Sobolev Spaces with Weights in Domains and Boundary Value Problems for Degenerate Elliptic Equations

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1 Sobolev Saces with Weights in Domains and Boundary Value Problems for Degenerate Ellitic Equations S. V. Lototsky Deartment of Mathematics, M.I.T., Room 2-267, 77 Massachusetts Avenue, Cambridge, MA , Published in Methods and Alications of Analysis, Vol.7, No. 1, , 2000 Abstract. A family of Banach saces is introduced to control the interior smoothness and boundary behavior of functions in a general domain. Interolation, embedding, and other roerties of the saces are studied. As an alication, a certain degenerate second-order ellitic artial differential equation is considered. 1. Introduction Let G be a domain in R d with a non-emty boundary G and ρ G (x) = dist(x, G). For 1 < and θ R define the sace L,θ (G) as follows: L,θ (G) = {u : u(x) ρ θ d (x)dx < }. G G Then we can define the saces H,θ m (G), m = 1, 2,..., so that H m,θ(g) = {u : u, ρ G Du,..., ρ m GD m u L,θ }, 1991 Mathematics Subject Classification. 35J70, 35K65. Key words and hrases. Interolation saces, ointwise multiliers, duality, embedding theorems. This work was artially suorted by the NSF grant DMS

2 where D k denotes generalized derivative of order k. The objective of the current aer is to define saces H,θ (G), R, so that, for ositive integer, the saces H,θ (G) coincide with the ones introduced above. It will be shown that these saces can be easily defined using the saces H (R d ) of Bessel otentials. Note that u H 1,d (G) if and only if u/ρ G, Du L (G), which means that, for bounded G, the sace H,d 1 (G) coincides with the sace H 1 (G). As a result, the saces H,θ (G) can be considered as a certain generalization of the usual Sobolev saces on G with zero boundary conditions. A major alication of the saces H,θ (G) is in the analysis of the Dirichlet roblem for stochastic arabolic equations [5, 7]. Some of the saces H,θ (G) have been studied before. Lions and Magenes [6] introduced what corresonds to H 2,d (G). They constructed the scale by interolating between the ositive integer for > 0 and used duality for < 0. Krylov [3] defined the saces H,θ (Rd +), where R d + is the half-sace. After that, if G is sufficiently regular and bounded, then H,θ (G) can be defined using the artition of unity, and this was done in [7]. Other related examles and references can be found in Chater 3 of [10]. In this aer, an intrinsic definition (not involving R d +) of the saces H,θ (G) is given for a general domain G, and the basic roerties of the saces are studied. Once a suitable definition of the saces is found, most of the roerties follow easily from the known results. Definition and roerties of the saces H,θ (G) are resented in Sections 2, 3, and 4. Roughly seaking, the index controls the smoothness inside the domain, and the index θ controls the boundary behavior. In articular, the sace H,θ (G) with sufficiently large and θ < 0 contains functions that are continuous in the closure of G and vanish on the boundary. In Section 5 some results are resented about solvability of certain degenerate ellitic equations in a general domain G. Throughout the aer, D m denotes a artial derivative of order m, that is, D m = m / x m 1 1 x m d d for some m m d = m. For two Banach saces, X, Y, notation X Y means that X is continuously embedded into Y. 2

3 2. Definition and main roerties of the weighted saces in domains Let G R d be a domain (oen connected set) with non-emty boundary G, and c > 1, a real number. Denote by ρ G (x), x G, the distance from x to G. For n Z and a fixed integer k 0 > 0 define the subsets G n of G by G n = {x G : c n k 0 < ρ G (x) < c n+k 0 }. Let {ζ n, n Z} be a collection of non-negative functions with the following roerties: ζ n C0 (G n ), D m ζ n (x) N(m)c mn, ζ n (x) = 1. n Z The function ζ n (x) can be constructed by mollifying the characteristic (indicator) function of G n. If G n is an emty set, then the corresonding ζ n is identical zero. If u D (G), that is, u is a distribution on C 0 (G), then ζ n u is extended by zero to R d so that ζ n u D (R d ). The sace H,θ (G) is defined as a collection of those u D (G), for which ζ n u is in H and the norms ζ n u H, n Z, behave in a certain way. Recall [10, Section 2.3.3] that the sace of Bessel otentials H is the closure of C 0 (R d ) in the norm F 1 (1 + ξ 2 ) /2 F L(R d ), where F is the Fourier transform with inverse F 1. Definition 2.1. Let G be a domain in R d, θ and, real numbers, and (1, + ). Take a collection {ζ k, n Z} as above. Then { H,θ (G) := u D (G) : u := } c nθ ζ H,θ (G) n (c n )u(c n ) <. H n Z (2.1) Since H 1 H 2 for 1 > 2, the definition imlies that H 1,θ (G) H 2,θ (G) for 1 > 2 and all θ R, 1 <. Still, it is necessary to establish correctness of Definition 2.1 by showing that the norms defined according to 3

4 (2.1) are equivalent for every admissible choice of the numbers c, k 0 and the functions ζ n. Proving this equivalence is the main goal of this section. Proosition If u is comactly suorted in G, then u H,θ (G) if and only if u H. 2. The set C 0 (G) is dense in every H,θ (G). 3. If = m is a non-negative integer, then H,θ (G) = { u : ρ k GD k u L,θ (G), 0 k m }, (2.2) where L,θ (G) = L (G, ρ θ d G (x)dx). 4. If {ξ n, n Z} is a system of function so that ξ n C 0 (G n ), D m ξ n (x) N(m)c mn, then n Z c nθ ξ n (c n )u(c n ) H N u H,θ with N indeendent of u, and if in addition n ξ(x) δ > 0 for all x G, then the reverse inequality also holds. Proof. 1. The result is obvious because, for comactly suorted u, the sum in (2.1) contains only finitely many non-zero terms. 2. Given u H,θ (G), first aroximate u by u K = u k K ζ k, and then mollify u K. 3. The result follows because, for all ν R and all x in the suort of ζ n, N 1 c νn ρ ν G (x) N 2 with N 1 and N 2 indeendent of n, ν, x. 4. Use that, by Theorem in [9], C 0 (R d ) functions are ointwise multiliers in every H. Remark 2.3. In the future we will also use a system of non-negative C 0 (R d ) functions {η n, n Z} with the following roerties: η n is suorted in {x : c n k 0 1 < ρ G (x) < c n+k 0+1 }, η(x) = 1 on the suort of ζ n, D m η n (x) N(m)c mn. By Proosition 2.2(4) the functions η n can relace ζ n in (2.1). 4

5 Proosition For every (1, ) and θ, R, the sace H,θ (G) is a reflexive Banach sace with the dual H,θ (G), where 1/ + 1/ = 1 and θ/ + θ / = d. 2. If 0 < ν < 1, = (1 ν) 0 + ν 1, 1/ = (1 ν)/ 0 + ν/ 1, and θ = (1 ν)θ 0 + νθ 1, then H,θ (G) = [H 0 0,θ 0 (G), H 1 1,θ 1 (G)] ν, (2.3) where [X, Y ] ν is the comlex interolation sace of X and Y (see [10, Section 1.9] for the definition and roerties of the comlex interolation saces). Proof. Let l θ (H ) be the set of sequences with elements from H and the norm {f n } l θ (H ) = n Z c nθ f n H. Define bounded linear oerators S,θ : H,θ (G) lθ (H ) and R,θ : l θ (H ) H,θ (G) as follows: (S,θ u) n (x) = ζ n (c n x)u(c n x), R,θ ({f n })(x) = n Z η n (x)f n (c n x). Note that R,θ S,θ = Id H,θ (G). Then, by Theorem in [10], the sace H,θ (G) is isomorhic to S,θ(H,θ (G)), which is a closed subsace of a reflexive Banach sace l(h θ ). This means that H,θ (G) is also a reflexive Banach sace. The interolation result (2.3) follows from Theorems and in [10]. Denote by (, ) the duality between H and H. If v H,θ (G), then, by the Hölder inequality, v defines a bounded linear functional on H,θ (G) as follows: u v, u = n c nd (v n, u n ), where u n (x) = ζ n (c n x)u(c n x) and v n (x) = η n (c n x)v(c n x). u, v C 0 (G), then v, u = G u(x)v(x)dx. Note that if Conversely, if V is a bounded linear functional on H,θ (G), then we use the Hahn-Banach theorem and the equality (l(h θ )) = l θ / (H ) to construct v H,θ (G) so that V (u) = v, u. 5

6 One consequence of (2.3) is the interolation inequality u H,θ (G) = ɛ u H 0,θ (G) + N(ν,, ɛ) u H 1 0,θ (G), ɛ > 0. (2.4) 1 Corollary 2.5. The sace H,θ does not deend, u to equivalent norms, on the secific choice of the numbers c and k 0 and the functions ζ n. Moreover, the distance function ρ G can be relaced with any measurable function ρ satisfying N 1 ρ G (x) ρ(x) N 2 ρ G (x) for all x G, with N 1, N 2 indeendent of x. Proof. By Proosition 2.2(3), we have the result for non-negative integer. For general > 0 the result then follows from (2.3), where we take 0 = 1 =, θ 0 = θ 1 = θ, and integer 0, 1. After that, the result for < 0 follows by duality. In view of Corollary 2.5, it will be assumed from now on that c = 2 and k 0 = 1. Remark 2.6. If X is a Banach sace of generalized functions on R d, then we can define the sace X θ (G) according to (2.1) by relacing the norm H with X. In articular, we can define the saces B,q;θ (G) and F,q;θ (G) using the saces B,q and F,q described in Section of [10]. Results similar to Proositions 2.2 and 2.4 can then be roved in the same way. Examle. (cf. [5, Definition 1.1].) Let G = R d + = {x = (x 1,..., x d ) R d : x 1 > 0} and ζ C0 ((b 1, b 2 )), 0 < b 1, b 2 > 3b 1. Define ζ(x) = ζ(x 1 ) and { H,θ = u D (G) : u := } e nθ ζu(e n ) <. H,θ H n Z It follows that H,θ = H,θ (Rd +) with H,θ (Rd +) defined according to (2.1), where c = e, ρ G (x) = x 1, ζ n (x) = ζ(e n x)/ k ζ(ek x), and k 0 is the smallest ositive integer for which b 1 > e k 0, b 2 < e k Pointwise multiliers, change of variables, and localization A function a = a(x) is a ointwise multilier in a liner normed function sace X if the oeration of multilication by a is defined and continuous in X. 6

7 To describe the ointwise multiliers in the sace H,θ (G), we need some reliminary constructions. For R define [0, 1) as follows. If is an integer, then = 0; if is not an integer, then is any number from the interval (0, 1) so that + is not an integer. The sace of ointwise multiliers in H is given by L (R d ), = 0 B + = C n 1,1 (R d ), = n = 1, 2,... C + (R d ), otherwise, where C n 1,1 (R d ) is the set of functions from C n 1 (R d ) whose derivatives of order n 1 are uniformly Lischitz continuous. In other words, if u H and a B +, then au H N(, d, ) a B + u H. For non-negative integer this follows by direct comutation, for ositive non-integer, from Corollary 4.2.2(ii) in [9], and for negative, by duality. For ν 0, define the sace A ν (G) as follows: (1) if ν = 0, then A ν (G) = L (G); (2) if ν = m = 1, 2,..., then A ν (G) = {a : a, ρ G Da,..., ρ m 1 G Dm 1 a L (G), ρ m GD m 1 a C 0,1 (G)}, a A ν (G) = m 1 k=0 ρ k G D k a L (G) + ρ m GD m a C 0,1 (G); (3) if ν = m + δ, where m = 0, 1, 2,..., δ (0, 1), then A ν (G) = {a : a, ρ G Da,..., ρ m GD m a L (G), ρ ν GD m a C δ (G)}, a A ν (G) = m ρ k G D m a L (G) + ρ ν GD m a C (G). δ k=0 Note that, for every a A ν (G) and n Z, ζ n (2 n )a(2 n ) B ν N a A ν (G) (3.1) 7

8 with N indeendent of n. Theorem 3.1. If a A + (G), then au H,θ (G) N(d,, ) a A + (G) u H,θ (G). Proof. We have to show that η n (2 n )a(2 n ) B + N a A + (G) with constant N indeendent of n. The result is obvious for = 0; for (0, 1] it follows from the inequality (with δ = + ) η n (x)a(x) η n (y)a(y) η n (x)ρ δ G (x) a(x)ρδ G(x) a(y)ρ δ G(y) + a(y) η n (x) η n (y) + η n (x)ρ δ G (x) a(y) ρδ G(x) ρ δ G(y) and the observation that both 2 n η n and ρ G are uniformly Lischitz continuous. If > 1, we aly the same arguments to the corresonding derivatives. Next, we study the following question: for what maings ψ : G 1 G 2 is the oerator u( ) u(ψ( )) continuous from H,θ (G 2) to H,θ (G 1)? Theorem 3.2. Suose that G 1 and G 2 are domains with non-emty boundaries and ψ : G 1 G 2 is a C 1 -diffeomorhism so that ψ( G 1 ) = G 2. For a ositive integer m define ν = max(m 1, 0). If Dψ A ν (G 1 ), then, for every [ ν, m] and u H,θ (G 2), with N indeendent of u. u(ψ( )) H,θ (G 1) N u H,θ (G 2) Proof. Denote by φ the inverse of ψ. If = 0, then u(ψ( )) = u(y) ρ θ d H,θ (G 1) G 1 (φ(y)) Dφ(y) dy G 2 and the result follows because uniform Lischitz continuity of ρ Gi, ψ, and φ imlies that the ratio ρ G1 (φ(x))/ρ G2 (x) is uniformly bounded from above and below. If = m, the comutation is similar. After that, for (0, m), the result follows by interolation, and for [ ν, 0), by duality. 8

9 The last result in this section is about localization. It answers the following question: for what collections of C (G) functions {ξ k, k = 1, 2,...} are the values of u and H,θ (G) n uζ n comarable? To begin with, let us H,θ (G) recall the corresonding theorem for H. Theorem 3.3. ([4, Lemma 6.7].) If {ξ k, k = 0, 1,..., } is a collection of C (R d ) functions so that su x k Dm ξ k (x) M(m), m 0, then k 0 ξ kv N v with N indeendent of v. If in addition inf x k ξ k(x) δ then the reverse inequality also holds: v H H H N k 0 ξ kv with N indeendent of v. H The following is the analogous result for H,θ (G). Theorem 3.4. Suose that {χ k, k 1} is a collection of C (G) functions so that su x G k ρm G (x) Dm χ k (x) N(m), m 0. Then k uχ k H,θ (G) N u H (G). If, in addition, inf x G k χ k(x) δ for some δ > 0, then,θ u N H,θ (G) k uχ k H (G).,θ Proof. With ˆχ 0,n = 1 η n, ˆχ k,n (x) = χ k (x)η n (x), k 1, we find k 1 uχ k H,θ (G) = n Z 2 nθ ˆχ k,n (2 n )ζ n (2 n )u(2 n ). H Both statements of the theorem now follow from Theorem 3.3. k 0 Examle. (cf. [7, Section 2].) Let G be a bounded domain of class C +2 with a artition of unity χ 0 C 0 (G), χ 1,... χ K C 0 (R d ) and the corresonding diffeomorhism ψ 1,..., ψ K that stretch the boundary inside the suort of χ 1,..., χ K (see, for examle, Chater 6 of [2] for details). Then an equivalent norm in H,θ (G) is given by u H,θ (G) = uχ 0 H + K m=1 u(ψ 1 m ( ))χ m (ψ 1 m ( )) H,θ (Rd + ). 9

10 Indeed, writing to denote the equivalent norms, we deduce from Proosition 2.2(1) and Theorems 3.2 and 3.4 that u H,θ (G) K m=0 uχ m H,θ (G) uχ 0 H + K m=1 u(ψ 1 m ( ))χ m (ψ 1 m ( )) H,θ (Rd + ). 4. Further roerties of the saces H,θ (G) Let ρ = ρ(x) be a C (G) function so that N 1 ρ G (x) ρ(x) N 2 ρ G (x) and ρ m G (x)dm+1 ρ(x) N(m) for all x G and for every m = 0, 1,.... In articular, ρ(x) = 0 on G and all the first-order artial derivatives of ρ are ointwise multiliers in every H,θ (G). An examle of the function ρ is ρ(x) = n Z 2 n ζ n (x), where the functions ζ n are as in Section 2 with c = 2. Theorem The following conditions are equivalent: u H,θ (G); u H 1,θ (G) and ρdu H 1,θ (G); u H 1,θ (G) and D(ρu) H 1,θ (G). In addition, under either of these conditions, the norm u H,θ (G) can be relaced by u H 1,θ (G) + ρdu H 1,θ (G) or by u H 1,θ (G) + D(ρu) H 1 (G).,θ 2. For every ν, R, ρ ν H,θ (G) = H,θ ν (G) and H,θ ν (G) is equivalent to ρ ν H,θ (G). (4.1) Proof. It is sufficient to reeat the arguments from the roofs of, resectively, Theorem 3.1 and Corollary 2.6 in [3]. Corollary If u H,θ (G), then Du H 1,θ+ (G) and Du H 1,θ+ (G) N(d,,, θ) u H,θ (G). 10

11 2. If ρ G is a bounded function (for examle, if G is a bounded domain), then H,θ 1 (G) H,θ 2 (G) for θ 1 < θ 2 and H (G) H,θ (G) for θ d. Recall the following notations for continuous functions u in G: u(x) u(y) u C(G) = su u(x), [u] C ν (G) = su, ν (0, 1). x G x,y G x y ν Theorem 4.3. Assume that d/ = k + ν for some k = 0, 1,... and ν (0, 1). If u H,θ (G), then m k=0 ρ k+θ/ D k u C(G) + [ρ m+ν+θ/ D m u] C ν (G) N(d,,, θ) u H,θ (G). Proof. It is sufficient to reeat the arguments from the roof of Theorem 4.1 in [3]. Note that if u H,θ (G) with > 1 + d/ and θ < 0, then, by Theorem 4.3, u is continuously differentiable in G and is equal to zero on the boundary of G. This is one reason why the saces H,θ (G) can be considered as a generalization of the usual Sobolev saces with zero boundary conditions. 5. Degenerate ellitic equations in general domains Throughout this section, G R d is a domain with a non-emty boundary but otherwise arbitrary, and ρ is the function introduced at the beginning of Section 4. Consider a second-order ellitic differential oerator L = a ij (x)d i D j + bi (x) ρ(x) D i c(x) ρ 2 (x), where D i = / x i and summation over the reeated indices is assumed. A related but somewhat different oerator is studied in Section 6 of [10]. The objective of this section is to study solvability in H,θ (G) of the equation Lu = f. It follows from Theorem 4.3 that, for aroriate θ and, the solution of the equation will also be a classical solution of the Dirichlet roblem 11

12 Lu = f, u G = 0. The values of R, 1 < <, and θ R will be fixed throughout the section. The following assumtions are made. Assumtion 5.1. Uniform elliticity: there exist κ 1, κ 2 > 0 so that, for all x G and ξ R d, κ 1 ξ 2 a ij (x)ξ i ξ j κ 2 ξ 2. Assumtion 5.2. Regularity of the coefficients: a A ν 1 (G) + b A ν 2 (G) + c A +1 + (G) κ 2, where ν 1 = max(2, 1 + ), ν 2 = max(1, + ). (See beginning of Section 3 for the definition of.) Note that under assumtion 5.2 the oerator L is bounded from H +1,θ (G) to H 1,θ+ (G). Therefore, we say that u H+1,θ (G) is a solution of Lu = f with f H 1,θ+ (G) if the equality Lu = f holds in H 1,θ+ (G). Theorem 5.1. Under Assumtions 5.1 and 5.2, there exists a c 0 > 0 deending only on d,, θ, the function ρ, and the coefficients a, b so that, for every f H 1,θ+ (G) and every c(x) satisfying c(x) c 0, the equation Lu = f has a unique solution u H +1,θ (G) and u H +1,θ (G) N f H 1,θ+ (G) with the constant N deending only on d,,, θ, the function ρ, and the coefficients a, b, c. To rove Theorem 5.1, we first establish the necessary a riori estimates, then rove the theorem for some secial oerator L, and finally use the method of continuity to extend the result to more general oerators. Lemma 5.2. If u H +1,θ (G) and Assumtions 5.1 and 5.2 hold, then ( ) u H +1,θ (G) N Lu H 1,θ+ (G) + u H 1,θ (G) with N indeendent of u. Proof. Assume first that b = c = 0. Define u n (x) = ζ n (2 n x)u(2 n x) and the oerator A n = (a ij (2 n x)η n (2 n x) + (1 η n (2 n x)δ ij ))D ij, 12

13 ( where η is as in Remark 2.3. Clearly, u n H +1 N Au n H 1 and, by (3.1), N is indeendent of n. On the other hand, A n u n (x) = 2 2n ( ζ n Lu + 2a ij D i ζ n D j u + a ij ud ij ζ n ) (2 n x). + u H 1 It remains to use the inequalities Du H 1 N u H ɛ u H +1 + Nɛ 1 u H 1 with sufficiently small ɛ, and then sum u the corresonding terms according to (2.1). If b, c are not zero, then a ij D ij u H 1,θ+ (G) Lu H 1,θ+ (G) + N u H,θ (G) + N u H 1,θ (G), and the result follows from the interolation inequality (2.4). Lemma 5.3. If Assumtions 5.1 and 5.2 hold, then there exists a c 0 > 0 deending on d,, θ, the function ρ, and the coefficients a, b, so that, for every c(x) satisfying c(x) c 0 and every u L,θ (G), with N indeendent of u. u L,θ (G) N ρ 2 Lu L,θ (G) Proof. It is enough to consider u C 0 (G). Writing f = ρ 2 Lu, we multily both sides by u 2 uρ θ d and integrate by arts similar to the roof of Theorem 3.16 in [3]. The result is f u 2 uρ θ d dx = G G ( ) c(x) + h(x) u ρ θ d dx, where h(x) N h and N h deends on d,, θ, and a A 2 (G) + b A 1 (G) + Dρ A 1 (G). It remains to take c 0 = 2N h and use the Hölder inequality. It follows from Lemmas 5.2 and 5.3 that if c(x) c 0 and 1, then u H +1,θ (G) N Lu H 1,θ+ (G). (5.1) Lemma 5.4. There exists a c > 0 deending on, θ,, and the function ρ so that the oerator ρ 2 (x) c is a homeomorhism from H +1,θ (G) to H 1,θ (G). ), 13

14 Proof. Keeing in mind that ρ C 0,1 (G) and ρ(x) = 0 on G, let ρ be a C 0,1 (R d ) extension of ρ so that ρ C (G G). Consider a family of diffusion rocesses (X x t, x R d, t 0) defined by X x t = x + 2 t 0 ρ(x x s )dw s, where (W t, t 0) is a standard d-dimensional Wiener rocess on some robability sace (Ω, F, P ) (see, for examle, Chater V of [1] or Chater I of [8]). Note that, by uniqueness, Xt x = x if x G, and Xt x G for all t > 0 as long as x G. Theorems (3.3) and (3.9) from Chater I of [8] imly that, with robability one, both DXt x and its inverse are in C(G) for all t 0. Further analysis shows that, for every > 1 and every ositive integer m, E DX x t A m (G) + E D(Xx t ) 1 A m (G) N 1e N 2t (5.2) with constants N 1 and N 2 deending on, m. Assume that f C 0 (G) and define u(x) = E 0 f(x x t )e ct dt. By Theorem in [1], there exists a c 1 > 0 deending only on d and ρ so that, for c > c 1, the function u is twice continuously differentiable in G and ρ 2 (x) u(x) cu(x) = f(x) for all x G. On the other hand, after reeating the roof of Theorem 3.2 and using (5.2), we conclude that there exists a c 2 deending on d,, ρ so that, for c > c 2 and for every R, the function u belongs to H,θ ν (G) and u H,θ (G) N f H,θ (G). The statement of Lemma 5.4 now follows. Proof of Theorem 5.1. Take c as in Lemma 5.4 and define the oerators L 0 = c/ρ 2 (x) and L 0 = ρ 2 (x) c. Lemmas 5.4 and Theorem 4.1(2) imly that, for all, θ R and 1 < <, these oerators are homeomorhisms from H +1,θ (G) to, resectively, H 1,θ+ (G) and H 1,θ (G). 14

15 Assume first that 1. Then a riory estimate (5.1) and the method of continuity (using the oerators λl+(1 λ)l 0, 0 λ 1) imly the conclusion of the theorem. If ν < 1, then assume first that 0 ν < 1. For f H ν 1,θ+ (G), define u = L 0 L 1 1 ( L 0 f) L 1 ( f), where f = (L L 0 L 0 L)L 1 1 ( L 0 f). Direct comutations show that f H ν,θ+ (G) and f H ν,θ+ (G) N f H ν 1,θ+ (G); u is well defined, u H ν+1,θ (G), u H ν+1,θ (G) Lu = f. This rocess can be reeated as many time as necessary. roved. N f H ν 1,θ+ (G), and Theorem 5.1 is Remark 5.5. It follows from Theorem 4.3 that, if the conditions of Theorem 5.1 hold with > d/ + 2 and θ <, then the function u is the classical solution of a ij (x)d ij u + bi (x) ρ(x) D iu c(x) ρ 2 (x) u = f, x G; u G = 0. Acknowledgment I wish to thank Professor David Jerison and Professor Daniel Stroock for very helful discussions. References [1] N. V. Krylov. Introduction to the Theory of Diffusion Processes. American Mathematical Society, Providence, RI, [2] N. V. Krylov. Lectures on Ellitic and Parabolic Equations in Hölder Saces. American Mathematical Society, Graduate Studies in Mathematics, v. 12, Providence, RI, [3] N. V. Krylov. Weighted Sobolev Saces and Lalace Equation and the Heat Equations in a Half Sace. Comm. Partial Differential Equations, 24 (1999), no. 9 10, [4] N. V. Krylov. An analytic aroach to SPDEs. In B. L. Rozovskii and R. Carmona, editors, Stochastic Partial Differential Equations. Six Persectives, Mathematical Surveys and Monograhs, ages AMS,

16 [5] N. V. Krylov and S. V. Lototsky. A Sobolev Sace Theory of SPDEs with Constant Coefficients in a Half Sace. SIAM J. Math. Anal. 31 (2000), no. 1, [6] J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Alications,I. Sringer-Verlag, Berlin, [7] S. V. Lototsky. Dirichlet Problem for Stochastic Parabolic Equations in Smooth Domains. Stochastics Stochastics Re. 68 (1999), no. 1 2, [8] D. W. Stroock. Toics in Stochastic Differential Equations. Tata Institute of Fundamental Research, Bombay, India, [9] H. Triebel. Theory of Function Saces II. Birkhauser, Basel, [10] H. Triebel. Interolation Theory, Function Saces, Differential Oerators. Johann Amrosius Barth, Heidelberg,