CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

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1 Electronic Journal of Differential Equations, Vol ), No. 39, pp ISSN: URL: or ftp ejde.math.swt.edu login: ftp) CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS ESPEN R. JAKOBSEN & KENNETH H. KARLSEN Astract. Using the maximum principle for semicontinuous functions [3, 4], we prove a general continuous dependence on the nonlinearities estimate for ounded Hölder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and Hölder continuity results for ounded viscosity solutions of such equations. Our results are general enough to encompass Hamilton-Jacoi-Bellman-Isaacs s equations of zero-sum, two-player stochastic differential games. An immediate consequence of the results otained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations. 1. Introduction We are interested in ounded continuous viscosity solutions of fully nonlinear degenerate elliptic equations of the form F x, ux), Dux), D 2 ux)) = 0 in R N, 1.1) where the usual assumptions on the nonlinearity F are given in Section 2 see also [4]). We are here concerned with the prolem of finding an upper ound on the difference etween a viscosity susolution u of 1.1) and a viscosity supersolution ū of F x, ūx), Dūx), D 2 ūx)) = 0 in R N, 1.2) where F is another nonlinearity satisfying the assumptions given in Section 2. The sought upper ound for u ū should in one way or another e expressed in terms of the difference etween the nonlinearities F F. A continuous dependence estimate of the type sought here was otained in [8] for first order time-dependent Hamilton-Jacoi equations. For second order partial differential equations, a straightforward applications of the comparison principle [4] 1991 Mathematics Suject Classification. 35J60, 35J70, 49L25. Key words and phrases. fully nonlinear degenerate elliptic equation, viscosity solution, Hamilton-Jacoi-Bellman-Isaacs equation, continuous dependence estimate, vanishing viscosity method, convergence rate. c 2002 Southwest Texas State University. Sumitted August 8, Pulished May 6, E. R. Jakosen is supported y the Norwegian Research Council NFR) grant no /410. K. H. Karlsen is supported y the project Nonlinear partial differential equations of evolution type - theory and numerics: BeMatA program of the Research Council of Norway. 1

2 2 ESPEN R. JAKOBSEN & KENNETH H. KARLSEN EJDE 2001/39 gives a useful continuous dependence estimate when, for example, F is of the form F = F + f for some function f = fx). In general, the usefulness of the continuous estimate provided y the comparison principle [4] is somewhat limited. For example, it cannot e used to otain a convergence rate for the vanishing viscosity method, i.e., an explicit estimate in terms of ν > 0) of the difference etween the viscosity solution u of 1.1) and the viscosity solution u ν of the uniformly elliptic equation F x, u ν x), Du ν x), D 2 u ν x)) = ν u ν x) in R N. 1.3) Continuous dependence estimates for degenerate paraolic equations that imply, among other things, a rate of convergence for the corresponding viscosity method have appeared recently in [2] and [6]. In particular, the results in [6] are general enough to include, among others, the Hamilton-Jacoi-Bellman equation associated with optimal control of a degenerate diffusion process. Continuous dependence estimates for the Hamilton-Jacoi-Bellman equation have up to now een derived via proailistic arguments, which are entirely avoided in [6]. The main purpose of this paper is to prove a general continuous dependence estimate for fully nonlinear degenerate elliptic equations. In addition, we estalish existence, uniqueness, and Hölder continuity results for ounded viscosity solutions. Although the results presented herein cannot e found in the existing literature, their proofs are mild) adaptions as are those in [2, 6]) of the standard uniqueness machinery for viscosity solutions [4], which relies in turn on the maximum principle for semicontinuous functions [3, 4]. In [2, 6], the results are stated for nonlinearities F, F with a particular form, and as such the results are not entirely general. In this paper, we avoid this and our main result Theorem 2.1) covers general nonlinearities F, F. We present examples of equations which are covered y our results. In particular, an explicit continuous dependence estimate is stated for the second order Hamilton- Jacoi-Bellman-Isaacs equations associated with zero-sum, two-player stochastic differential games see, e.g., [9] for a viscosity solution treatment of these equations). For these equations such a result is usually derived via proailistic arguments, which we avoid entirely here. Also, it is worthwhile mentioning that a continuous dependence estimate of the type derived herein is needed for the proof in [1] of the rate of convergence for approximation schemes for Hamilton-Jacoi-Bellman equations. The rest of this paper is organized as follows: In Section 2 we state and prove our main results. In Section 3 we present examples of equations covered y our results. Finally, in Appendix A we prove some Hölder regularity results needed in section 2. Notation. Let e defined as follows: x 2 = m i=1 x i 2 for any x R m and any m N. We also let denote the matrix norm defined y M = sup e R p Me e, where M R m p is a m p matrix and m, p N. We denote y S N the space of symmetric N N matrices, and let B R and B R denote alls of radius R centered at the origin in R N and S N respectively. Finally, we let denote the natural orderings of oth numers and square matrices. Let USCU), CU) and C U) denote the spaces of upper semicontinuous functions, continuous functions, and ounded continuous functions on the set U. If f : R N R m p is a function and µ 0, 1], then define the following semi)

3 EJDE 2001/39 CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS 3 norms: f 0 = sup x R N fx), [f] µ = sup x,y R N x y fx) fy) x y µ, and f µ = f 0 + [f] µ. By C 0,µ R N ) we denote the set of functions f : R N R with finite norm f µ. 2. The Main Result We consider the fully nonlinear degenerate elliptic equation in 1.1). The following assumptions are made on the nonlinearity F : R N R R N S N R: C1) F CR N R R N S N ). C2) For every x, r, p, if X, Y S N, X Y, then F x, r, p, X) F x, r, p, Y ). C3) For every x, p, X, and for R > 0, there is γ R > 0 such that F x, r, p, X) F x, s, p, X) γ R r s), for R s r R. Our main result is stated in the following theorem: Theorem 2.1 Continuous Dependence Estimate). Let F and F e functions satisfying assumptions C1) C3). Moreover, let the following assumption hold for some η 1, η 2 0, µ 0, 1], and K > 0: F y, r, αx y) ɛy, Y ) F x, r, αx y) + ɛx, X) K x y µ + η 1 + α x y 2 + η2) 2 + ɛ 1 + x 2 + y 2) ), 2.1) for α, ɛ > 0, x, y R N, r R, r K, and X, Y S N satisfying ) ) ) 1 X 0 I I I 0 α + ɛ. 2.2) K 0 Y I I 0 I If u, ū C 0,µ0 R N ), µ 0 0, 1], satisfy in the viscosity sense F [u] 0 and F [ū] 0, then there is a constant C > 0 such that: sup R N u ū) C γ ) η1 + η µ µ0 2, where γ is defined in C3) with R = max u 0, ū 0 ), and µ µ 0 = minµ, µ 0 ). Remark 2.2. For simplicity, we consider only equations without oundary conditions. However, the techniques used herein can e applied to the classical Dirichlet and Neumann prolems, at least on convex domains. We refer to [5, 2] for the handling of classical oundary conditions. Finally, note that we are not ale to treat so-called oundary conditions in the viscosity sense [4, section 7C]. Before giving the proof, we state and prove the following technical lemma: Lemma 2.3. Let f USCR N ) e ounded from aove and define m, m ɛ 0, x ɛ R n as follows: m ɛ = max x R n{fx) ɛ x 2 } = fx ɛ ) ɛ x ɛ 2, Then as ɛ 0, m ɛ m and ɛ x ɛ 2 0. m = sup x R n fx).

4 4 ESPEN R. JAKOBSEN & KENNETH H. KARLSEN EJDE 2001/39 Proof. Choose any η > 0. By the definition of supremum there is an x R N such that fx ) m η. Pick an ɛ so small that ɛ x 2 < η, then the first part follows since m m ɛ = fx ɛ ) ɛ x ɛ 2 fx ) ɛ x 2 m 2η. Now define k ɛ = ɛ x ɛ 2. This quantity is ounded y the aove calculations since f is ounded from aove. Pick a converging susequence {k ɛ } ɛ and call the limit k 0). Note that fx ɛ ) k ɛ m k ɛ, so going to the limit yields m m k. This means that k 0, that is k = 0. Now we are done since if every susequence converges to 0, the sequence has to converge to 0 as well. Proof of Theorem 2.1. We start y defining the following quantities φx, y) := α 2 x y 2 + ɛ x 2 + y 2), 2 ψx, y) := ux) ūy) φx, y), σ := sup ψx, y) := ψx 0, y 0 ), x,y R N where the existence of x 0, y 0 R N is assured y the continuity of ψ and precompactness of sets of the type {φx, y) > k} for k close enough to σ. We shall derive a positive upper ound for σ, so we may assume that σ > 0. We can now apply the maximum principle for semicontinuous functions [4, Theorem 3.2] to conclude that there are symmetric matrices X, Y S N such that D x φx 0, y 0 ), X) J 2,+ ux 0 ), D y φx 0, y 0 ), Y ) J 2, ūy 0 ), where X and Y satisfy inequality 2.2) for some constant K. So y the definition of viscosity suand supersolutions we get 0 F y 0, ūy 0 ), D y φx 0, y 0 ), Y ) F x 0, ux 0 ), D x φx 0, y 0 ), X). 2.3) Since σ > 0 it follows that ux 0 ) ūy 0 ). We can now use C3) on F ) and the fact that ux 0 ) ūy 0 ) = σ + φx 0, y 0 ) σ to introduce σ and to rewrite 2.3) in terms of ūy 0 ): F x 0, ux 0 ), D x φx 0, y 0 ), X) F x 0, ūy 0 ), D x φx 0, y 0 ), X) so that 2.3) ecomes γux 0 ) ūy 0 )) γσ, γσ F y 0, ūy 0 ), D y φx 0, y 0 ), Y ) F x 0, ūy 0 ), D x φx 0, y 0 ), X). Now since u, ū are ounded, D y φx 0, y 0 ) = αx 0 y 0 ) ɛy 0, and D x φx 0, y 0 ) = αx 0 y 0 ) + ɛx 0, we may use 2.1) to get the estimate [ γσ K x 0 y 0 µ + η 1 + α x 0 y η2) 2 + ɛ 1 + x0 2 + y 0 2) ]. 2.4) By considering the inequality 2ψx 0, y 0 ) ψx 0, x 0 ) + ψy 0, y 0 ), and Hölder continuity of u and ū, we find α x 0 y 0 2 ux 0 ) uy 0 ) + ūx 0 ) ūy 0 ) Const x 0 y 0 µ0, which means that x 0 y 0 Const α 1/0). Furthermore, y Lemma 2.3 there is a continuous nondecreasing function m : [0, ) [0, ) satisfying m0) = 0 and ɛ x 0 2, ɛ y 0 2 mɛ). 2.5)

5 EJDE 2001/39 CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS 5 The last two estimates comined with 2.4) yield ] γσ Const [α µ 0 + η 1 + α µ αη2 2 + mɛ). 2.6) Without loss of generality, we may assume η 2 2 < 1 and µ µ 0 = µ 0. Now we choose α such that α µ0/0) = αη 2 2, and oserve that this implies that α > 1, which again means that α µ/0) α µ0/0). Thus we can ound the the smaller term y the larger term. By the definition of σ, ux) ūx) ɛ x 2 σ for any x R N, so sustituting our choice of α into 2.6), leads to the following expression γux) ūx)) Const [ η 1 + η µ µ0 2 + mɛ) ] + γɛ x 2, and we can conclude y sending ɛ to 0. Next we state results regarding existence, uniqueness, and Hölder continuity of ounded viscosity solutions of 1.1). To this end, make the following natural assumptions: C4) There exist µ 0, 1], K > 0, and γ 0R, γ 1R, K R > 0 for any R > 0 such that for any α, ɛ > 0, x, y R N, R r R, X, Y S N satisfying 2.2), F x, r, αx y) ɛy, Y ) F y, r, αx y) + ɛx, X) γ 0R x y µ + γ 1R α x y 2 + K R ɛ 1 + x 2 + y 2). C5) M F := sup R N F x, 0, 0, 0) <. Theorem 2.4. Assume that C1) C5) hold and that γ R = γ is independent of R. Then there exists a unique ounded viscosity solution u of 1.1) satisfying γ u 0 M F. Proof. Under conditions C1) C4) we have a strong comparison principle for ounded viscosity solutions of 1.1) see also [4]). By assumptions C3) and C5) we see that M F /γ and M F /γ are classical supersolution and susolution respectively of 1.1). Hence existence of a continuous viscosity solution satisfying the ound γ u 0 M F follows from Perron s method, see [4]. Uniqueness of viscosity solutions follows from the comparison principle. Remark 2.5. The condition that γ R e independent of R and condition C5) are not necessary for having strong comparison and uniqueness. Theorem 2.6. Assume that C1) C5) hold and that γ R = γ is independent of R. Then the ounded viscosity solution u of 1.1) is Hölder continuous with exponent µ 0 0, µ]. Proof. This theorem is consequence Lemmas A.1 and A.3, which are stated and proved in the appendix. The final result in this section concerns the rate of convergence for the vanishing viscosity method, which considers the uniformly elliptic equation 1.3). Existence, uniqueness, oundedness, and Hölder regularity of viscosity solutions of 1.3) follows from Theorems 2.4 and 2.6 under the same assumptions as for 1.1). Theorem 2.7. Assume that C1) C5) hold and that γ R = γ is independent of R. Let u and u ν e C 0,µ0 R N ) viscosity solutions of 1.1) and 1.3) respectively. Then u u ν 0 Const ν µ0/2.

6 6 ESPEN R. JAKOBSEN & KENNETH H. KARLSEN EJDE 2001/39 Proof. It is clear from Theorem 2.4, Lemma A.1, and the proof of Lemma A.3 that µ 0 µ and that u ν µ0 can e ounded independently of ν. Now we use Theorem 2.1 with F [u] = F [u] ν u. This means that F x, r, αx y) ɛy, Y ) F y, r, αx y) + ɛx, X) νtr[y ] + γ 0R x y µ + γ 1R α x y 2 + K R ɛ 1 + x 2 + y 2), with R = M F /γ. From 2.2) it follows that if e i is a standard asis vector in R N, then e i Y e i Kα + ɛ), so tr[y ] NKα + ɛ). This means that 2.1) is satisfied with η 1 = 0 and η2 2 = NKν. Now Theorem 2.1 yield u u ν Const ν µ0/2. Interchanging u, F and u ν, F in the aove argument yields the other ound. 3. Applications In this section, we give three typical examples of equations handled y our assumptions. It is quite easy to verify C1) C5) for these prolems. We just remark that in order to check C4), it is necessary to use a trick y Ishii and the matrix inequality 2.2), see [4, Example 3.6]. Example 3.1 Quasilinear equations). tr[σx, Du)σx, Du) T D 2 u] + fx, u, Du) + γu = 0 in R N, where γ > 0, for any R > 0, σ matrix-valued) and f real-valued) are uniformly continuous on R N B R and R N [ R, R] B R respectively, and for any R > 0 there are K, K R > 0 such that the following inequalities hold: σx, p) σy, p) K x y, fx, t, p) fy, t, p) K R p x y + x y µ ), for t R, fx, t, p) fx, s, p) when t s, fx, 0, 0) K, for any x, y, p R N and t, s R. Example 3.2 Hamilton-Jacoi-Bellman-Isaacs equations). In R N, { sup inf tr [ σ α,β x)σ α,β x) T D 2 u ] } α,β x)du+c α,β x)u+f α,β x) = 0, 3.1) α A β B where A, B are compact metric spaces, c γ > 0, and [σ α,β ] 1, [ α,β ] 1, [c α,β ] µ, [f α,β ] µ + f α,β 0 are ounded independent of α, β. Example 3.3 Sup and inf of quasilinear operators). In R N, { sup inf tr [ σ α,β x, Du)σ α,β x, Du) T D 2 u ] } + f α,β x, u, Du) + γu = 0, α A β B where A, B are as aove, γ > 0, and σ, f continuous satisfies the same assumptions as in Example 3.1 uniformly in α, β. We end this section y giving an explicit continuous dependence result for second order Hamilton-Jacoi-Bellman-Isaacs equations associated with zero-sum, twoplayer stochastic differential games with controls and strategies taking values in A and B see Example 3.2). We refer to [9] for an overview of viscosity solution theory and its application to the partial differential equations of deterministic and stochastic differential games.

7 EJDE 2001/39 CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS 7 Theorem 3.4. Let u and ū e viscosity solutions to 3.1) with coefficients σ,, c, f) and σ,, c, f) respectively. Moreover, assume that oth sets of coefficients satisfy the assumptions stated in Example 3.2. Then there is a µ 0 0, µ] such that u, ū C 0,µ0 R N ) and u ū 0 C for some constant C. sup A B + sup A B [ σ α,β σ α,β µ0 0 + α,β ] α,β µ0 0 [ c α,β c α,β 0 + f α,β f α,β 0 ]), Proof. With η 1 = sup [ c α,β c α,β 0 + f α,β f ] [ α,β 0, η2 2 = sup A B A B σ α,β σ α,β 2 0+ α,β α,β 2 0 we apply Theorem 2.1 to u ū and then to ū u to otain the result. Appendix A. Hölder Regularity We consider the two cases γ > 2γ 1 and 0 < γ < 2γ 1 separately. Lemma A.1. Assume that C1) C5) hold and that u is a ounded viscosity solution of 1.1). Let R = u 0, define γ := γ R, and similarly define γ 0, γ 1, K. If γ > 2γ 1 then u C 0,µ, and for all x, y R N, ux) uy) γ 0 γ 2γ 1 x y µ. Proof. This proof is very close to the proof of Theorem 2.1, and we will only indicate the differences. Let σ, φ, x 0, y 0 e defined as in Theorem 2.1 when ψx, y) = ux) uy) 2φx, y). Note the factor 2 multiplying φ. We need this factor to get the right form of our estimate! A consequence of this is that we need to change α, ɛ to 2α, 2ɛ whenever we use C4) and 2.2). Now we proceed as in the proof of Theorem 2.1. We use the maximum principle for semicontinuous functions and the definition of viscosity su- and supersolutions u is oth!), we use C3) together with ux 0 ) uy 0 ) = σ + α x 0 y ɛ x y 0 2) σ + α x 0 y 0 2, and finally we use C4) and all the aove to conclude that γσ γ 0 x 0 y 0 µ γ 2γ 1 )α x 0 y ωɛ), ], A.1) for some modulus ω. Here we have also used the ounds 2.5) on x 0, y 0. Compare with 2.4). Note that for any k 1, k 2 > 0, max r 0 { k1 r µ k 2 αr 2} = c 1 k 2 1 αk 2 ) µ where c 1 = Furthermore for fixed α, Lemma 2.3 yields lim σ = sup ux) uy) α x y 2 ) := m. ɛ 0 x,y R N µ ) µ µ )

8 8 ESPEN R. JAKOBSEN & KENNETH H. KARLSEN EJDE 2001/39 So let k 1 = γ 0 and k 2 = γ 2γ 1 > 0 y assumption), and go to the limit ɛ 0 for α fixed in A.1). The result is m k 2 1 µ γk 2 c 1 α µ γ 2γ 1 γ γ 0 γ 2γ 1 ) 2 ) 2 where k = γ0 γ 2γ 1 c 1. Since, in view of A.2), c 1 α µ kα µ ux) uy) m + α x y 2 kα µ + α x y 2, we can minimize with respect to α otain where c 2 = ux) uy) min α 0 µ ) 2 + µ {kα µ + α x y 2 } = c 2 k 2 x y µ, ) µ 2., A.2) Now we can conclude y sustituting for k and oserving that c 2 c Remark A.2. Lemma A.1 is not sharp. It is possile to get sharper results using a test function of the type φx, y) = L x y δ + ɛ x 2 + y 2) and playing with all three parameters L, δ, ɛ. However, C4) is adapted to the test function used in this paper, so changing the test function means that we must change C4) too. We will now use the previous result and an iteration technique introduced in [7] for first order equations) to derive Hölder continuity for solutions of 1.1) for 0 < γ < 2γ 1. Note that since Lemma A.1 is not sharp, our next result will not e sharp either. We also note that in the case γ = 2γ 1 the Hölder exponent is of course at least as good as for γ = 2γ 1 ɛ, ɛ > 0 small. Lemma A.3. Assume that C1) C5) hold and that u is a ounded viscosity solution of 1.1). Let R = u 0, define γ := γ R, and similarly define γ 0, γ 1, K. If 0 < γ < 2γ 1 then u C 0,µ0 R N ) where µ 0 = µ γ 2γ 1. Proof. Let λ > 0 e such that γ + λ 2γ and let v C 0,µ R N ) e in the set X := { f CR N ) : f 0 M F /γ }. Then note that ±M F /γ are respectively super- and susolutions of the following equation: F x, ux), Dux), D 2 ux)) + λux) = λvx) x R N. A.3) Let T denote the operator taking v to the viscosity solution u of A.3). It is well-defined ecause y Theorem 2.4 there exists a unique viscosity solution u of equation A.3). Furthermore y Theorem A.1 and the fact that ±M F /γ are respectively super- and susolutions of A.3), we see that For v, w C 0,µ R N ) X we note that T : C 0,µ R N ) X C 0,µ R N ) X. T w w v 0 λ/γ + λ) and T v w v 0 λ/γ + λ)

9 EJDE 2001/39 CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS 9 are oth susolutions of A.3) ut with different right hand sides, namely λv and λw respectively. So y using the comparison principle Theorem 2.4 twice comparing with T v and T w respectively) we get: T w T v 0 λ γ + λ w v 0 w, v C 0,µ R N ) X. A.4) Let u 0 x) = M F /γ and u n x) = T u n 1 x) for n = 1, 2,... Since C 0,µ R N ) X is a Banach space and T a contraction mapping A.4) on this space, Banach s fix point theorem yields u n u C 0,µ R N ) X. By the staility result for viscosity solutions of second order PDEs, see Lemma 6.1 and Remark 6.3 in [4], u is the viscosity solution of 1.1). Since F [u]+λu = 0 and F [u n ]+λu n = λu n 1, the continuous dependence result Theorem 2.1 yields u u n 0 λ ) n λ λ + γ u un 1 0 u u 0 0. A.5) λ + γ Furthermore y Theorem A.1 we have the following estimate on the Hölder seminorm of u n : [u n ] µ γ 0 + λ[u n 1 ) n ] µ λ [u 0 ] µ + K ), A.6) γ + λ 2γ 1 γ + λ 2γ 1 where the constant K does not depend on n or λ 1). Now let x, y R N and note that ux) uy) ux) u n x) + u n x) u n y) + u n y) uy). Using A.5) and A.6) we get the following expression: { ) n ) n } λ λ ux) uy) Const + x y µ. A.7) γ + λ γ + λ 2γ 1 Now let t = x y and ω e the modulus of continuity of u. Fix t 0, 1) and define λ in the following way: λ := 2γ 1 µ n log 1 ). t Note that if n t is a sufficiently large numer, then n n t implies that γ+λ 2γ Using this new notation, we can rewrite A.7) in the following way: { ωt) Const 1 + µγ ) ) n 1 1 log µ γ 2γ ) ) } n log t µ. 2γ 1 t n 2γ 1 t n Letting n, we otain ωt) Const { t µγ/2γ1 + t µγ/2γ1 µ t µ}. Now we can conclude since this inequality must hold for any t 0, 1).

10 10 ESPEN R. JAKOBSEN & KENNETH H. KARLSEN EJDE 2001/39 References [1] G. Barles and E. R. Jakosen. On the convergence rate for approximation schemes for the Hamilton-Jacoi-Bellman equation. M2AN Math. Model. Numer. Anal., 361):33 54, [2] B. Cockurn, G. Gripenerg, and S.-O. Londen. Continuous dependence on the nonlinearity of viscosity solutions of paraolic equations. J. Differential Equations, 1701): , [3] M. G. Crandall and H. Ishii. The maximum principle for semicontinuous functions. Differential Integral Equations, 36): , [4] M. G. Crandall, H. Ishii, and P.-L. Lions. User s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. N.S.), 271):1 67, [5] G. Gripenerg Estimates for viscosity solutions of paraolic equations with Dirichlet oundary conditions. Proc. Amer. Math. Soc. To appear. [6] E. R. Jakosen and K. H. Karlsen. Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate paraolic equations. J. Differential Equations. To appear. [7] P.-L. Lions. Existence results for first-order Hamilton-Jacoi equations. Ricerche Mat., 321):3 23, [8] P. E. Souganidis. Existence of viscosity solutions of Hamilton-Jacoi equations. J. Differential Equations, 563): , [9] P. E. Souganidis. Two-player, zero-sum differential games and viscosity solutions. In Stochastic and differential games, pages Birkhäuser Boston, Boston, MA, Espen R. Jakosen Department of Mathematical Sciences Norwegian University of Science and Technology N 7491 Trondheim, Norway address: erj@math.ntnu.no URL: erj Kenneth H. Karlsen Department of Mathematics University of Bergen Johs. Brunsgt. 12 N 5008 Bergen, Norway address: kennethk@mi.ui.no URL: kennethk