Non-divergence Elliptic Equations of Second Order with Unbounded Drift

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1 Unspecified Boo Proceedings Series Dedicated to Nina N. Ural tseva Non-divergence Elliptic Equations of Second Order with Unbounded Drift M. V. Safonov Abstract. We consider uniformly elliptic equations and inequalities of second order in the nondivergence form Lu = a ij D ij u + b i D i u = 0 ( 0, 0), in a domain Ω R n. We derive an interior Harnac inequality, a boundary Harnac inequalities and a comparison theorem for Lipschitz boundaries in the case b i L n, and a Hopf-Oleini type estimate near flat boundary in the case when the normal component of the vector (b 1,..., b n) belongs to L q, q > n. Our main tools are special Landis-type growth lemmas. 1. Introduction In this paper, we derive interior and boundary pointwise estimates for solutions to the equation (1.1) Lu := a ij D ij u + b i D i u = 0 in a domain Ω R n, and the inequalities Lu 0 or Lu 0, under minimal assumptions on the functions a ij and b i. We use the summation convention over repeated indices and the notations D i := / x i, D ij := D i D j. We assume that a ij and b i are measurable functions on R n, a ij satisfy the uniform ellipticity condition (1.2) a ij (x) = a ji (x), ν ξ 2 a ij (x)ξ i ξ j ν 1 ξ 2 ξ, x R n, with a constant ν (0, 1], and b i, f L n (Ω). Throughout the paper, the operator L in (1.1) is applied to functions u in the class W (Ω) := W 2,n loc (Ω) C(Ω), which implies, in particular, that u, D i u, D ij u belong to the Lebesgue space L n (Ω ) for any bounded open set Ω Ω Ω. The following estimate is crucial for our considerations. It can be considered as a particular case (when m = n) of the estimate (5.6) in the paper by A.D. Alesandrov [A63] (see also [GT83], (9.14)). Theorem 1.1. Let Ω be a bounded open set in R n, and let u be a function in W (Ω) such that Lu f in Ω. Suppose that the coefficients a ij satisfy (1.2), and b i, f L n (Ω). Then (1.3) sup Ω where f := max{ f, 0}, (1.4) S := S(Ω) := u sup u + N diam Ω e NS f n,ω, Ω Ω b n dx, b := (b 1,..., b n ), 1991 Mathematics Subject Classification. Primary 35J15, 35J67; Secondary 35B05. Key words and phrases. Second-order elliptic equations, Harnac inequality, boundary Hopf lemma, measurable coefficients. 211 c 0000 (copyright holder)

2 212 M. V. SAFONOV ( 1/n g n,ω := g dx) n the norm of g in L n (Ω), Ω and N is a positive constant depending only on n and ν. Corollary 1.2. Let Ω be a bounded open set in R n, and let u be a function in W (Ω) such that Lu 0 in Ω. Then the maximum of u on Ω is attained on Ω. Remar 1.3. Note that the equality Lu = f in (1.1) and the inequality Lu f in Theorem 1.1 hold true almost everywhere in Ω with respect to the Lebesgue measure on R n. It is easy to see that the function u(x) := 1 x 2 satisfies u + b i D i u = 0 in B 1 := {x R n : x < 1}, where b i (x) := nx i x 2 L n ε (B 1 ) for arbitrary small ε > 0, and the maximum of u on B 1 is not attained on B 1. Therefore, the assumption b i L n is essential for the whole theory, which is based on the maximum principle. Remar 1.4. Note that the quantity S(Ω) in (1.4) is scaling invariant in the following sense. If u = u(x) satisfies the equation Lu = 0 in Ω, then for any constant > 0, the function ũ(y) := u(y) satisfies D i ũ = D i u, D ij ũ = 2 D ij u, hence Lũ := ã ij D ij ũ + b i D i ũ = 0 in Ω := {y R n : y Ω}, where ã ij (y) := a ij (y), b i (y) := b i (y). Therefore, b := ( ) 1/2 b2 i satisfies S( Ω) := b(y) n dy = b(y) n n dy = b(x) n dx =: S(Ω). Ω Ω Ω Obviously, this argument also wors for inequalities Lu 0 or Lu 0 in Ω. In the next section, we will use the scaling invariance of S(Ω), together with the maximum principle, in order to prove some special growth lemmas (Lemmas 2.1 and 2.5) and a doubling property of solutions (Lemma 2.2) in the case b L n, i.e. S(Ω) <. The growth lemmas were first introduced by E.M. Landis [La67, La71]. Among other applications of these lemmas, Landis gave alternative proofs of results by De Giorgi and Moser on Hölder regularity and Harnac inequalities for solutions to second order elliptic equations in the divergence form. These results were extended to the equations in the non-divergence form with b L in our joint wor with N.V. Krylov [KS80, S80] (see also the boo [K85]). For this purpose, we also used some variants of growth lemmas. Now this technique became standard, see the paper by H. Aimar, L. Forzani, and R. Toledano [AFT01], in which they treat Hölder and Harnac properties from an abstract point of view. In Section 3, we use the growth lemmas and the doubling property of solutions in order to derive the interior and boundary Harnac inequalities and the Hölder estimates for solutions of equations (1.1) with b L n. Note that the Hölder regularity in the case b L n was proved earlier by O.A. Ladyzhensaya and N.N. Ural tseva [LU85]. However, the Hölder constants in that paper depend on the modulus of continuity of b in L n, or more precisely, on the constant ρ > 0 for which the norms of b in L n (Ω B ρ (x)) are small enough for all x Ω. In our estimates, all the constants depend on b only through the quantity S(Ω) in (1.4). Finally, in Section 4, we prove a Hopf-Oleini type estimate near flat boundary {x n = 0} for positive solutions of the equation a ij D ij u + b i D i u = 0 with b L n, b n L q, q > n. A close result is contained in the paper by O.A. Ladyzhensaya

3 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 213 and N.N. Ural tseva [LU88], Lemma 4.4, under a bit stronger assumption b L q, q > n. See also a recent wor by A.I. Nazarov and N.N. Ural tseva [NU09] for further results concerning eliptic and parabolic, divergence and nondivergence, equations with unbounded lower-order coefficients. Notations. x = (x 1,..., x n 1, x n ) = (x, x n ) are vectors or points in R n, (x, y) := x i y i the scalar product of x, y R n (the summation convention is implied), and x := (x, x) 1/2 the length of x R n. The standard orthonormal basis in R n consists of n vectors {e 1,..., e n }, so that x = x i e i for all x R n. tr a := a ii the trace of a n n matrix a = [a ij ]. For c R 1, [c] denotes its i integer part (the maximal integer c), c + := max{c, 0}, c := max{ c, 0}. B r (x) := {y R n : y x < r} a ball of radius r > 0 centered at x R n. B r(x ) := {y R n 1 : y x < r} is a similar ball in R n 1 centered at x R n 1. B r := B r (0), B r := B r(0). Ω is the boundary of an open set Ω R n, Ω its Lebesgue measure. The notation A := B, or B =: A, means A = B by definition. Throughout the paper, N, c (with indices or without) denote different constants depending only on the prescribed quantities, such that n, ν, etc. This dependence is indicated in the parentheses: N = N(n, ν,...), c = c(n, ν,...). 2. Growth Lemmas The first growth lemma can be treated as a growth lemma for narrow domains. Lemma 2.1 (First growth lemma). Let Ω be an open set in R n, and let u W (Ω), x 0 R n, and r > 0 be such that u 0, Lu 0 in Ω; and u = 0 on ( Ω) B 2r (x 0 ). We claim that for arbitrary constant β 1 (0, 1), there is a constant µ 1 (0, 1), depending only on n, ν, and S, such that from the estimate for the Lebesgue measure (2.1) Ω B 2r (x 0 ) µ 1 B 2r it follows (2.2) M r := sup u β 1 M 2r. Ω B r(x 0) Proof. Using translation in R n, and multiplying u by a constant, we can assume x 0 = 0, M r > 0, and M 2r = 1. Moreover, since S(Ω) is invariant with respect to rescaling (Remar 1.4), we can choose any convenient value r > 0. (a) We first tae r = 1 and assume that S := S(Ω) 1. By Corollary 1.2, the maximum of u on Ω B 1 is attained at some point y 0 Ω ( B 1 ). Consider the function v(x) := u(x) x y 0 2 on the set Ω := Ω B 1 (y 0 ). Since v u = 0 on ( Ω) B 2, and v = u 1 0 on Ω ( B 1 (y 0 )), we have v 0 on Ω. Moreover, Lv = Lu 2 tr a 2 (b, x y 0 ) f := 2nν 1 2 b in Ω. Applying Theorem 1.1 to the function v in Ω, and having in mind that S(Ω) 1 and Ω µ 1 Ω B 2 µ 1 B 2, we get (µ 1/n (2.3) M 1 := sup u = u(y 0 ) = v(y 0 ) N 0 f n,ω N S 1/n), Ω B 1

4 214 M. V. SAFONOV with constants N 0, N 1 1, depending only on n and ν. Now fix an arbitrary constant β 1 (0, 1), and set µ 0 := S 0 := (β 1 /2N 1 ) n. If S S 0, we can tae µ 1 = µ 0, and then the desired estimate M 1 β 1 follows from (2.3). (b) In the remaining case S > S 0, we set m := [S/S 0 ] + 1 > S/S 0 > 1. It is convenient to tae r := 4m. Then we divide Ω (B 8m \ B 4m ) into m disjoint sets Ω := Ω ( B 4m+4 \ B 4m+4 4 ) for = 1, 2,..., m. We have m b n dx Ω =1 Ω b n dx =: S. Since S < ms 0, at least one of the integrals on the left side S 0. Fix the corresponding index = 0 for which this is the case. Finally, we set µ 1 := (4m) n µ 0. By the maximum principle, (2.4) M r := sup u sup u = u(y) Ω B 4m Ω B 4m+40 2 for some y Ω ( B 4m+40 2). Note that the set Ω B 2 (y) is contained in Ω 0 hence the integral of b n over this set S 0, and moreover, from (2.1) it follows Ω B 2 (y) Ω B 8m µ 1 B 8m = µ 0 B 2. From the previous part (a) of the proof it follows that u(y) sup Ω B 1(y) u β 1 Together with (2.4), this completes the proof. sup Ω B 2(y) u β 1 M 2r. Lemma 2.2 (Doubling property). Let x 0 R n, r > 0, and a function v in W (B 4r (x 0 )) be such that (2.5) v 0, Lv 0 in B 4r (x 0 ); and v 1 in B r (x 0 ). Then (2.6) v λ = λ(n, ν, S) > 0 in B 2r (x 0 ). Proof. Without loss of generality, we assume x 0 = 0, r = 1. In the part (a) below, we consider the case n = 1, part(b) deals with n 2 and small S, and part (c) with general S <. We need to treat the case n = 1 separately, because in our approach, we use the fact that the spheres B r R n are connected. This fails if n = 1, because in this case B r consists of two disjoint point ±r. (a) In the case n = 1, we have v 0, av + bv 0 in ( 4, 4); and v 1 on [ 1, 1], where 0 < ν a ν 1, and b L 1 ( 4, 4). Using properties of wea derivatives (the product formula (7.18) and the chain rule in Lemma 7.5, [GT83]), one can rewrite the above differential inequality as follows: [ ] x (2.7) (Av ) b(y) 0, where A(x) := exp a(y) dy. 0

5 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 215 The function A(x) satisfies (2.8) N 1 0 A(x) N 0 := exp(ν 1 S) in ( 4, 4), S := 4 4 b(x) dx. If suffices to get the estimate v λ = λ(n, ν, S) > 0 on the interval [ 2, 1], because a similar estimate on the symmetric interval [1, 2] can be obtained by replacing x with x. Denote κ := min We can assume κ 1/2, because [ 2, 1] v. otherwise the estimate v λ holds true with λ = 1/2. By the maximum principle, i.e. Corollary 1.2 applied to the function v, we have κ = v( 2). Since v is continuous and v( 2) = κ 1/2 < 1 v( 1), there is h (0, 1] such that v( 2 + h) = 1, and v(x) < 1 for all x [ 2, 2 + h). Furthermore, for each y ( 4, 2 + h) the minimum of v(x) on [y, 2 + h] is attained at the point y. This means that the function v(x) is non-decreasing on ( 4, 2 + h], and therefore v 0 in ( 4, 2 + h). From (2.7) it follows that the function Av is non-increasing. Together with (2.8), this implies κ = v( 2) 2 1 N 2 0 v dx 2 h 2+h 2 1 N h Av dx This proves the estimate (2.6) in the case n = h Av dx N 0 2 v dx = 1 N0 2 [v( 2 + h) v( 2)] 1 2N0 2 =: λ. (b) In the rest of the proof of this lemma, we assume n 2. First consider the case when S := S(Ω) S 0 = S 0 (n, ν) a small positive constant to be chosen below in (2.10). Note that [ ] ( L 0 x γ 0 ) ( := aij D ij x γ 0 ) (γ0 + 2)a ij x i x j = γ0 x 2 tr a x γ0 2 γ 0 [(γ 0 + 2)ν nν 1] x γ0 2 0 for x 0, provided γ nν 2. Fix such γ 0 = γ 0 (n, ν) > 0, and consider the function w(x) := x γ0 4 γ0 1 4 γ0 in Ω 1 := {x R n : 1 < x < 4}. We have L 0 w 0 in Ω 1, w v on Ω 1. Therefore, and by Theorem 1.1, L(w v) Lw = L 0 w + (b, Dw) (b, Dw) in Ω 1, (2.9) sup Ω 1 (w v) N 0 (b, Dw) n,ω1 N 1 b n,ω = N 1 S 1/n, with some constants N 0, N 1 1 depending only on n and ν. Next, fix the constants (2.10) λ 0 := 2 γ0 4 ( γ0 2 (1 ) 0, 1 ) ( ) n λ0, and S 0 :=, 4 γ0 2 which also depend only on n and ν. Then from S S 0 and (2.9) it follows w v λ 0 in Ω 1. On the other hand, obviously w 2λ 0 on the set {1 x 2} Ω 1. Therefore, v(x) λ 0 for 1 x 2. Since also v(x) 1 for x < 1, the desired estimate (2.6) holds true with λ = λ 0 (n, ν) > 0, provided S S 0 = S 0 (n, ν). N 1

6 216 M. V. SAFONOV (c) Now it remains to consider the case n 2, S > S 0. As in the proof of Lemma 2.1, we set m := [S/S 0 ] + 1 > S/S 0 > 1. Choose ρ 0 (0, 1/2) of order 1/2 and y 1,..., y m B 1/2 such that the balls B ρ0 (y ) are disjoint for = 1,..., m. Then the tubes ( ) T := B 4 B ρ0 (ty ), = 1,..., m, t 1 are also disjoint. Since S < ms 0, we can fix = 1 such that the integral of b n over T 1 does not exceed S 0. Further, divide B 4 \ B 2 into m disjoint shells Ω := B r \ B r 1, where r := ; = 1,..., m, m and choose = 2 such that the integral of b n over Ω 2 does not exceed S 0. Finally, set R := (2 m + 1 m, 4 1 ), ρ := 1 { m 4 min ρ 0, 1 }. m Fix an arbitrary y B R. One can move the ball B 4ρ (z) continuously inside T 1 from z = y 1 B 1/2 to z = 2R y 1 B R, and then inside Ω 2 to z = y. Therefore, there is a sequence such that z 0 := y 1, z 1,..., z j0, z j0+1,..., z m0 := y, (2.11) B 4ρ (z j ) T 1 for j j 0 ; B 4ρ (z j ) Ω 2 for j j 0 + 1, z j+1 z j ρ for all j = 0, 1,..., m 0 1, and m 0 is of order m. We claim that (2.12) v λ j 0 in B ρ (z j ) for j = 0, 1,..., m 0, where λ 0 = λ 0 (n, ν) > 0 is a constant in part (b), i.e. in (2.10), which corresponds to S = S 0. For j = 0, this inequality is contained in (2.5), because B ρ (z 0 ) = B ρ (y 1 ) B 1/2 (y 1 ) B 1. Moreover, if (2.12) is true for some j, then the function v j := λ j 0 v satisfies (2.5) with x 0 = z j and r = ρ. By our construction of the sets T 1 and Ω 2, from (2.11) it follows S(B 4ρ (z j )) S 0 for all j. Therefore, we can apply the preceding part (b) of this proof, which yields v j λ 0 in B 2ρ (z j ). Since z j+1 z j ρ, we have B ρ (z j+1 ) B 2ρ (z j ), hence v λ j+1 0 on B ρ (z j+1 ). By induction, (2.12) holds true for all j = 0, 1,..., m 0. Here m 0 does not exceed a constant m 1 depending only on n, ν, and S. Therefore, v(y) = v(z m0 ) λ m0 0 λ m1 0 =: λ = λ(n, ν, S) > 0. Here y is an arbitrary point in B R. From the inequalities Lv 0 in B R and v λ on B R it follows v λ on B R. Since R > 2 = 2r, the desired estimate (2.6) follows. Corollary 2.3. Let x 0 R n, r > 0, and a function v in W (B r (x 0 )) be such that (2.13) v 0, Lv 0 in B r (x 0 ); and v 1 in B εr (x 0 ), where 0 < ε 1/2. Then (2.14) v ε γ in B r/2 (x 0 ), where γ = γ(n, ν, S) = log 2 λ > 0, and λ is the constant in the previous lemma.

7 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 217 Proof. We assume x 0 = 0. Set r j := 2 j r for j = 1, 2,..., and choose natural such that 2 1 < ε 2, so that r +1 < εr r. By our assumptions, v 1 on B εr B r+1. The previous lemma with r = r +1 yields v λ in B r. Repeatedly using this lemma again, we get v λ 2 in B r 1,..., v λ in B r1 = B r/2. Since λ = 2 γ ε γ, the corollary is proved. The following technical lemma will help us to deal with functions defined on a ball B r rather than on a general open set Ω R n. Lemma 2.4 (Extension lemma). Let Ω be an open set in R n, and let u be a function in W (Ω), such that u 0, Lu 0 in Ω; and u = 0 in ( Ω) B r, where B r := B r (x 0 ) for some r > 0 and x 0 R n. We claim that there are functions u ε W (B r ) defined for each ε > 0, such that u ε 0, Lu ε 0 in B r ; u ε 0 in B r \ Ω, and u ε u as ε 0 + uniformly on Ω B r. Proof. We partially follow [CS07], pp Fix a standard function 0 η C (R 1 ), such that η(t) 0 for t 1, and η(t) dt = 1; R 1 and set η ε (t) := ε 1 η(ε 1 t 2) for ε > 0, t R 1. These are smooth functions vanishing on R 1 \[ε, 3ε]. Further, by repeated integration of η ε, define the functions G ε C (R 1 ) satisfying the properties G ε 0 on (, ε], G ε η ε on R 1. Now we set u ε := G ε (u) in Ω, u ε 0 on B r \ Ω. From the properties of η ε it follows G ε 0, G ε 0, and (u 3ε) + u ε (u ε) +. Since u = 0 on the set ( Ω) B r, the functions u ε vanish near this set. Hence we have u ε W (B r ), u ε 0, and u ε u 3ε on Ω. Finally, Lu ε = LG ε (u) = G ε(u) Lu + G ε (u) a ij D i u D j u 0 in Ω. Lemma is proved. In comparison with the first growth lemma (Lemma 2.1), the next lemma states that, roughly speaing, from (2.1) with µ 1 < 1 it follows (2.2) with β 1 < 1. Lemma 2.5 (Second growth lemma). Let Ω be an open set in R n, and let u W (Ω), x 0 R n, and r > 0 be such that u 0, Lu 0 in Ω; and u = 0 on ( Ω) B 2r (x 0 ). We claim that for arbitrary µ 2 (0, 1), there is a constant β 2 = β 2 (n, ν, S, µ 2 ) (0, 1), such that from (2.15) Ω B r (x 0 ) µ 2 B r it follows (2.16) M r := sup u β 2 M 2r. Ω B r(x 0)

8 218 M. V. SAFONOV Replacing Ω by Ω B 2r (x 0 ), and u by const u we can assume that 0 u M 2r = 1 in Ω. Taing v := 1 u, µ := 1 µ 2 (0, 1), and β := 1 β 2 (0, 1), we see that this lemma follows from the following one. Lemma 2.6. Let Ω be an open set in R n, and let v W (Ω), x 0 R n, and r > 0 be such that v 0, Lv 0 in Ω; and v 1 on ( Ω) B 2r (x 0 ). We claim that for arbitrary µ (0, 1), there is a constant β = β(n, ν, S, µ) (0, 1) such that from B r (x 0 ) \ Ω µ B r it follows v β on Ω B r (x 0 ). Proof. We follow the lines of the proof of Lemma 2.3 in [S80], though some details are different. As before, we assume x 0 = 0. Moreover, applying the extension lemma (Lemma 2.4) to the function u := 1 v, we can also assume that the function v belongs to W (B 2r ) and satisfies v 0, Lv 0 in B 2r. One needs to show that from B r {v 1} µ B r with 0 < µ < 1 it follows v β = β(n, ν, S, µ) (0, 1) on B r. First consider the case (2.17) B r {v 1} µ 0 B r, where µ 0 := 1 µ 1 (n, ν, S, 1/2), i.e. µ 1 is the constant in Lemma 2.1 corresponding to β 1 = 1/2. Then B r {u > 0} = B r {v < 1} = B r B r {v 1} (1 µ 0 ) B r = µ 1 B r. By Lemma 2.1 applied to Ω := Ω B r {u > 0} and r/2 in place of r, we get u 1/2, and v = 1 u 1/2 on B r/2. Now by the doubling property (Lemma 2.2), v β 0 = β 0 (n, ν, S) := λ/2 > 0 on B r. We have proved that from (2.17) it follows v β 0 = β 0 (n, ν, S) > 0 on B r, so that the estimate v β > 0 holds true for µ µ 0 with β = β 0. Now consider the remaining case when the set Γ := B r {v 1} satisfies µ B r Γ < µ 0 B r. Almost every point x in the set Γ is its density point, which implies that B ρ (x) B r and Γ B ρ (x) > µ 0 B ρ for small ρ > 0. One can include B ρ (x) into a monotone continuous family of balls B θ, 0 θ 1, such that B 0 = B ρ (x) and B 1 = B r. Then ϕ(θ) := Γ B θ / B θ is a continuous function on [0, 1] with the boundary values ϕ(0) > µ 0 > ϕ(1). Therefore, for some intermediate value θ 0 (0, 1) we have ϕ(θ 0 ) = µ 0, and the corresponding ball B := B θ0 satisfies (2.18) B B r, B Γ = B {v 1} = µ 0 B. Denote by Γ 1 the union of all the balls B satisfying (2.18). By the previous argument v β 0 > 0 on Γ 1. Obviously Γ \ Γ 1 = 0, because Γ 1 contains all the density points of Γ. Further, by the simple Vitali lemma, there is a finite collection of disjoint balls B (1), B (2),..., B (m), satisfying (2.18), such that m B (j) c 0 Γ 1, where c 0 = c 0 (n) > 0. j=1

9 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 219 This implies m m Γ 1 \ Γ B (j) (Γ 1 \ Γ) = B (j) \ Γ and j=1 = (1 µ 0 ) j=1 m B (j) (1 µ 0 )c 0 Γ 1 (1 µ 0 )c 0 Γ, j=1 (2.19) B r {v β 0 } Γ 1 c 1 Γ = c 1 B r {v 1} c 1 µ B r, where c 1 = c 1 (n, ν, S) := 1 + (1 µ 0 )c 0 > 1. If the left side is < µ 0 B r, then we can apply this estimate again with the function β0 1 v in place of v, then to β 2 0 v, etc, which gives us B r {v β 0 } c 1µ B r for = 1, 2,.... This iteration stops when the left side becomes µ 0 B r. Since c 1µ 1, we must have 0 = 0 (n, ν, S, µ) := [ ln µ/ ln c 1 ]. Finally, since the function β 0 0 v satisfies (2.17), it follows β 0 0 v β 0 > 0 on B r, and the estimate v β > 0 on B r holds true with β = β(n, ν, S, µ) := β > 1. Lemma is proved. Hölder regularity of solutions to non-homogeneous equations Lu = f with f L n follows from Theorem 1.1 and Lemma 2.5 in the same way as Theorem IV.2.5 in [K85], or Theorem 4.1 in [S80], are derived from the corresponding statements. The next theorem is quite similar to these results, therefore we formulate it without proof. We only note that the proof uses approximation of u by solutions of equations with regular coefficients, so that some auxiliary boundary value problems have solutions. This part is provided be the approximation Lemma 4.2 below. Theorem 2.7. Let u be a function in W (B 2r ), r > 0, such that Lu = f in B 2r, where f L n (B 2r ). Then there are constants α (0, 1) and N > 0, depending only on n, ν, and S, such that (2.20) sup x,y B r u(x) u(y) x y α N r α ( ) sup u + r f n,b2r. B 2r 3. Interior and boundary Harnac inequalities Theorem 3.1 (Interior Harnac inequality). Let u be a function in W (B 8r ) satisfying u > 0, Lu := a ij D ij u + b i D i u = 0 in B 8r for some r > 0. Then (3.1) sup u N 1 inf u, where N 1 = N 1 (n, ν, S) 1, S := b n dx. B r B r B 8r Proof. We partially follow the proof of Theorem 3.1 in [S80]. Without loss of generality, we assume r = 1. Let γ = γ(n, ν, S) > 0 be the constant in Corollary 2.3. Since 2 x 1 in the ball B 1 := B 1 (0), we have (3.2) sup u M := sup(2 x ) γ u = (2 x 0 ) γ u(x 0 ) B 1 B 2 for some x 0 B 2. Further, consider the function u 0 := u u(x 0) 2 in the ball B ρ (x 0 ), where ρ := 2 x 0. 2

10 220 M. V. SAFONOV Since 2 x ρ in B ρ (x 0 ), we also have ( ) γ 2 x sup u 0 < sup u sup u ρ γ M = 2 γ u(x 0 ) = 2 γ+1 u 0 (x 0 ), B ρ(x 0) B ρ(x 0) B ρ(x 0) ρ and sup u 0 u 0 (x 0 ) > β 1 sup u 0, where β 1 = β 1 (n, ν, S) := 2 γ 1 > 0. B ρ/2 (x 0) B ρ(x 0) Now we can use Lemma 2.1 in an equivalent form if (2.2) fails, then (2.1) fails, with Ω := B 8 {u 0 > 0}, r := ρ/2, and u 0 in place of u. By this lemma, there is a constant µ 1 > 0 depending only on n, ν, and S, such that (3.3) B ρ (x 0 ) {u 0 > 0} > µ 1 B ρ. Next, the function v := u/u 0 (x 0 ) satisfies v > 0, Lv = 0 in Ω := B 8 {v < 1} = B 8 \ Ω, and v = 1 on ( Ω ) B 8. Moreover, by virtue of (3.3), B ρ (x 0 ) \ Ω = B ρ (x 0 ) {v 1} = B ρ (x 0 ) {u 0 0} > µ 1 B ρ. Applying Lemma 2.6 to the function v in Ω, with r := ρ, we obtain the estimate By the choice of x 0 and ρ, v β = β(n, ν, S, µ 1 ) > 0 in B ρ (x 0 ). u = u(x 0 ) v c 1 := β (2ρ) γ M in B ρ (x 0 ). Finally, we apply Corollary 2.3 to the function c 1 1 u with r = 6 and ε := ρ/6 (0, 1/6). In our case, B 3 (x 0 ) = B r/2 (x 0 ) B 6 (x 0 ) B 8, so that the estimate (2.14) implies c 1 1 u εγ in the ball B 3 (x 0 ), which contains B 1 := B 1 (0). Therefore, inf u B 1 inf B 3(x 0) u εγ c 1 = ( ρ 6 ) γ β (2ρ) γ M = βm 12 γ. This estimate together with (3.2) imply the Harnac inequality (3.1) with N 1 = N 1 (n, ν, S) := 12 γ β 1 1. As a standard consequence of the interior Harnac inequality, we have Theorem 3.2 (Liouville). Let u be a bounded from above or from below function in W (R n ) := W 2,n (R n ) C(R n ) such that Lu := a ij D ij u + b i D i u = 0 in the entire space R n, with a ij satisfying (1.2) and b L n (R n ). Then u = const in R n. Indeed, replacing u by const ± u is necessary, we can reduce the proof to the case inf u = 0. In this case, taing the limit in (3.1) as r, we get u 0 in R Rn. n The following theorem is a more general form of the interior Harnac inequality, which is convenient for applications. Theorem 3.3. Let Ω be a bounded open set in R n, and let u be a function in W (Ω) satisfying u > 0, Lu = 0 in Ω. Then for arbitrary constant δ > 0, such that the set Ω δ := {x Ω : dist(x, Ω) > δ} is nonempty and connected, we have (3.4) sup u N 2 inf u, where N 2 = N 2 (n, ν, S, δ/diam Ω), S := b n dx. Ω δ Ω δ Ω

11 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 221 Proof. Fix points x, y Ω δ. One can choose a sequence x (0), x (1),..., x (m) in Ω δ, such that x (0) = x, x (m) = y, and x () x ( 1) < δ/8 for all = 1,..., m; and m does not exceed a number m 0 which depends only on n and δ/diam Ω. Applying Theorem 3.1 to the balls B r (x () ), = 1,..., m m 0, we obtain u(x) = u(x (0) ) N 1 u(x (1) )... N m 1 u(x (m) ) = N m 1 u(y) N m0 1 u(y), where N 1 = N 1 (n, ν, S) is the constant in (3.1). Since the points x, y Ω δ can be selected in an arbitrary way, the inequality (3.4) follows with N 2 := N m0 1. We now proceed to the boundary estimates for positive solutions vanishing at the bottom of a Lipschitz cylinder. Here ψ is a Lipschitz function on R n 1 with a Lipschitz constant K 0, i.e. (3.5) ψ(x ) ψ(y ) K x y for all x, y R n 1. For r > 0, denote (3.6) Q r := {x = (x, x n ) R n : x < r, 0 < x n ψ(x ) < r}, Γ r := {x = (x, x n ) R n : x r, x n = ψ(x )} Q r. The following theorem contains a Carleson type estimate for equations (1.1) with b L n. Theorem 3.4 (Boundary Harnac inequality). Let ψ be a function on R n 1 satisfying the Lipschitz condition (3.5), ψ(0) = 0, and let u be a function in W (Q 2r ) for some r > 0, such that (3.7) u > 0, Lu = 0 in Q 2r ; and u = 0 on Γ 2r, where Q 2r and Γ 2r are defined according to (3.6). Then (3.8) sup Q r u N 3 u(0, r), where N 3 = N 3 (n, ν, S, K) 1. Proof. Since all the conditions here are invariant with respect to rescaling (see Remar 1.4), we can assume that r = 1. (a) First we show that there is a constant γ = γ(n, ν, S, K) > 0, such that (3.9) M := sup Q 2 d γ u N u(p 1 ), where d = d(x) := dist(x, Q 2 ), the constant N = N(n, ν, S, K) 1, and P 1 := (0, 1). Fix x Q 2. The ball of radius d(x) centered at x touches Q 2 at some point y. A simple geometrical consideration shows that there is a smooth curve parameterized by the arc length C := {z = z(s), 0 s s 0 } Q 2, connecting the points y = z(0) and P 1 = z(s 0 ), passing through the point x, i.e. z(s 1 ) = x for some s (0, s 0 ], and such that (3.10) cs d(z(s)) s for all s [0, s 0 ], with a constant c (0, 1) depending only on K. (This property means that Q 2 is a John domain) Set θ := 1 c/8. Then for arbitrary s (0, s 0 ] and t [θs, s], we have z(t) z(s) t s (1 θ)s = cs 8 d(z(s)). 8

12 222 M. V. SAFONOV From Theorem 3.1, with r = d(z(s))/8), it follows (3.11) N 1 u(z(s)) u(z(t)), 0 < θs t s s 0. Fix a constant γ = γ(n, ν, S, K) > 0 such that 1 θ γ N 1. Then ϕ(s) := s γ u(z(s)) N 1 (θs) γ u(z(s)) (θs) γ u(z(θs)) = ϕ(θs), 0 < s s 0. For each s (0, s 0 ], there is an integer 0, such that θs 0 < θ s s 0. Using the previous inequalities, including (3.11) with s = s 0, we get ϕ(s) ϕ(θ 1 s) ϕ(θ s) s γ 0 N 1u(z(s 0 )) = s γ 0 N 1u(P 1 ). Note that s 0 c 1 d(z(s 0 )) c 1. Therefore, at the point x = z(s 1 ), d γ u(x) = d γ u(z(s 1 )) s γ 1 u(z(s 1)) = ϕ(s 1 ) Nu(P 1 ), where N := c γ N 1. Since x Q 2 can be chosen in an arbitrary way, the estimate (3.9) follows. (b) Our next step is to prove the estimate (3.12) M 0 := sup Q 2 d γ 0 u N 0M, where d 0 = d 0 (x) := dist(x, ( Q 2 ) \ Γ 2 ), with a constant N 0 = N 0 (n, ν, S, K) 1. Note that d γ 0 u = 0 on Q 2, hence the supremum in (3.12) is attained at some point x 0 Q 2, i.e. d γ 0 u(x 0) = M 0. We claim that (3.13) d(x 0 ) ε 0 d 0 (x 0 ) with ε 0 = ε 0 (n, ν, S, K) (0, 1/4]. The constant ε 0 will be specified below. Suppose (3.13) fails, i.e. ρ := d(x 0 ) < ε 0 ρ 0, where ρ 0 := d 0 (x 0 ), 0 < ε 0 1/4. Since 4ρ < 4ε 0 d 0 (x 0 ) < d 0 (x 0 ), the intersection ( Q 2 ) B 4ρ (x 0 ) lies in Γ 2, so that u = 0 on this set. Further, the ball B ρ (x 0 ) touches Q 2 at some point y 0 Γ 2. By (3.6), Γ 2 is the graph of a Lipschitz function x n = ψ(x ) restricted to x 2. It is easy to see that the measure B 2ρ (x 0 ) \ Q 2 B ρ (y 0 ) \ Q 2 µρ n with µ = µ(n, K) (0, 1). Now we can apply Lemma 2.5 with r := 2ρ and µ 2 := 1 2 n µ. By this lemma, u(x 0 ) sup Q 2 B 2ρ(x 0) u β By the triangle inequality, sup Q 2 B 4ρ(x 0) u, where β = β(n, ν, S, K) (0, 1). d 0 (x) ρ 0 4ρ > (1 4ε 0 )ρ 0 on Q 2 B 4ρ (x 0 ). Combining these inequalities, we obtain M = ρ γ 0 u(x 0) (1 4ε 0 ) γ β sup Q 2 B 4ρ(x 0) d γ 0 u (1 4ε 0) γ β M. For small enough ε 0 = ε 0 (n, ν, S, K) (0, 1), the right side is strictly less that M, and we get the desired contradiction. The above argument proves the estimate (3.13), which in turn implies (3.12) with N 0 := ε γ 0 1 as follows: M 0 = d γ 0 u(x 0) ε γ 0 dγ u(x 0 ) ε γ 0 M.

13 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 223 (c) Both top and bottom portions of Q 2 are graphs of Lipschitz functions x n = ψ(x ) + c, with c = 0 or 2. An elementary geometric reasoning shows that d 0 (x) (1 + K 2 ) 1/2 on Q 1. Hence sup u (1 + K 2 ) γ/2 sup d γ 0 u (1 + K2 ) γ/2 M 0. Q 1 Q 1 This estimate together with (3.12) and (3.9) yields the desired estimate (3.4). Lemma is proved. In the following Theorem 3.6, which is preceded with a technical Lemma 3.5, we deal with ratios u 1 /u 2 of positive solutions. Note that only the numerator u 1 vanishes on Γ 2r, while u 2 is just a positive solution. In particular, in the case u 2 1, Theorem 3.6 is reduced to Theorem 3.4. Let ψ = ψ(x ) be a function on R n 1 satisfying the Lipschitz condition (3.5), and ψ(0) = 0. For r > 0 and h 0, denote (3.14) Q r,h := {x = (x, x n ) R n : x < r, Q + r,h := {x = (x, x n ) R n : x < r, Γ r,h := {x = (x, x n ) R n : x r, S r,h := {x = (x, x n ) R n : x = r, 0 < x n ψ(x ) < h}, h/2 < x n ψ(x ) < h}, x n = ψ(x ) + h}, 0 < x n ψ(x ) < h}. Comparing these notations with (3.6), we see that Q r = Q r,r, Γ r = Γ r,0. For r, h > 0, the boundary Q r,h of the cylinder Q r,h is the union of three disjoint sets: the top Γ r,h, the bottom Γ r, and the lateral side S r,h. If ψ 0, this terminology is understood in the usual sense. Lemma 3.5. Let w be a function in W (Q r,h ) for some 0 < h r, such that (3.15) Lw = 0 in Q r,h, w 0 on Γ r, and (3.16) inf Γ r,h w sup S r,h ( w) +. We claim that there is a constant ε 1 = ε 1 (n, ν, S, K) (0, 1/4], such that from h ε 1 r it follows (3.17) w(0, x n ) 0 for 0 x n h. Proof. Without loss of generality, we assume r = 1, and the left side of (3.16) is equal to 1. Then (3.18) w 1 on Γ 1,h, w 0 on Γ 1, and w 1 on S 1,h. (a) Consider the function Obviously, it satisfies u := w on the set Ω := Q 1,h {w < 0}. 0 < u 1, Lu = 0 in Ω; u = 0 on ( Ω) Q 1,h. For each x 0 Ω S 3/4,h, the measure Ω B 1/4 (x 0 ) Q 1,h N 0 h with N 0 = N 0 (n) > 0,

14 224 M. V. SAFONOV so that we can apply Lemma 2.1 with x 0 Ω S 3/4,h and r = 1/8 to the function u := w. Since also u 0 on the remaining part of Q 3/4,h, we obtain the estimate (3.19) sup Q 3/4,h ( w) + = sup S 3/4,h ( w) + β 1 = β 1 (n, ν, S, h) 0 + as h 0 +. (b) Next, set w 1 := w + β 1. By the maximum principle, (3.18), and (3.19), it follows w 1 0, Lw 1 = 0 in Q 3/4,h ; w 1 0 on Γ 3/4, w 1 1 on Γ 3/4,h. For an arbitrary x 0 = (x 0, x 0n ) Γ 1/2,h and ρ := (1 + K 2 ) 1/2 h/2, the intersection ( Q 3/4,h ) B 2ρ (x 0 ) lies in the set Γ 3/4,h, which is the graph of a Lipschitz function. Therefore, the measure B ρ (x 0 ) \ Q 3/4,h µ B ρ with a constant µ = µ(n, K) (0, 1). Using Lemma 2.6 with Q 3/4,h, w 1, ρ in place of Ω, v, r respectively, we get the estimate w 1 β = β(n, ν, S, K) > 0 on Q 3/4,h B ρ (x 0 ). Further, by Theorem 3.3, applied to the function w 1 on the intersection Q 3/4,h { x x 0 < ρ}, it follows w 1 (x 0 te n ) β 0 = β 0 (n, ν, S, K) > 0 for 0 t h 2. Since x 0 is an arbitrary point in Γ 1/2,h, we get the estimate (3.20) w 1 β 0 > 0 in Q + 1/2,h. (c) It is important that the constant β 0 in (3.20) does not depend on h. By virtue of (3.19), one can choose the constant ε 1 = ε 1 (n, ν, s, K) (0, 1/4] in such a way that from h (0, ε 1 ] it follows 2β 1 β 0. Then the estimates (3.20) and (3.19) imply (3.21) inf w = inf w 1 β 1 β 1 sup ( w) +. Q + Q + Q 1/2,h 1/2,h 1/2,h Note that Q + 1/2,h contains the set Γ 1/2,h/2 = Γ r/2,h/2, and Q 1/2,h contains S 1/2,h/2 = S r/2,h/2. Therefore inf w sup ( w) +. Γ r/2,h/2 S r/2,h/2 This simply means that the inequality (3.16) remains true with r, h being replaced by r/2, h/2. Iterating this procedure, we can replace r, h by 2 r, 2 h for = 1, 2,.... Correspondingly, the first inequality in (3.21) implies w 0 on Q r,2 h for = 0, 1, 2,..., and (3.17) follows by continuity of w. Lemma is proved. Theorem 3.6 (Comparison theorem). Let ψ be a function on R n 1 satisfying the Lipschitz condition (3.5), ψ(0) = 0, and let u 1 and u 2 be functions in W (Q 3r ), r > 0, such that (3.22) u 1,2 > 0, Lu 1,2 = 0 in Q 3r ; and u 1 = 0 on Γ 3r. Then (3.23) sup Q r u 1 u 2 N 4 u1(0, r) u 2 (0, r), where N 4 = N 4 (n, ν, S, K) 1.

15 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 225 Proof. Multiplying u 1,2 by appropriate constants if necessary, we can assume u 1 (0, r) = u 2 (0, r) = 1. By Theorem 3.4 (3.24) u 1 N 3 in Q r, where N 3 = N 3 (n, ν, S, K) 1 is the constant in this theorem. Then set h := ε 1 r, where ε 1 = ε 1 (n, ν, S, K) (0, 1) is the constant in the previous lemma. Applying the Harnac inequality, Theorem 3.3, to the function u 2, we obtain (3.25) u 2 c 0 = c 0 (n, ν, S, K) > 0 on Q r \ Q r,h. Finally, we set Then N 4 := 2N 3 /c 0, and w := N 4 u 2 u 1. w N 4 c 0 N 3 N 3 on Q r \ Q r,h, and w u 1 N 3 in Q r. Therefore, the function w satisfies all the assumption of the previous lemma, which implies w(0, x n ) 0 for 0 x n r. This construction remains valid if we move the origin 0 R n to any point y = (y, y n ) Γ r. Under this translation, the set Q 2r will be replaced by Q 2r (y) := {x = (x, x n ) R n : x y < 2r, 0 < x n ψ(x ) < 2r}, which is a subset of Q 3r. As a result, we have w 0, or equivalently, u 1 /u 2 N 4, on the whole set Q 1. Theorem is proved. Corollary 3.7. Let u 1 and u 2 be functions in W (Q 3r ), r > 0, satisfying (3.22), and in addition, u 2 = 0 on Γ 3r. Then u 1 (3.26) sup N 2 u 1 4 inf, Q r u 2 Q r u 2 where N 4 = N 4 (n, ν, S, K) 1 is the constant in (3.23). Proof. Since both functions u 1 and u 2 vanish on Γ 3r, we can interchange u 1 and u 2 in (3.23), so that ( u ) 1 1 u 2 inf = sup N 4 u2(0, r) Q r u 2 Q r u 1 u 1 (0, r). Multiplying both sides of this inequality by the corresponding sides of (3.23), we get the desired estimate (3.26). 4. Boundary Hopf-Oleini estimates In a particular case ψ 0 and b 0, the function u 2 (x) = u 2 (x, x n ) := x n satisfies Lu = 0 and vanishes on the {x n = 0}. In this case Q r := { x < r, 0 < x n < r}, and the estimate (3.26) provides both upper and lower bounds for the ratio u 1 (x)/x n near the origin in R n. In 1952, the lower bounds of such ind for solutions of uniformly elliptic equations Lu = 0 with b L were independently obtained by E. Hopf [H52] and O.A. Oleini [O52]. They considered domains Ω satisfying the interior sphere condition at a point y 0 Ω, i.e. there exists a ball B r (x 0 ) Ω such that ( Ω) B r (x 0 ) = {y 0 }. The Hopf-Oleini estimates state that for any function u C 2 (Ω) C(Ω) satisfying u > 0, Lu 0 in Ω, and u(y 0 ) = 0, and any vector v R n such that (v, x 0 y 0 ) > 0, we have t 1 u(y 0 + tv) const > 0 for small t > 0. See the boos by M.H. Protter and H.F. Weinberger [PW67], and by D. Gilbarg and N.S. Trudinger [GT83], for further references on this subject.

16 226 M. V. SAFONOV Here we only mention that in a majority of sources, such ind of estimates are obtained by means of more or less standard barrier technique, which requires the boundedness (at least locally) of coefficients. Our approach uses special iterations based on Theorem 1.1 and Lemma 2.1. It allows us to derive a Hopf-Oleini type estimate in the case b L q, q > n. On the other hand, this estimate fails if b L n, as the following example shows. Example 4.1. Consider the functions x n u 1 (x) := ln x and u 2 (x) := x n ln x in the cylinder Q := {x = (x, x n ) R n : x < 1/2, 0 < x n < 1/2}, extended as u 1 = u 2 = 0 on ( Q) {x n = 0}. Each of these two functions can be considered as a positive solution to the equation u + (b, Du) := u + b i D i u = 0 in Q, where the vector function b := u Du 2 Du satisfies b = u Du const x ln x Ln (Q) for n 2. However, in any neighborhood of 0 R n, inf(u 1 /x n ) = 0 and sup(u 2 /x n ) =. In fact, if b L q with q > n, then any positive solution of the equation Lu = 0 in Q 2r vanishing on Γ 2r (we follow notations (3.6) with ψ 0) satisfies a two-sided estimate u u (4.1) 0 < inf sup <. Q r x n Q r x n Here both the upper estimate sup(u/x n ) <, and the lower estimate inf(u/x n ) > 0 are contained the in paper by O.A. Ladyzhensaya and N.N. Ural tseva [LU88], Lemmas 2.3 and 4.4 correspondingly). In Theorem 4.3 below, the lower bound is extended to a bit more general case b L n, b n L q with q > n. The upper estimate can be obtained similarly, with some simplifications. The following lemma helps to reduce the proofs of such ind of results to the case when the coefficients of L are smooth. Lemma 4.2 (Approximation lemma). Let Ω be a bounded open set in R n satisfying an interior cone condition, i.e. there a constants K > 0 and r 0 > 0 such that for each y Ω, there is a Cartesian coordinate system centered at the point y = 0, such that the cone (4.2) C := {x = (x, x n ) R n : K x < x n < Kr 0 } Ω c := R n \ Ω. Let u be a function in W 2,n (Ω) C(Ω) satisfying the inequality (4.3) Lu := a ij D ij u + b i D i u f in Ω, where a ij satisfy (1.2), b L n (Ω), f L n (Ω). We claim that there are approximations of functions a ij, b i, f by functions a ε ij, bε i, f ε C (Ω), ε > 0, such that a ε ij satisfy (1.2), (4.4) a ε ij a ij a.e. in Ω; b ε i b i, f ε f in L n (Ω) as ε 0 +, and solutions u ε C (Ω) C(Ω) of the Dirichlet problems (4.5) L ε u ε := a ε ijd ij u ε + b ε i D ε i u = f ε in Ω, u ε = u on Ω,

17 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 227 satisfy (4.6) sup(u u) 0 as ε 0 +. Ω The solvability of the problems (4.5) in Lipschitz domains for equations with smooth, or even Hölder or just continuous, coefficients is well nown; see e.g. [M67], Theorem 3. Proof. (a) We first consider the case b L (Ω). The coefficients a ij are defined on the whole space R n, and also b i, f can be extended from Ω to R n by setting b i, f 0 on R n \ Ω. One can approximate a ij, b i, f by convolutions a ε ij := a ij η ε, b ε i := b i η ε, f ε := f η ε with standard ernels η ε, ε > 0, satisfying 0 η ε C (R n ), η ε 0 on R n \ B ε, and η ε (x) dx = 1. R n Then a ε ij, bε i, f ε C (R n ), a ε ij satisfy (1.2), and (4.7) a ε ij a ij, b ε i b i as ε 0 + a.e. in Ω. This convergence follows from the properties of the Lebesgue sets (see [St70], Sec. I.1.8). Moreover, f ε f in L n (R n ) as ε 0 + (see [GT83], Sec. 7.2). Having in mind the application of Theorem 1.1 to u ε u, we need to estimate from below the functions We have L ε (u ε u) = L ε u ε Lu + (L L ε )u F ε := f ε f + (L L ε )u. (L L ε )u := (a ij a ε ij)d ij u + (b i b ε i )D i u 0 in L n (Ω) as ε 0 +. (Here we use our additional assumption b i L ). By Theorem 1.1, from the property (4.7) it follows: sup(u u) N F ε n,ω 0 as ε 0 +. Ω (b) In the general case b L n (Ω), fix a small constant δ > 0, and choose an open subset Ω δ Ω, such that b L (Ω \ Ω δ ), and the norms in L n (Ω δ ), Define the functions b i, f as follows a ij D ij u n,ωδ, b i n,ωδ, f n,ωδ δ. b i 0, f := a ij D ij u on Ω δ, b i b i, f f on Ω \ Ω δ. The inequality (4.3) remains valid with the modified b i and f: By the choice of Ω δ, we also have Lu := a ij D ij u + b i D i u f. (4.8) b i b i n,ω = b i n,ωδ δ, f f n,ω = a ij D ij u f n,ωδ 2δ. Since b i L, we can apply the argument in (a) correspondingly with b δ,ε i := b i η ε, f δ,ε := f η ε, u δ,ε, in place of b ε i,, f ε, u ε. One can go through this construction for each δ := 2, = 1, 2,.... Therefore, there is a sequence ε 0 such that b δ,ε i b i n,ω δ, f δ,ε f n,ω δ ; and sup(u δ,ε u) δ Ω

18 228 M. V. SAFONOV for all ε (0, ε ]. Here b i and f are defined above for δ = δ. Together with (4.8), the first two of these inequalities imply b δ,ε i b i n,ω 2δ, f δ,ε f n,ω 3δ. Finally, for each ε > 0, we tae a ε ij := a ij η ε, as in the part (a), and define b ε i, f ε in two steps: (i) find from the relations ε +1 < ε ε ; then (ii) tae b ε i := bδ,ε i, f ε := f δ,ε. This choice of a ε ij, bε i, f ε satisfies all the requirements of our lemma. Lemma is proved. Theorem 4.3. Let Ω be an open set in R n, and let u be a function in W (Ω) satisfying u > 0 and Lu = 0 in Ω. Let y 0 Ω and r > 0 be such that in a Cartesian coordinate system centered at y 0 = 0, the cylinder (4.9) Q 2r := B 2r (0, 2r) = {x = (x, x n ) R n : x < 2r, 0 < x n < 2r} is contained in Ω, and u(0) = 0. Suppose that (4.10) S := b n dx <, and S 1 := r q n b n q dx <, Q 2r Q 2r where q = const > n. Then (4.11) x 1 n u(0, x n ) c 1 r 1 u(0, r) for 0 < x n r, where c 1 = c 1 (n, ν, S, S 1, q) (0, 1]. Remar 4.4. This theorem provides a Hopf-Oleini type estimate for equations with b L q, q > n. One can modify this formulation in different directions. (i) One can replace the condition Q 2r Ω with y 0 ( Q 2r ) ( Ω) by the interior sphere condition B r Ω with y 0 ( B r ) ( Ω), in both cases u(y 0 ) = 0. One case is reduced to another by an appropriate C 2 -transformation of coordinates. (ii) Using Lemma 4.2, and the comparison principle, one can replace the equality Lu = 0 in Ω by the inequality Lu 0 in Ω. (iii) By the interior Harnac inequality, from (4.11) it follows that for any vector v = (v 1,..., v n ) R n such that v n > 0, we have t 1 u(tv) const > 0 for small t > 0. Proof of Theorem 4.3. All the quantities in the above formulation are invariant with respect to transformations x const x and u const u. Without loss of generality, we assume r = 1 and u(0, 1) = 1. By the previous lemma, u lim sup u ε, where u ε are solutions to the Dirichlet problem (4.5) with coefficients a ε ij, bε i C (Ω), and f ε 0. Therefore, we can also assume a ij, b i C (Ω). ε 0 + Set δ := 1 n/q (0, 1), α := δ/2, and for = 1, 2,..., r := 2, h := r 1+α, Ω := Q r := B r (0, r ), Ω + := Ω {x n > h }, Ω := Ω {x n < h }. By the Harnac inequality, the quantities (4.12) m := inf x 1 Ω + n u > 0, = 1, 2,..., are estimated from below by a positive constants depending on the prescribed quantities and. Our goal is to eliminate the dependence on. From the definition of m it follows v := m x n u 0 on Ω ( {x n = 0} {x n = h } ).

19 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 229 We can split v into two functions: v = w + z on Ω, where w and z are solutions of the problems (4.13) Lw = Lv = m b n in Ω, w = 0 on Ω ; Lz = 0 in Ω, z = v on Ω. These problems have classical solutions because of our smoothness assumptions on the coefficients a ij, b i. By Theorem 1.1, sup w Nr m b n n,ω Ω with N = N(n, ν, S). The last factor is estimated by Hölder s inequality (1/n = 1/q + 1/p): b n n,ω b n q,ω Since n/p = 1 n/q = δ > 0, we get 1 p,ω (4.14) w Nr 1+δ m in Ω. NS 1/q 1 r n/p, N = N(n). Here and in the rest of the proof, N denotes different constants depending only on n, ν, S, S 1, q. Our next step is to evaluate, for 0 < ρ r and fixed, M(ρ) := sup(z ) +, where D(ρ) := B ρ (0, h ). D(ρ) Consider the case M(ρ) > 0, and 0 < ρ r h. Since z = v 0 on {x n = 0} {x n = h }, by the maximum principle, the value M(ρ) is attained by z on the lateral side of D(ρ): M(ρ) = z (x(ρ)) = z (x (ρ), x n (ρ)), where x (ρ) = ρ, 0 x n (ρ) h. In a subcase 0 x n (ρ) h /2, one can apply Lemma 2.5 to the function z with Ω := Ω {z > 0}, x 0 := (x (ρ), 0), and r := h /2. Obviously, the measure condition (2.15) holds true with µ 2 = 1/2. From this lemma and the maximum principle, it follows M(ρ) = z (x(ρ)) sup z β Ω B h /2(x 0) sup Ω B h (x 0) z β M(ρ + h ), where β = β(n, ν, S) (0, 1). If h /2 x n h, then these inequalities remain valid with x 0 := (x (ρ), h ). Let be large enough, so that the integer part j := [r +1 /h ] = [2 α 1 ] 1. Then r +1 + jh 2r +1 = r, and iterating the previous estimate, we get M(r +1 ) β M(r +1 + h )... β j M(r +1 + jh ) β j M(r ) β j m h. Here β j β 1 β 2α 1 = β 1 exp ( cr α ), where c := (ln β)/2 > 0. This estimate together with (4.14) yield the estimate for m x n u =: v = w +z : ( (4.15) m x n u N r 1+δ + exp ( cr α ) ) m in D(r +1 ). From the definition (4.12) of m it follows that if m > m +1, then (4.16) m +1 = inf x 1 n u, where := ( Ω + +1 \ ) Ω+ D(r+1 ),

20 230 M. V. SAFONOV Since x n h +1 = r 1+α +1 = 2 1 α r 1+α, and δ = 2α, the properties (4.15) and (4.16) imply ( m m +1 N r α + r 1 α exp ( cr α ) ) m. We proved the last inequality in the case m > m +1. In the opposite case m m +1 it is obvious, so this inequality holds true without any restrictions. Since the exponential term converges to 0 as faster than any positive power of r, there are large natural numbers 0 and N 0, depending only on n, ν, S, S 1, q, such that m m +1 N 0 r α m, and N 0 r α 1/2 for all 0. From m +1 ( 1 N 0 r α ) m, together with an elementary inequality ln(1 t) 2t for 0 t 1/2, we obtain For any integer 1 > 0, ln m +1 ln m ln ( 1 N 0 r α ) 2N0 r α, 0. ln m 1 ln m 0 = This implies the estimate 1 1 = 0 ( ln m+1 ln m ) 2N0 m c 0 = c 0 (n, ν, S, S 1, q) > 0 for all > 0. = 0 r α =: N 1. In particular, x 1 n u(0, x n ) c 0 for all x n (0, r 0+1). By the Harnac inequality, this estimate holds true for x n r 0+1 as well, possibly with a different constant, because 0 depends only on the prescribed quantities. This proves the desired estimate (4.11). Remar 4.5. In our recent paper [S08], the Oleini-Hopf type estimate (4.11) was extended to the case when Q r is represented in the form (3.6) with (4.17) ψ(x ) := ψ 0 ( x ), where ψ 0 0, ψ 0 0, and 1 0 t 2 ψ(t) dt <. This condition is sharp in a sense that if the integral in (4.17) diverges, then (4.11) fails. However, in [S08] we assumed b i 0. In the present paper, we impose the complementary conditions ψ 0, and b L q, q > n. In our forthcoming wor, we plan to combine the results and techniques of these two papers in order to cover the general case: ψ satisfying (4.17), and b L q, q > n. Acnowledgements. The author is thanful to Nicolai V. Krylov for stimulating conversations. A part of this paper was completed during June 2009, when the author too part in the INdAM intensive period in Milan, Italy, organized by Ugo Gianazza, Sandro Salsa, and Vincento Vespri. The author thans the organizers for invitation and useful discussions. Special thans are due to Nina N. Ural tseva, who raised some open problems which motivated the whole research related to this paper. Finally, the author wishes to than the referee for corrections and valuable advices.

21 ELLIPTIC EQUATIONS WITH UNBOUNDED DRIFT 231 References [AFT01] H. Aimar, L. Forzani, and R. Toledano, Hölder regularity of solutions of PDE s: a geometrical view. Comm. Partial Differential Equations 26 (2001), no. 7 8, [A63] A.D. Alesandrov, Uniqueness conditions and estimates for the solution of the Dirichlet problem, Vestni Leningrad Univ. 18 (1963), no. 3, 5 29 (in Russian). English transl. in Amer. Math. Soc. Transl. (2) 68 (1968), [CS07] S. Cho and M.V. Safonov, Hölder regularity of solutions to second order elliptic equations in non-smooth domains, Journal of Boundary Value Problems, Special Volume 2007, 24 pp. [GT83] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin Heidelberg New Yor Toyo, [H52] E. Hopf, A remar on linear elliptic differential equations of second order, Proc. Amer. Math. Soc., 3 (1952), [K85] N.V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Naua, Moscow, 1985 (in Russian). English transl.: Reidel, Dordrecht, [KS80] N.V. Krylov and M.V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Izvestia Aad. Nau SSSR, ser. Matem. 44(1980), (in Russian). English transl. in Math. USSR Izvestija, 16 (1981), [La67] E.M. Landis, A new proof of E. De Georgi s theorem, Trudy Mosov. Mat. Obshch. 16 (1967), (in Russian). English transl. in Trudy Moscow Math. Soc (1968). [La71] E.M. Landis, Second Order Equations of Elliptic and Parabolic Type, Naua, Moscow, 1971 (in Russian). English transl.: Amer. Math. Soc., Providence, RI, [LU85] O.A. Ladyzhensaya and N.N. Ural tseva, Estimates of the Hölder constant for functions satisfying a uniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Stelov (LOMI) 147 (1985), (in Russian). English transl. in J. Soviet Math. 37 (1987), [LU88] O.A. Ladyzhensaya and N.N. Ural tseva, Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality, Trudy Mat. Inst. Stelov 179 (1988), (in Russian). English transl. in Proc. Stelov Inst. Math. 179 (1989), [M67] K. Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. (4) 76 (1967), [NU09] A.I. Nazarov and N.N. Ural tseva, Qualitative properties of solutions to elliptic and parabolic equations with unbounded lower-order coefficients, Preprints of the St. Petersburg Math. Soc , 9 pp. [O52] O.A. Oleini, On properties of solutions of certain boundary problems for equations of elliptic type, Mat. Sb. (N. S.) 30 (1952), (in Russian). [PW67] M.H. Protter and H.F. Weinberger, Maximim Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., [S80] M.V. Safonov, Harnac inequality for elliptic equations and the Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Stelov (LOMI) 96 (1980), (in Russian). English transl. in J. Soviet Math. 21, no. 5 (1983), [S08] M.V. Safonov, Boundary estimates for positive solutions to second order elliptic equations, arxiv: v2 [math.ap], 20 p. [St70] E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton. Princeton University Press, School of Mathematics, University of Minnesota address: safonov@math.umn.edu