classical subsolution of (3). In the cases of Monge-Amp`ere equations and Hessianequations, this fact has been proved in [5] and [18]. Corollary 1.2.
Assume that ψ ( z, r ) does not depend on r and lim R →∞ f ( R, R, ..., R ) > sup Ω ψ. ULLY NONLINEAR ELLIPTIC EQUATIONS 3
Then a function u ∈ U SC (Ω) is a viscosity subsolution of (3) iff for every open set U ⋐ Ω , there exists a decreasing sequence { u j } of smooth Γ -subharmonic functions on U such that u j → u as j → ∞ and F ( Hu j ( z )) ≥ ψ ( z ) in Ω for every j . Our second purpose is to study the comparison principle for (3). It follows from [12]that the comparison principle holds if ψ ( z, r ) − ǫr is non-decreasing in r for some ǫ > F , one can permit ǫ = 0 (see [4]).In this paper, we establish a version of the comparison principle with weaker conditionsfor F : Theorem 1.3.
Let Ω ⊂ C n be a bounded domain. Let u ∈ U SC ∩ L ∞ (Ω) and v ∈ LSC ∩ L ∞ (Ω) , respectively, be a bounded subsolution and a bounded supersolution of theequation (4) F ( Hw ) = ψ ( z, w ) , in Ω . Assume that u ≤ v in ∂ Ω and lim R →∞ f ( R, R, ..., R ) > sup K ψ ( z, v ( z )) , for every K ⋐ Ω . Then u ≤ v in Ω . By Theorem 1.3 and the Perron method, if v is a viscosity supersolution of (3) satisfyingsome suitable conditions then the functionΦ v = sup { w : w is a subsolution of (3), w ≤ v } , is a discontinuous viscosity solution of (3) (see Proposition 4.2). In this paper, we showthat Φ v is further locally approximated by a decreasing sequence of viscosity solutions. Theorem 1.4.
Assume that v is a viscosity supersolution of (3) in Ω satisfying thefollowing conditions: a) for all ˆ z ∈ ∂ Ω , lim Ω ∋ z → ˆ z v ( z ) = v (ˆ z );b) sup ∂ Ω v < ∞ and lim R →∞ f ( R, R, ..., R ) > ψ ( z, M ) , for every z ∈ Ω , where M > sup ∂ Ω u is a constant.Then, for every relatively compact open subset U of Ω , there exists a decreasing sequence u j of viscosity solutions of (3) in U such that lim j →∞ u j = Φ v in U . Acknowledgements.
The authors would like to thank Lu Hoang Chinh for fruitfuldiscussions on discontinuous viscosity solutions. This research began while the first nameauthor was visiting Vietnam Institute for Advanced Study in Mathematics(VIASM). Hewould like to thank the institution for the hospitality.2.
Preliminaries
In this section, we recall the definitions and some properties of viscosity sub/super-solutions.
Definition 2.1. (Test functions) Let w : Ω −→ R be any function defined in Ω and z ∈ Ω a given point. An upper test function (resp., a lower test function) for w at thepoint z is a C -smooth function q in a neighbourhood of z such that w ( z ) = q ( z ) and w ≤ q (resp., w ≥ q ) in a neighbourhood of z . HOANG-SON DO AND QUANG DIEU NGUYEN
Definition 2.2.
1. A function u ∈ U SC (Ω) is said to be a (viscosity) subsolution of (5) F ( Hu ) = ψ ( z, u ) , in Ω if for any point z ∈ Ω and any upper test function q for u at z , we have F ( Hq ( z )) ≥ ψ ( z , u ( z )) (and then Hq ( z ) ∈ M (Γ , n ) ). In this case, we also say that F ( Hu ) ≥ ψ ( z, u ) in the viscosity sense in Ω .2. A function v ∈ LSC (Ω) is said to be a (viscosity) supersolution of (5) in Ω if for anypoint z ∈ Ω and any lower test function q for v at z , we have F ( Hq ( z )) ≤ ψ ( z , u ( z )) . In this case, we also say that F ( Hv ) ≤ ψ ( z, v ) in the viscosity sense in Ω .3. A function u ∈ C (Ω) is said to be a (viscosity) solution of (5) in Ω if it is a subsolutionand a supersolution of (5) in Ω .4. A function u ∈ L ∞ (Ω) is said to be a discontinuous viscosity solution of (5) in Ω if u ∗ is a subsolution and u ∗ is a supersolution of (5) in Ω . It follows from the definition directly that if u, v are viscosity subsolutions of (5) thenmax { u, v } is a viscosity subsolution of (5). Furthermore, we also have: Proposition 2.3.
Assume that G ( Ω is an open set. Suppose that v is a viscositysubsolution of (5) in G and u is a viscosity subsolution of (5) in Ω such that lim sup G ∋ z → z v ( z ) ≤ u ( z ) ,for every z ∈ ∂G ∩ Ω . Then, the function ˜ u = ( u in Ω \ G, max { u, v } in G, is a viscosity subsolution of (5) in Ω . In the case of Hessian equations (i.e., f ( x ) = ( σ k ( x )) /k and Γ = Γ k ), we use thenotation F k instead of F . The following result has been proved in [18]: Proposition 2.4.
Let u ∈ U SC (Ω) . Then the following conditions are equivalent: a) u is a viscosity subsolution of the equation F k ( Hw ) = ψ ( z, w ) in the sense ofDefinition 2.2; b) for every z ∈ Ω , for every upper test function q for u at z , we have σ k ( λ ( Hq ( z )) ≥ ψ k ( z , u ( z )) (it does not require that Hq ( z ) ∈ M (Γ k , n ) ). Actually, in [18], a function u ∈ U SC (Ω) is called a viscosity subsolution of the givenHessian equation if it satisfies the condition b) in the above proposition. By the proof of[18, Lemma 3.7], if b) is satisfied then, for every z ∈ Ω and for every upper test function q for u at z , the Hessian matrix Hq ( z ) = ( ∂ q∂z α ∂z β ( z )) is k -positive, i.e., Hq ( z ) ∈ M (Γ k , n ).Then b ) ⇒ a ). The fact a ) ⇒ b ) is obvious.If u ∈ U SC (Ω) (resp., u ∈ LSC (Ω)) then for every x ∈ Ω, the set J ± u ( z ) = { ( Dw ( z ) , Hw ( z )) ∈ R n × H n : w is an upper test function (resp. a lower testfunction) for u at z } is called the super-(resp., sub-)differential of u at z . The set J ± u ( z ) = { ( p, Z ) ∈ R n × H n : ∃ z m → z and ( p m , Z m ) ∈ J ± u ( z m ) such that( p m , Z m ) → ( p, Z ) and u ( z m ) → u ( z ) } is called the limiting super-(resp. sub-)differential of u at z . By the continuity of F and ψ ,the limiting super/sub-differentials can be used to identify viscosity sub/super-solutionsas follows: ULLY NONLINEAR ELLIPTIC EQUATIONS 5
Proposition 2.5. a) Let u ∈ U SC (Ω) . Then u is a viscosity subsolution of the equation (5) iff for any point x ∈ Ω , for every ( p, X ) ∈ J + u ( z ) , we have F ( X ) ≥ ψ ( z, u ) . b) Let u ∈ LSC (Ω) . Then u is a viscosity supersolution of the equation (5) iff for anypoint x ∈ Ω , for every ( p, X ) ∈ J − u ( z ) , we have F ( X ) ≤ ψ ( z, u ) . The following proposition is deduced by combining Proposition 2.5 and [2, Proposition4.3]:
Proposition 2.6. a) Assume that { u α } is a family of viscosity subsolutions of the equation (6) F ( Hw ) = ψ ( z, w ) , in Ω . If u = sup α u α is locally bounded from above then its usc regularization u ∗ is aviscosity subsolution of (6) in Ω .b) Assume that { u α } is a family of viscosity supersolutions of (6) in Ω . If u = inf α u α islocally bounded from below then its lsc regularization u ∗ is a viscosity supersolution of (6) in Ω .b) Assume that u j is a decreasing (resp., increasing) sequence of viscosity subsolutions(resp., supersolutions) to (6) . Then u = lim j →∞ u j is either a viscosity subsolutions(resp., supersolutions) to (6) or identically −∞ (resp., ∞ ). Approximation of subsolutions
In this section, we will prove the Theorem 1.1 and Corollary 1.2. First, we have thefollowing lemma:
Lemma 3.1.
There exists a mapping
Φ : M (Γ , n ) → M (Γ n , n ) H e H = Φ( H ) depending on F such that a) For all B ∈ M (Γ , n ) , (7) F ( B ) = inf { ∆ e H B + F ( H ) − ∆ e H H : H ∈ M (Γ , n ) } , where ∆ e H B = trace ( e HB ) = n P j,k =1 e h jk b kj . b) For all B ∈ H n , if (8) inf { ∆ e H B + F ( H ) − ∆ e H H : H ∈ M (Γ , n ) } ≥ , then B ∈ M (Γ , n ) .Proof. a) By the concavity of F in M (Γ , n ), for every H ∈ M (Γ , n ), the subdifferential ∂ ( − F ( H )) is nonempty, i.e., there exists ˜ H ∈ H n \ { } such that(9) F ( B ) − F ( H ) ≤ ∆ ˜ H ( B − H ) = ∆ ˜ H ( B ) − ∆ ˜ H ( H ) , for all B ∈ M (Γ , n ). Moreover,(10) F ( B ) = lim H → B ( F ( H ) + ∆ ˜ H ( B − H )) . Combining (9) and (10), we have(11) F ( B ) = inf { ∆ ˜ H ( B ) + F ( H ) − ∆ ˜ H ( H ) : H ∈ M (Γ , n ) } . HOANG-SON DO AND QUANG DIEU NGUYEN
By (2) and (9), for every N ∈ M (Γ n , n ) \ { } and for each H ∈ M (Γ , n ), we have(12) ∆ ˜ H ( N ) = ∆ ˜ H ( H + N − H ) ≥ F ( H + N ) − F ( H ) > . Hence, e H ∈ M (Γ n , n ) for every H ∈ M (Γ , n ).b) Assume that B / ∈ M (Γ , n ) and the condition (8) is satisfied. Let t > B + t I ∈ ∂M (Γ , n ). Then, for every t > t , we have B + tI ∈ M (Γ , n ). By theassumption, we get∆ ^ B + tI B + F ( B + tI ) − ∆ ^ B + tI ( B + tI ) ≥ inf H ∈ M (Γ ,n ) { ∆ e H B + F ( H ) − ∆ e H H } ≥ , for every t > t . Then F ( B + tI ) − t ∆ ^ B + tI I ≥ , for every t > t . Letting t ց t , we get(13) lim sup t → t +0 ∆ ^ B + tI I ≤ lim t → t +0 F ( B + tI ) t = F ( B + t I ) t = 0 . Moreover, it follows from (9) that F ( B + (1 + t ) I ) − F ( B + tI ) ≤ (1 + t − t )∆ ^ B + tI I, for every t < t < t + 1. Letting t ց t , we get(14) lim inf t → t +0 ∆ ^ B + tI I ≥ lim t → t +0 F ( B + (1 + t ) I ) − F ( B + tI )1 + t − t = F ( B + (1 + t ) I ) > . By (13) and (14), we get a contradiction.Thus, the condition (8) implies that B ∈ M (Γ , n ). (cid:3) Corollary 3.2.
Let t ∈ [0 , and ≤ ψ , ψ ∈ C (Ω) . Assume that u j is a viscositysubsolution of the equation F ( Hw ) = ψ j ( z ) in Ω for j = 1 , . Then, the function tu +(1 − t ) u is a subsolution of the equation F ( Hw ) = tψ ( z ) + (1 − t ) ψ ( z ) .Proof. By Lemma 3.1, we have, for j = 1 , e H u j + F ( H ) − ∆ e H H ≥ ψ j , in the viscosity sense in Ω for every H ∈ M (Γ , n ). Then, it follows from [10, Proposition3.2.10’, page 147] that (15) holds in the distribution sense. Therefore, we have(16) ∆ e H ( tu + (1 − t ) u ) + F ( H ) − ∆ e H H ≥ tψ ( z ) + (1 − t ) ψ ( z ) , in the distribution sense. Using again [10, Proposition 3.2.10’, page 147], we get (16)holds in the viscosity sense. Thus, by Lemma 3.1, we obtain F ( H ( tu + (1 − t ) u )) ≥ tψ ( z ) + (1 − t ) ψ ( z ) , in the viscosity sense. (cid:3) Corollary 3.3.
Let u ∈ U SC (Ω) . Then the following conditions are equivalent a) u is subharmonic and for every ǫ > , u ∗ χ ǫ is Γ -subharmonic in Ω ǫ . Here χ ǫ is thestandard modifier, ∗ is the convolution operator and Ω ǫ = { z ∈ Ω : d ( z, ∂ Ω) > ǫ } . b) u is Γ -subharmonic. c) F ( Hu ) ≥ in the viscosity sense, i.e., for any point z ∈ Ω and any upper testfunction q for u at z , we have Hq ( z ) ∈ M (Γ , n ) . ULLY NONLINEAR ELLIPTIC EQUATIONS 7
Proof. ( a ) ⇒ ( b ) and ( b ) ⇒ ( c ) are clear. It remains to show ( c ) ⇒ ( a ).Assume that F ( Hu ) ≥ ⊂ Γ , we have ∆ u ≥ u ∈ SH (Ω).Moreover, it follows from Lemma 3.1 that(17) ∆ e H u + F ( H ) − ∆ e H H ≥ , in the viscosity sense for every H ∈ M (Γ , n ). Then, it follows from [10, Proposition3.2.10’, page 147] that (17) holds in the distribution sense. Hence∆ e H ( u ∗ χ ǫ ) + F ( H ) − ∆ e H H ≥ , in the classical sense in Ω ǫ for every H ∈ M (Γ , n ). Using again Lemma 3.1, we have H ( u ∗ χ ǫ )( z ) ∈ M (Γ , n ) for every ǫ > z ∈ Ω ǫ . Thus u ∗ χ ǫ is Γ-subharmonic in Ω ǫ .The proof is completed. (cid:3) Proof of Theorem 1.1.
Assume that u is a viscosity subsolution of the equation F ( Hw ) = ψ ( z ) , in Ω. By Corollary 3.3, we have u and u ∗ χ ǫ are Γ-subharmonic ( ǫ > e H ( u ∗ χ ǫ ) + F ( H ) − ∆ e H H ≥ ψ ∗ χ ǫ , in the classical sense in Ω ǫ for every H ∈ M (Γ , n ). Hence, it follows from Lemma 3.1 that F ( H ( u ∗ χ ǫ )) ≥ ψ ∗ χ ǫ in Ω ǫ in the classical sense. (cid:3) Proof of Corollary 1.2.
If there exists a decreasing sequence { u j } of smooth Γ-subharmonicfunctions on U such that u j → u as j → ∞ and F ( Hu j ( x )) ≥ ψ ( z ) in Ω for every j then,by Proposition 2.6, u is a viscosity subsolution of (3).For the converse, assume that u is a viscosity subsolution of (3) and U is a relativelycompact open subset of Ω. By Theorem 1.1, we have u ∗ χ ǫ ց u as ǫ ց F ( Hu ∗ χ ǫ ) ≥ ψ ∗ χ ǫ in U for every 0 < ǫ < ǫ , where ǫ > U ⋐ Ω ǫ . Since ψ ∗ χ ǫ converges uniformly to ψ in U , there exists 0 < ... < ǫ j +1 < ǫ j < ... < ǫ < ǫ suchthat lim j →∞ ǫ j = 0 and | ψ ∗ χ ǫ − ψ | < j , in U . By the assumption, there exists R ≫ f ( R, R, ..., R ) > sup U ψ + r forsome 0 < r <
1. For every j > − log r , we denote: u j ( z ) = u ∗ χ ǫ j + R | z | j +1 r . Then, u j is a decreasing sequence of smooth Γ-subharmonic functions in U satisfyinglim j →∞ u j = u . Moreover, for every j > − log r and for each z ∈ U , we have F ( Hu j ) ≥ (1 − j r ) F ( Hu ∗ χ ǫ j ) + 12 j r F ( H ( u ∗ χ ǫ j ) + R | z | ≥ (1 − j r ) ψ ∗ χ ǫ + 12 j r F ( RI ) ≥ (1 − j r )( ψ ( z ) − j ) + 12 j r ( ψ ( z ) + r ) > ψ ( z ) , in U . (cid:3) HOANG-SON DO AND QUANG DIEU NGUYEN Comparison principle and applications
Now we prove the second main theorem of this paper:
Theorem 4.1.
Let u ∈ U SC ∩ L ∞ (Ω) and v ∈ LSC ∩ L ∞ (Ω) , respectively, be a boundedsubsolution and a bounded supersolution of the equation (18) F ( Hw ) = ψ ( z, w ) , in Ω . Assume that u ≤ v in ∂ Ω and (19) lim R →∞ f ( R, R, ..., R ) > sup Ω ψ ( z, v ( z )) . Then u ≤ v in Ω .Proof. First, we consider the case where u − δ | z | is a subsolution of (18) for some δ > z ∈ Ω such that(20) ( u − v )( z ) = max Ω ( u − v ) > . For each
N >
0, we denote φ N ( z, w ) = u ( z ) − v ( w ) − N | z − w | , for all ( z, w ) ∈ Ω . Since Ω is compact and φ N is upper semicontinuous, there exists( z N , w N ) ∈ Ω such that φ N ( z N , w N ) = max Ω φ N .Moreover, by [2, Lemma 3.1], we can assume that z N and w N converge to z as N → ∞ .In particular, there exists N > z N , w N ∈ B ( z , R ) for every N > N , where0 < R < d ( z , ∂ Ω). By the maximum principle [2, Theorem 3.2], there exist Z N , W N ∈ H n such that (2 N ( z N − w N ) , Z N ) ∈ J + u ( z N ), (2 N ( z N − w N ) , W N ) ∈ J − v ( w N ) and Z N ≤ W N for all N > N . Hence, we have(21) F ( Z N − δI ) ≥ ψ ( z N , u ( z N )) , and(22) F ( W N ) ≤ ψ ( w N , v ( w N )) , and(23) F ( Z N ) ≤ F ( W N ) . Combining (22) and (23), we get(24) F ( Z N ) ≤ M, for all N > N , where M := sup { ψ ( z, v ( z )) : x ∈ B ( z , R ) } . Since lim R →∞ f ( R, ..., R ) >M , there exist R ≫ < r ≪ F ( RI ) = f ( R, ..., R ) > M + r. for all R > R . Then, by (24) and by the concavity of F , we have, for every 0 < ǫ < F ( Z N ) − F ( Z N − δI )2 δ ≥ F ( Z N + R I ) − F ( Z N ) R ≥ ǫF ( Z N /ǫ ) + (1 − ǫ ) F ( R I/ (1 − ǫ )) − MR ≥ (1 − ǫ )( M + r ) − MR , ULLY NONLINEAR ELLIPTIC EQUATIONS 9 for all
N > N . Letting ǫ = r M + r , we get(25) F ( Z N ) ≥ F ( Z N − δI ) + 2 M δ (2 M + r ) R , for every N > N . Combining (21), (22), (23) and (25), we obtain(26) ψ ( z N , u ( z N )) + 2 M δ (2 M + r ) R ≤ ψ ( w N , v ( w N )) , for every N > N . Since z N , w N → z and ψ is uniformly continuous, we also have(27) lim N →∞ ( ψ ( z N , u ( z N )) − ψ ( w N , u ( z N ))) = 0 . Moreover, it follows from [2, Lemma 3.1] that lim N →∞ ( u ( z N ) − v ( w N )) = max Ω ( u − v ) > ψ is non decreasing in the last variable, we get(28) lim inf N →∞ ( ψ ( w N , u ( z N )) − ψ ( w N , v ( w N ))) ≥ . Combining (26), (27) and (28), we get0 ≤ lim inf N →∞ ( ψ ( z N , u ( z N )) − ψ ( w N , v ( w N ))) ≤ − M δ (2 M + r ) R , and this is a contradiction. Thus, (20) is not true.In the general case, for each δ >
0, we denote u δ ( z ) = u ( z ) + δ ( | z | − A ), where A = max {| w | : w ∈ Ω } . By the above argument, we have u δ ≤ v in Ω for all δ > δ ց
0, we get u ≤ v in Ω.The proof is completed. (cid:3) By using the Perron method [2] and Theorem 1.3, we obtain the following result:
Proposition 4.2.
Let v ∈ LSC ∩ L ∞ (Ω) be a bounded supersolution of the equation (29) F ( Hw ) = ψ ( z, w ) , in Ω such that lim Ω ∋ z → ˆ z v ( z ) = v (ˆ z ) , for all ˆ z ∈ ∂ Ω and lim R →∞ f ( R, R, ..., R ) > sup Ω ψ ( z, v ( z )) . Denote by S the set of all viscosity subsolutions w to (29) satisfying w ≤ v . Then thefunction u ( z ) = sup { w ( z ) : w ∈ S } , is a discontinuous viscosity solution of (29) with u = u ∗ ∈ S .Proof. By Proposition 2.6, we have u ∗ is a viscosity subsolution of the equation F ( D w ) = ψ ( x, w ) in Ω. Moreover, since u ≤ v in Ω, we havelim sup Ω ∋ z → ˆ z u ∗ ( z ) = lim sup Ω ∋ z → ˆ z u ( z ) ≤ lim Ω ∋ z → ˆ z v ( z ) = v (ˆ z ) , for all ˆ z ∈ ∂ Ω. Then, it follows from Theorem 1.4 that u ∗ ≤ v . Hence, u ∗ ∈ S and u = u ∗ .It remains to show that u ∗ is a viscosity supersolution.Assume that there exist a point z ∈ Ω, an open neighbourhood U ⊂ Ω of z and afunction η ∈ C ( U ) such that η ( z ) = u ∗ ( z ), η ≤ u ∗ | U , Hη ( z ) ∈ M (Γ , n ) and F ( Hη ( z )) > ψ ( z , η ( z )). By the continuity of
F, ψ, η and Hη , there exist r, s > B ( z , r ) ⊂ U , Hη ( x ) − sI ∈ M (Γ , n ) for all z ∈ B ( z , r ) and F ( Hη ( z ) − sI ) > ψ ( z, η ( z ) + s ) , for every z ∈ B ( z , r ). Denote˜ η ( z ) = η ( z ) − s | z − z | + min { s, sr } .We have(30) F ( H ˜ η ( z )) ≥ ψ ( z, ˜ η ( z )) , ∀| z − z | ≤ r, and(31) ˜ η ( z ) ≤ u ( z ) , ∀ r/ ≤ | z − z | ≤ r. Denote ˜ u ( z ) = ( u ( z ) if z ∈ Ω \ B ( z , r ) , max { u ( z ) , ˜ η ( z ) } if z ∈ B ( z , r ) . Then ˜ u ∈ S and ˜ u ≥ u . Since u = sup { w : w ∈ S } , we have ˜ u = u . Moreover,˜ u ∗ ( z ) ≥ ˜ η ( z ) ≥ u ∗ ( z ) + min { s, sr } > u ∗ ( z ) . and it implies that ˜ u is not identical to u . We get a contradiction. Thus, u ∗ is a superso-lution of (29). (cid:3) Note that every harmonic function is a supersolution of (29). By using Theorem 1.3and Proposition 4.2, we obtain the following result which will be used in the next section:
Proposition 4.3.
Assume that Ω is a bounded smooth domain and ϕ is a continuousfunction on ∂ Ω satisfying lim R →∞ f ( R, R, ..., R ) > ψ ( z, sup ∂ Ω ϕ ) , for every z ∈ Ω . Suppose that there exists u ∈ U SC (Ω) such that u | ∂ Ω = ϕ and F ( Hu ) ≥ ψ ( z, u ) in the viscosity sense in Ω . Then, there exists a unique u ∈ C (Ω) such that u | ∂ Ω = ϕ and F ( Hu ) = ψ ( z, u ) in the viscosity sense in Ω . Maximal viscosity subsolutions
In this section, we study some properties of maximal viscosity subsolutions (see belowfor the defintion). Theorem 1.4 is deduced by combining Proposition 5.1 and Theorem5.2.Similar to the concept of maximal plurisubharmonic functions [19] (see also [15]), wesay that a viscosity subsolution u for (3) is maximal if u satisfies the following condition:For every open set U ⋐ Ω and for each v ∈ U SC ( U ) such that v is a subsolution for (3)in U and v ≤ u in ∂U , we have v ≤ u in U . Proposition 5.1.
Under the assumption of Theorem 1.4, the function Φ v is a maximalviscosity subsolution for (3) .Proof. By Proposition 4.2, we have Φ v is a viscosity subsolution of (3). We will show thatΦ v is maximal.Let U be a relatively open subset of Ω. Let w ∈ U SC ( U ) such that w is a subsolutionfor (3) in U and w ≤ Φ v in ∂U . By Proposition 2.3, the function ULLY NONLINEAR ELLIPTIC EQUATIONS 11 u ( z ) = ( Φ v ( z ) if z ∈ Ω \ U, max { w ( z ) , Φ v ( z ) } if z ∈ U, is a subsolution of (3) in Ω. Since u = Φ v ≤ v in Ω \ U , it follows from Theorem 1.3 that u ≤ v in Ω. Then, by the definition of Φ v , we get u ≤ Φ v in Ω.Thus Φ v is a maximal viscosity subsolution of (3). (cid:3) Theorem 5.2.
Assume that u is a maximal viscosity subsolution for (3) in Ω satisfying sup Ω u < ∞ and (32) lim R →∞ f ( R, R, ..., R ) > ψ ( z, M ) , for every x ∈ Ω , where M > sup Ω u is a constant. Then, for every relatively compactopen subset U of Ω , there exists a decreasing sequence u j of viscosity solutions of (3) in U such that lim j →∞ u j = u in U . In order to prove Theorem 5.2, we need the following lemma:
Lemma 5.3.
For every ǫ > , there exists an open set U with smooth boundary such that Ω ǫ ⋐ U ⋐ Ω , where Ω ǫ = { z ∈ Ω : d ( z, ∂ Ω) > ǫ } . Proof.
Consider the function g ( z ) = ( d ∗ χ ǫ/ )( z ), where χ ǫ/ is the standard modifier, ∗ is the convolution operator and d ( z ) = − d ( z, ∂ Ω) . We have g is well-defined and smoothin Ω ǫ/ . Moreover, for every ǫ/ < t < ǫ/ ǫ ⋐ U t ⋐ Ω ǫ/ ,where U t = { z ∈ Ω ǫ/ : g ( z ) < − t } . In particular, we have ∂U t = g − ( t ) ⋐ Ω ǫ/ for every ǫ/ < t < ǫ/
4. By Sard’s Theorem, there exists t ∈ ( ǫ/ , ǫ/
4) such that Dg ( z ) = 0 forevery z ∈ g − ( t ). Then, U := U t is a smooth open set satisfying Ω ǫ ⋐ U ⋐ Ω.The proof is completed. (cid:3)
Proof of Theorem 5.2.
By Lemma 5.3, there exists a smooth open set V such that U ⋐ V ⋐ Ω. By the compactness of U , we can assume that V has finite (open) connectedcomponents. Then the problem is reduced to the case where U is a smooth domain.Since u is Γ-subharmonic, we have u ∗ χ ǫ ց u as ǫ ց j ≫ V + j B n ⋐ Ω, where B n is the unit ball with respect to theEuclidean norm in C n = R n . For every j ∈ Z + , we denote v j = u ∗ χ ( j + j ) − . Then v j iswell-defined in U and sup U v j < M. By Proposition 4.3, for each j ∈ Z + , there exists aunique u j ∈ C ( U ) such that u j | ∂U = v j | ∂U and F ( Hu j ) = ψ ( z, u j ) in the viscosity sensein U . We will show that u j decreases to u as j → ∞ .Since u j is Γ-subharmonic in U , we havesup U u j = sup ∂U u j < M. It follows from Theorem 1.3 that u j ≥ u j +1 ≥ u , and then(33) ˜ u := lim j →∞ u j ≥ u, in U . It follows from Proposition 2.6 that ˜ u is a viscosity subsolution of the equation F ( Hw ) = ψ ( z, w ). Moreover, we have˜ u | ∂U = lim j →∞ u j | ∂U = lim j →∞ v j | ∂U = u | ∂U . Since u is a maximal viscosity subsolution of the equation F ( Hw ) = ψ ( z, w ), we get(34) u ≥ ˜ u, in U . Combining (33) and (34), we obtain u = ˜ u = lim j →∞ u j , in U . (cid:3) Corollary 5.4.
Under the assumption of Theorem 5.2, if u is bounded then u is also adiscontinuous viscosity solution of (3) .Proof. By Theorem 5.2, for every relatively compact open subset U of Ω, there exists adecreasing sequence u j of viscosity solutions of (3) in U such that lim j →∞ u j = u in U .Then, by Proposition 2.6, u ∗ = (inf j u j ) ∗ is a viscosity supersolution of (3) in U . Since U is arbitrary, we get u ∗ is a viscosity supersolution of (3) in Ω. Hence, u is a discontinuousviscosity solution of (3). (cid:3) Remark 5.5.
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Institute of Mathematics, Vietnam Academy of Science and Technology, 18 HoangQuoc Viet, Hanoi, Vietnam
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