WEIGHTED SOBOLEV'S INEQUALITIES FOR

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DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN. Abstract. Using a ... See also [2, 4, 17, 18] for other ways of proving (1.4) and further ...
WEIGHTED SOBOLEV’S INEQUALITIES FOR BOUNDED DOMAINS AND SINGULAR ELLIPTIC EQUATIONS DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN Abstract. Using a method developed by P´erez and Wheeden and the representation of smooth functions by integral operators whose kernels are gradients of the Green functions we obtain weighted Sobolev’s inequalities for bounded domains which improve and unify several kinds of inequalities. From these results we establish Green functions and the existence, uniqueness and regularity results for a class of singular elliptic equations.

1. Introduction Let v and w be two weights, i.e., nonnegative locally integrable functions, defined on a bounded domain Ω ⊂ Rn , n ≥ 3. In this paper we study explicit sufficient conditions on v and w for the validity of the following two-weight Sobolev’s inequality   1q   p1 q (1.1) |u(x)| w(x)dx ≤ C |∇u(x)|p v(x)dx , Ω

Ω

where 1 < p ≤ q < ∞ and u varies in Cc∞ (Ω), the class of compactly supported and infinitely differentiable functions on Ω. To establish (1.1) one has used one of the following two representations of u in Cc∞ (Rn ):  x1 ∂u (1.2) u(x1 , · · · , xn ) = (t, x2 , · · · , xn )dt, −∞ ∂x1 and (1.3)

 u(x) = c(n) Rn

x−y · ∇u(y)dy, |x − y|n

where c(n) is a constant depending only on the dimension n. 1991 Mathematics Subject Classification. Primary 26D10, 46E35,42B20. Key words and phrases. Weighted Sobolev’s Inequalities, Fractional Integrals, Green Functions, H¨ older regularity, Singular Elliptic Equations . 1

2 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

On one hand, using (1.2) one has obtained Hardy’s inequalities of the form   p −p |u(x)| δ(x) dx ≤ C |∇u(x)|p dx, u ∈ Cc∞ (Ω), (1.4) Ω

Ω

where 1 < p < +∞ and δ(x) is the distance of x from the boundary ∂Ω of Ω. These results are convenient to study partial differential equations strongly degenerate or singular at the boundary of Ω (see, e.g., [8, 9, 15]). See also [2, 4, 17, 18] for other ways of proving (1.4) and further discussions. On the other hand, using (1.3) and weighted norm inequalities for fractional integrals one has established weighted Sobolev’s inequalities that include the following Hardy’s inequality:   |u(x)|p dx ≤ C |∇u(x)|p dx, u ∈ Cc∞ (Ω), (1.5) p |x − x | 0 Ω Ω where 1 < p < n and x0 ∈ Ω. These results could be used to study equations strongly degenerate or singular inside the domain Ω (see, e.g., [5, 10, 19, 22]). However, they can not be applied to equations with strong singularities at the boundary of Ω as the weight functions are required to be defined and at least integrable on some open neighborhood of Ω . One of the main goals of this paper is to unify these results by replacing representations (1.2) and (1.3) by the following one:  ∇y G(x, y) · ∇u(y)dy (1.6) u(x) = Ω

for u ∈ Cc∞ (Ω). Here G is the Green function for the Laplacian with homogeneous Dirichlet condition on Ω. Such a representation enables one to employ the following estimate on gradients of Green functions: (1.7)

|∇y G(x, y)| ≤ K(n, Ω)

min{δ(x), |x − y|} , |x − y|n

which holds for all x, y ∈ Ω, x = y and for bounded domains Ω with appropriately smooth boundary, say C 1,α -boundary or boundary that satisfies an exterior sphere condition uniformly; see [13, 26]. From (1.7) x−y we see that the kernel ∇y G(x, y) is more convenient than |x−y| n in order ∞ to establish (1.1) for functions in Cc (Ω) since the factor min{δ(x), |x− y|} can be used to absorb singularities not only in the interior but also at the boundary of Ω. Combining representation (1.6), estimate (1.7) and using the method developed in [19] and [21] we obtain (1.1) for weight functions v and w which could be singular in the interior or at the boundary of Ω. In

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

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particular, as a special case of Theorem 3.6 in Section 3, we obtain the following unification of (1.4) and (1.5):   |u(x)|p |∇u(x)|p β−p+t δ(x) dx ≤ C δ(x)t dx (1.8) β−η −η Ω |x − x0 | Ω |x − x0 | for u ∈ Cc∞ (Ω), where 0 ≤ β ≤ p < n, −n + β < η < n(p − 1), −1 − β < t < p − 1, and x0 ∈ Ω. In fact, we prove a more general result by considering an integral operator of the form  K(x, y)f (y)dy T f (x) = Ω

for a kernel K : Ω × Ω → [0, ∞) that satisfies the estimate K(x, y) ≤ κ min{δ(x), |x − y|}α |x − y|−n for all x, y ∈ Ω, x = y and for some α, 0 < α < n, where κ is a positive constant independent of x and y. We obtain the boundedness of T as a linear operator from Lp (Ω, v) to Lq (Ω, w) under explicit conditions on the pair of weights (v, w); see Theorem 3.1 and Remark 3.2. Our study of the boundedness of T is also motivated by the study of a class of singular elliptic partial differential equations associated to the following operator n n   Di (aij Dj u) + bj Dj u + cu, (1.9) Lu = − i,j=1

j=1

where the coefficients bj and c may be singular inside Ω or at its boundary. Precisely, from the boundedness of T we obtain Green functions for L as well as the existence, uniqueness and uniform H¨older continuity for weak solutions of the Dirichlet problem  Lu = f in Ω u = 0 on ∂Ω. The details of this analysis will be carried out in Section 4 where our main results are Theorem 4.5, Theorem 4.7 and Theorem 4.8. Throughout the paper we consider only cubes with sides parallel to the axes. The side-length of a cube Q is denoted by (Q), and rQ, r > 0, denotes the cube concentric with Q whose side-length is r(Q). By a dyadic cube we mean a half open cube Q of the form Q = 2i (k + [0, 1)n ) for some i ∈ Z and k ∈ Zn . Given a measurable set E, the Lebesgue measure of E is denoted by |E| and its characteristic function is denoted by χE . Finally, for each p > 1 we will denote by p p the conjugate of p, i.e., p = p−1 .

4 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

2. Preliminaries In this section we provide some background from the theory of Banach function spaces that will be needed later. For a more complete account of the theory we refer the readers to [3]. Let (R, μ) be a measure space, and let M+ (R) be the cone of μ-measurable functions on R whose values lie in [0, ∞]. A mapping ρ : M+ (R) → [0, ∞] is called a Banach function norm if, for all f , g, fk , k ∈ N, in M+ (R), for all constants a ≥ 0, and for all μ-measurable subsets E of R, the following properties hold: (P1) ρ(f ) = 0 if and only if f = 0 μ-a.e.; ρ(af ) = aρ(f ); ρ(f + g) ≤ ρ(f ) + ρ(g); (P2) 0 ≤ g ≤ f μ-a.e. implies ρ(g) ≤ ρ(f ); (P3) 0 ≤ fk ↑ f μ-a.e. implies ρ(fk ) ↑ ρ(f ); (P4) μ(E) < ∞ implies ρ(χ  E ) < ∞; (P5) μ(E) < ∞ implies E f dμ ≤ CE ρ(f ), for some constant CE , 0 < CE < ∞, depending on E and ρ but independent of f . Let M(R) denote the collection of all μ-measurable functions on R. The collection X = X(ρ) of all functions f ∈ M(R) for which ρ(|f |) < ∞ is called a Banach function space. For each f ∈ X we define ||f ||X = ρ(|f |) which makes (X, || · ||) a complete normed space. The most important property of Banach function spaces that we shall use is as follows. Given a Banach function space X there is another Banach function space X  , the associate space of X, for which the following generalized H¨older’s inequality holds for all f ∈ X and g ∈ X :  (2.1) |f (x)g(x)|dx ≤ ||f ||X ||g||X  . R

Examples of Banach function spaces include Lebesgue Lp -spaces, Lorentz spaces, as well as Orlicz spaces that we shall describe next. Let (R, μ) be a measure space and let B : [0, ∞) → [0, ∞) be a Young function, i.e., B is a continuous, convex and increasing function for which B(0) = 0 and B(t) → ∞ as t → ∞. We shall require that B satisfy the Δ2 condition, i.e., there exist constants C > 0 and t0 ≥ 0 such that B(2t) ≤ CB(t) for all t ≥ t0 . The Orlicz space LB (R, μ) consists of all μ-measurable functions f such that   |f (y)|  B dμ(y) < +∞ λ R

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

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for some λ > 0. The space LB (R, μ) is equipped with the Luxemburg norm defined by    |f (y)|  dμ(y) ≤ 1 . B ||f ||B = inf λ > 0 : λ R Note that each Young function has an associated complementary Young function B that satisfies −1

t ≤ B −1 (t)B (t) ≤ 2t for all t > 0. Moreover, B = B and the associate space (LB (R, μ)) of LB (R, μ) is LB (R, μ). In what follows, we shall consider only Banach function spaces X over Rn with respect to the Lebesgue measure. Given any measurable function f and a cube Q ⊂ Rn , we define the X-average of f over Q by ||f ||X,Q = ||τ(Q) (f χQ )||X , where τδ , δ > 0, is the dilation operator τδ f (x) = f (δx), and χE is the characteristic function of a set E. In the case X = Lr (Rn ) this  1 average is equal to ( |Q| |f |r dx)1/r . More generally, if X = LB (Rn ) is Q the Orlicz space defined by a Young function B then the X-average of f ∈ X is given by    |f (y)|  1 dy ≤ 1 . B ||f ||X,Q = inf λ > 0 : |Q| Q λ For any Banach function space X, we associate the following maximal operator defined for each locally integrable function f by MX f (x) = sup ||f ||X,Q . Qx

Thus if, for example, X = Lr (Rn ) then MX f (x) = [M (|f |r )]1/r , where M is the Hardy-Littlewood maximal function defined for a locally integrable function g by  1 |g(y)|dy, M g(x) = sup Qx |Q| Q where the supremum is taken over all cubes Q that contain x. In the case MX : Ls (Rn ) → Ls (Rn ) we denote by ||MX ||Ls →Ls the smallest constant C > 0 such that ||MX f ||Ls ≤ C||f ||Ls for all f ∈ Ls (Rn ). When X = Lr (Rn ), then MX : Ls (Rn ) → Ls (Rn ) for 1 < r < s < ∞ follows from the well-known Hardy-Littlewood

6 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

maximal theorem. In the case X = LB (Rn ) for a Young function B then MX : Ls (Rn ) → Ls (Rn ) if and only if there is a positive constant c for which  ∞ B(t) dt (2.2) < +∞, ts t c or equivalently,  ∞  s s−1 t dt < +∞. (2.3) t B(t) c For a proof of this fact see [20]. Typical examples of Young functions B that satisfy (2.3) are given by 



B(t) = ts [log(e + t)]s −1+δ , δ > 0, and a “weaker” one, 





B(t) = ts [log(e + t)]s −1 [log log(e + t)]s −1+δ , δ > 0. 3. Weighted Sobolev’s inequalities on bounded domains Let Ω be a bounded domain in Rn and let d(Ω) be its diameter, i.e., d(Ω) = sup{|x − y| : x, y ∈ Ω}. In this section we consider the following integral operator  K(x, y)f (y)dy T f (x) = Ω

whose kernel K is a nonnegative measurable function on Ω × Ω such that for some α, 0 < α < n, and for some κ > 0, (3.1)

K(x, y) ≤ κ min{δ(x), |x − y|}α |x − y|−n

for all x, y ∈ Ω, x = y. Here δ(x) denotes the distance of x from the boundary ∂Ω of Ω. The main result of this section is the following theorem concerning the boundedness of T from Lp (Ω, v) to Lq (Ω, w) for a pair of weights (v, w). Its proof is an adaptation of the techniques due to P´erez and Wheeden in [19] and [21] which borrow some ideas appeared in [14] and [22]. Theorem 3.1. Let 1 < p ≤ q < ∞ and and let v, w be nonnegative measurable functions on Ω. Suppose that for some γ, 0 < γ ≤ α, (3.2)

sup Q:(Q)≤4d(Ω)

γ

1

1

1

q |Q| n + q − p ||hα−γ Q w χΩ ||X,Q ||v

−1 p

χΩ ||Y,Q < +∞,

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

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where hQ (x) = min{δ(x), (Q)} and X, Y are two Banach function   spaces for which MX  : Lq (Rn ) → Lq (Rn ) and MY  : Lp (Rn ) → Lp (Rn ). Then there exists a constant C > 0 such that   1q   p1 q p (3.3) [T f (x)] w(x)dx ≤ C f (x) v(x)dx , Ω

Ω

for every nonnegative bounded measurable function f compactly supported in Ω. Remark 3.2. Obviously, condition (3.2) in the above theorem is weaker than the condition (3.4)

α

1

1

1

sup|Q| n + q − p ||w q ||X,Q ||v Q

−1 p

||Y,Q < +∞,

which was introduced in [19]. Moreover, as discussed in [19], condition (3.4) in its turn sharpens and unifies those such as Fefferman–Phong condition in [11] and its variants in [5], [6] and [22]. From the discussion in Section 2 one can take for example X =  qr L (Rn ) and Y = Lp r (Rn ) for r > 1 or more generally, X = LA (Rn ) and Y = LB (Rn ) for two Young functions A and B that satisfy  ∞  p p−1  ∞  q q −1 t t dt dt < +∞ and < +∞ A(t) t B(t) t c c for a constant c > 0. On the other hand, with an appropriate choice of γ in (0, α], the hypothesis of Theorem 3.1 holds also for v ≡ 1, w = δ −β , 0 ≤ β ≤ qα + n(1 − pq ), which includes the interesting case p = q = αβ provided Ω is a bounded Lipchitz domain. This can be seen from the following lemma. Lemma 3.3. Let 0 ≤ β < 1 and assume that Ω is a bounded Lipschitz domain in Rn . Then for any cube Q we have  δ −β dy ≤ C (Q)n−β . (3.5) Q∩Ω

Proof. Let Q = Q(x, r) where x is the center of Q and r is its diameter. Since inequality (3.5) is obvious in the case r ≤ δ(x) , we may assume 2 that x ∈ ∂Ω. It is also enough to prove (3.5) for cubes Q such that r ≤ r0 for some r0 > 0. Thus by choosing r0 small enough we may assume that B(x, r) ∩ Ω = B(x, r) ∩ {(ξ, η) ∈ Rn−1 × R : φ(ξ) < η} and B(x, r) ∩ ∂Ω = B(x, r) ∩ {(ξ, η) ∈ Rn−1 × R : φ(ξ) = η},

8 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

where x = (ξx , ηx ) and φ is a Lipschitz function such that φ(ξx ) = ηx and φ(ξ1 )−φ(ξ2 ) ≤ L|ξ1 −ξ2 | for a constant L > 0 that could be chosen to be the same for every x ∈ ∂Ω. From a result in [25], Chapter VI, there exists a constant c > 0 depending only on Ω such that δ(ξ, η) ≥ c[η − φ(ξ)] for any (ξ, η) ∈ B(x, r) ∩ Ω. Thus we have   −β δ dy ≤ δ −β dy Q∩Ω

B(x,r)∩Ω



≤ C



|ξ−ξx | kΩ there are at most 3n dyadic cubes of side-length 2m that intersect Ω . Hence for a constant C(n, γ) > 0,   γ |Qkj | n −1 ≤ 3n 2m(γ−n) = C(n, γ)d(Ω)γ−n . k,j:(Qkj )>2kΩ Qkj ∩Ω =∅

m>kΩ

From this we have   γ−n I1 ≤ C d(Ω) f (y)σ(y)χΩ (y)dy Q0

hQ0 (x)α−γ g(x)w(x)χΩ (x)dx

Q0

for a cube Q0 of side-length 4d(Ω) that contains Ω. Thus by generalized H¨older’s inequality (2.1) and condition (3.2) on v and w, along with the boundedness of MX  and MY  , one obtains the following estimate for I1 : 1

1

(3.12) I1 ≤ C d(Ω)γ−n ||gw q ||X  ,Q0 ||f σ p ||Y  ,Q0 × 1

1

2 q p ×||hα−γ Q0 w χΩ ||X,Q0 ||σ χΩ ||Y,Q0 |Q0 |   1    p1 1 1  q q  ≤ C ||gw q ||X  ,Q0 dy ||f σ p ||pY  ,Q0 dy × Q0

Q0

1 γ 1 + 1q − p1 q χ || p χ || n w ||σ |Q | ×||hα−γ Ω X,Q Ω Y,Q 0 0 0 Q0



 1  

 p1 1 q MX  (gw ) dy MY  (f σ p )p dy Rn Rn 1  1    p q q p ≤ C g wdx f σdy . Ω Ω

To estimate I2 we let I2 denote the sum taken over all j, k such that (Qkj ) ≤ 2kΩ . Again by generalized H¨older’s inequality (2.1) and condition (3.2) on the pair of weights (v, w) we have ≤ C

I2 ≤

1 q



q

1

1

||gw q ||X  ,3Qkj ||hα−γ w q χΩ ||X,3Qkj × Qk j

I2 1

1

γ

×||f σ p ||Y  ,3Qkj ||σ p χΩ ||Y,3Qkj |Qkj | n +1  1 1 1 +1 ≤ C ||gw q ||X  ,3Qkj ||f σ p ||Y  ,3Qkj |Qkj | p q . I2

12 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

Thus we obtain from H¨older’s inequality for sums and the fact that p ≤ q, (3.13)

I2 ≤C

 I2

≤C



||gw

1 q

p

 ||pX  ,3Qk |Qkj | q j

 1   p

||f σ

1 p

I2

 p1

||pY  ,3Qk |Qkj | j

 1    p1 1 1  q ||gw q ||qX  ,3Qk |Qkj | ||f σ p ||pY  ,3Qk |Qkj | . j

I2

j

I2

Now for each k we let Dk = ∪j Qkj . Also for each k, j we set Ejk = Qkj \ Qkj ∩ Dk+1 . Then {Ejk }k,j is a disjoint family of sets since Dk+1 ⊂ Dk . Moreover,  (3.14) |Qkj ∩ Dk+1 | = |Qk+1 | i i:Qk+1 ⊂Qkj i





ak+1

i:Qk+1 ⊂Qkj i

≤ ≤



1

Qk+1 i



1 ak+1

Qkj

hα−γ (x)w(x)g(x)dx Qk+1 i

hα−γ (x)w(x)g(x)dx Qk j

n

2 |Qk |, a j

where the last inequality follows from the maximality of Qkj . Hence a |E k |, a − 2n j

|Qkj | ≤

(3.15)

since a > 2n . At this point, combining (3.13), (3.14) and (3.15) we obtain the following chain of inequalities: I2 ≤C



||gw

I2

≤C

 I2

≤C

 I2

≤C

Ejk



Rn

Ejk

1 q

 1  

 ||qX  ,3Qk |Ejk | j

||gw

1 q

q

 ||qX  ,3Qk dy j 1 q

q

MX  (gw )(y) dy

Ejk

I2

q

q

 p1

||pY  ,3Qk |Ejk | j

 1   

MX  (gw )(y) dy 1 q

||f σ

I2

1 p

||f σ

 1    q

 1   q

Rn

I2

Ejk

1 p

||pY  ,3Qk dy j 1

 p1

MY  (f σ p )(y)p dy 1 p

p

MY  (f σ )(y) dy

 p1

.

 p1

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

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Thus the boundedness of the maximal operators MX  and MY  then gives   1    p1 q q g(y) w(y)dy f (y)p σ(y)dy . (3.16) I2 ≤ C Ω

Ω

Finally, combining the estimates (3.11), (3.12) and (3.16) we get the required inequality (3.6) which completes the proof of Theorem 3.1.  Remark 3.4. By keeping track of the constants occurring in the proof of Theorem 3.1 we see that the constant C in (3.3) can be chosen so that C = BAM , where A is the quantity in the left-hand side of (3.2), M = ||MX  ||Lq →Lq ||MY  ||Lp →Lp , and B is a positive constant independent of v, w and the Banach function spaces X and Y . In the case v ≡ 1, from (3.6) and Fubini Theorem we see that the inequality (3.3) is equivalent to  Ω



T (gw)(y)f (y)dy ≤ C



p

f (x) dx

 p1  

Ω



g(x)q w(x)dx

 1 q

Ω ∗

for all appropriate functions f and g, where T is defined by 



(3.17)

T f (y) =

K(x, y)f (x)dx. Ω

Thus by Theorem 3.1 and duality we get the following corollary which will be needed in the next section. Corollary 3.5. Let 1 < p ≤ q < ∞ and let b be a nonnegative measurable function on Ω. Suppose that for some γ, 0 < γ ≤ α, γ

1

1

|Q| n + q − p || min{δ, (Q)}α−γ bχΩ ||X,Q < +∞,

sup Q:(Q)≤4d(Ω)



where X is a Banach function space for which MX  : Lq (Rn ) →  Lq (Rn ). Then for every nonnegative bounded measurable function g compactly supported in Ω,   1   1 p q ∗ p q [T (bg)(x)] dx ≤C g(x) dx , Ω

Ω



where T is defined by equation (3.17). The following weighted Sobolev’s inequality is a consequence of Theorem 3.1, representation (1.6) and estimate (1.7) for gradients of Green functions.

14 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

Theorem 3.6. Let Ω ⊂ Rn be a bounded domain whose boundary is of class C 1,α or satisfies an exterior sphere condition uniformly and let 1 < p ≤ q < ∞. Suppose that (v, w) is a pair of weights such that, for some γ, 0 < γ ≤ 1, γ

1

1

1

|Q| n + q − p || min{δ, (Q)}1−γ w q χΩ ||X,Q ||v

sup

−1 p

χΩ ||Y,Q < +∞,

Q:(Q)≤4d(Ω)

where X and Y are two Banach function spaces for which MX  :   Lq (Rn ) → Lq (Rn ) and MY  : Lp (Rn ) → Lp (Rn ). Then   1q   p1 q (3.18) |u(x)| w(x)dx ≤C |∇u(x)|p v(x)dx for all u ∈

Ω ∞ Cc (Ω).

Ω

Remark 3.7. Using Lemma 3.3 we can deduce from Theorem 3.6 inequality (1.8) mentioned in the Introduction. Moreover, Theorem 3.6 also covers and improves earlier results obtained in [8, Theorem 1.1] by  taking X = Lqr (Rn ), Y = Lp r (Rn ), (v, w) = (1, gδ −β ), g ∈ Ls,∞ (Ω), s ∈ (1, ∞], β ∈ [0, q + n(1 − pq − 1s )], with an appropriate choice of γ > 0 and r > 1. Here Ls,∞ (Ω) denotes the weak-Ls space (see e.g., [25]). Note that the results in [8, Theorem 1.1] require that g ∈ Ls (Ω) and in the case p = q = β and s = ∞ we also obtain Hardy’s inequality (1.4). 4. Applications to singular elliptic equations In this section we study the Green function, existence, uniqueness and regularity of generalized solutions for the following elliptic operator: n n   ij Di (a Dj u) + bj Dj u + cu, Lu = − i,j=1

j=1

where the coefficients a , 1 ≤ i, j ≤ n, bj , 1 ≤ j ≤ n and c are real measurable functions on a bounded domain Ω ⊂ Rn that satisfies an exterior sphere condition uniformly. In what follows, the following conditions are imposed on these functions: (C1) aij = aji for i, j = 1, . . . , n and there are positive real numbers λ1 , λ2 such that n  2 aij (x)ξi ξj ≤ λ2 |ξ|2 λ1 |ξ| ≤ ij

i,j=1

holds for all ξ = (ξ1 , . . . , ξn ) ∈ Rn and almost every x ∈ Ω.

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

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(C2) |aij (x)−aij (y)| ≤ ϕ(|x−y|) for any x, y ∈ Ω. Here ϕ : R+ → R+ is supposed to be nondecreasing and to satisfy ϕ(2t) ≤ Cϕ(t)  for some C > 0 and all t > 0; also 0 ϕ(t) dt < ∞. t (C3) There are γ1 , γ2 ∈ (0, 1] and a nonnegative measurable function n ω on Ω such that the following holds for some σ ∈ (1, n−1 ). For n each t ∈ [σ, n−1 ) there are Banach function spaces Xt , Yt for   which MXt : Lt (Rn ) → Lt (Rn ), MYt : Lt (Rn ) → Lt (Rn ) and (4.1)

(4.2)

||MXt ||Lt →Lt ≤ A1 ;

||MYt ||Lt →Lt ≤ A1 ,

γ1

n 

|Q| n || min{δ, (Q)}1−γ1

sup Q:(Q)≤4d(Ω)

|bj |χΩ ||Xt ,Q ≤ A2 ,

j=0

and (4.3)

γ2

sup Q:(Q)≤4d(Ω)

|Q| n || min{δ, (Q)}1−γ2 ωχΩ ||Yt ,Q ≤ A2

n for some A1 , A2 > 0 independent of t ∈ [σ, n−1 ), where we set b0 = cω −1 in (4.2).

Remark 4.1. Condition (C3) holds if, for example, |bj | ≤ gδ −β1 , j = 1, . . . , n,

|c| ≤ gf δ −β1 −β2 ,

where g ∈ Ls,∞ (Ω), s ∈ (n, ∞], 0 ≤ β1 ≤ 1 − ns , 0 ≤ β2 ≤ 1, and n ,∞ s n , n−1 ), f ∈ L 1−β2 (Ω). This can be seen by taking ω = f δ −β2 , σ ∈ ( s−r  s n rt n (n−1)t n Xt = L (R ), 1 < r < n , Yt = L (R ) for each t ∈ [σ, n−1 ), and choosing appropriate γ1 , γ2 ∈ (0, 1]. It was shown in [13] that under conditions (C1) and (C2) on aij there exists

a unique Green function G on Ω×Ω for the elliptic operator L0 = − ni,j=1 Di (aij Dj ) which has the following properties: there exists a constant K = K(n, λ1 , λ2 , Ω) > 0 such that for each x, y and z in Ω, (G1) G(x, y) = G(y, x) ≥ 0. (G2) G(·, y) ∈ W01,t (Ω) and supy∈Ω ||G(·, y)||W 1,t (Ω) < ∞ for every 0 n ). t ∈ [1, n−1 n (G3) For each t ∈ [1, n−1 ) and each  > 0 there is ρ > 0 such that ||G(·, y)||W 1,t (Ω) ≤  for all y in the set {z ∈ Ω : δ(z) ≤ ρ}. 0

16 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

(G4) For each φ ∈ Cc∞ (Ω),   n ∂G ∂φ aij (x) (x, y) (x)dx = φ(y). ∂xi ∂xj Ω i,j=1 (G5) G(x, y) ≤ K|x − y|2−n . (G6) |G(x, y) − G(z, y)| ≤ K|x − z|α {|x − y|2−n−α + |z − y|2−n−α } for some α ∈ (0, 1). (G7) |∇x G(x, y)| ≤ K|x − y|1−n . (G8) |∇x G(x, y)| ≤ Kδ(y)|x − y|−n . Let S be an integral operator whose kernel is the Green function G of L0 defined for each x ∈ Ω and appropriate function u by   n j b (y)Dj u(y) + c(y)u(y) G(y, x)dy. Su(x) = Ω

j=1

We first establish a lemma concerning the boundedness of S on the n Sobolev’s space W01,t (Ω) for t near n−1 . Lemma 4.2. There exists a constant A0 > 0 such that for each t ∈ n [σ, n−1 ) and u ∈ W01,t (Ω), ||∇[Su]||Lt (Ω) ≤ A0 A1 A2 (1 + A0 A1 A2 )||∇u||Lt (Ω) , where σ is as in (C3). Proof. From the definition of S, for any u ∈ Cc∞ (Ω) we have ∇[Su](x) = =

  n Ω

j=1

  n Ω

bj (y)Dj u(y) + c(y)u(y) ∇x G(y, x)dy b (y)Dj u(y) + b (y)ω(y)u(y) ∇x G(y, x)dy. j

0

j=1

Note that (G7) and (G8) can be restated as |∇x G(y, x)| ≤ K |x − y|−n min{δ(y), |x − y|}. Thus from (4.1), (4.2), Corollary 3.5 and Remark 3.4 we can estimate (4.4)

||∇[Su]||Lt (Ω) ≤ CA1 A2 (||∇u||Lt (Ω) + ||uω||Lt (Ω) ).

On the other hand, it follows from (4.1), (4.3), Theorem 3.6 and Remark 3.4 that (4.5)

||uω||Lt (Ω) ≤ CA1 A2 ||∇u||Lt (Ω) .

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

Thus combining (4.4) and (4.5) we obtain the lemma.

17



The following two lemmas are needed later in oder to establish integrability of Green functions for the operator L on Ω × Ω. Lemma 4.3. The mapping y → G(·, y) is uniformly continuous from n Ω into W01,t (Ω) for each t ∈ [1, n−1 ). n Proof. Let  > 0 be a given. Since t < n−1 , by (G2) and H¨older’s inequality there exists a number s > 0 such that

 ||∇x G(·, y) − ∇x G(·, z)||tLt (B(z,s)∩Ω) ≤ . 2 s ∞ Now for ν ∈ (0, 4 ) we choose a C -function ζ on Ω such that 0 ≤ ζ ≤ 1, ζ = 1 on Ω \ B(y, 3ν), ζ = 0 on B(y, 2ν)) and ||∇ζ||L∞ (Ω) ≤ Cν −1 for a constant C > 0. Then for z ∈ B(y, ν) and φ = ζ 2 [G(·, y) − G(·, z)] we get from (G4),   n ∂φ ∂ G(x, y) − G(x, z) aij (x) dx = 0. ∂xi ∂xj Ω i,j=1 (4.6)

Since B(z, ν) ⊂ B(y, 2ν) and |x − z| ≤ 32 |x − y| for x ∈ Ω \ B(y, 2ν), from (G6) and the above equation it follows that  (4.7) Ω\B(y,3ν)

|∇x G(x, y) − ∇x G(x, z)|2 dx



|∇ζ(x)|2 |G(x, y) − G(x, z)|2 dx Ω  −2 |G(x, y) − G(x, z)|2 dx ≤Cν Ω\B(y,2ν)  −2 2α ≤ C ν |y − z| |x − z|4−2n−2α dx ≤C

Ω\B(z,ν) 2α

≤ C(n, α, ν, Ω) |y − z| .

Lemma 4.3 then follows from estimates (4.6) and (4.7) above.



Lemma 4.4. Suppose that K is a function on Ω × Ω for which the mapping y → K(·, y) is uniformly continuous from Ω into Lp (Ω) for

in Lp (Ω × Ω) such that K(·, y) some p ≥ 1. Then there is a function K

y) in Lp (Ω) for every y ∈ Ω. = K(·, Proof. For each integer k > 0 we denote by IΩk the set of dyadic cubes of side-length 2−k that intersect Ω. Thus for each cube Q ∈ IΩk we can

18 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

choose a point yQ ∈ Q ∩ Ω. With that choice of {yQ }Q∈IΩk we define for each positive integer k and (x, y) ∈ Ω × Ω,  Kk (x, y) = χQ (y)K(x, yQ ), k Q∈IΩ

i.e., Kk (x, y) = K(x, yQ ), where Q is the dyadic cube of side-length 2−k that contains y. Since the mapping y → K(·, y) is uniformly continuous from Ω into Lp (Ω) we have lim sup ||Ki (·, y) − Kj (·, y)||Lp (Ω) = 0.

k→∞ y∈Ω i,j≥k

Thus it is possible to choose a subsequence {Kkj (·, y)}j of {Kk (·, y)}k such that sup ||Kkj+1 (·, y) − Kkj (·, y)||Lp (Ω) ≤ 2−j y∈Ω

for j ∈ N. We now let

y) = Kk1 (x, y) + K(x,

∞ 

[Kkj+1 (x, y) − Kkj (x, y)].

j=1

in Lp (Ω × Ω) and for each y ∈ Ω It could be seen then that Kk → K

y) in Lp (Ω). Thus K(·, y) = Kk (·, y) converges to both K(·, y) and K(·,

y) in Lp (Ω) for every y ∈ Ω. K(·,  We now get the Green function for the operator L together with its properties in the following theorem extending some results in [13]. Theorem 4.5. Suppose that (4.8)

A0 A1 A2 (1 + A0 A1 A2 ) < 1,

where A0 is as in Lemma 4.2. Then there exists a unique measurable function H on Ω × Ω such that H(·, y) + S(H(·, y)) = G(·, y) for each y ∈ Ω. Moreover, H enjoys the following properties: (H1) H(·, y) ∈ W01,t (Ω) and supz∈Ω ||H(·, z)||W 1,t (Ω) < ∞ for each 0 n y ∈ Ω and t ∈ [1, n−1 ). n ) and  > 0 there is ρ > 0 such that (H2) For each t ∈ [1, n−1 ||H(·, y)||W 1,t (Ω) ≤  for all y in the set {z ∈ Ω : δ(z) ≤ ρ}. 0

n (H3) H ∈ Ls (Ω × Ω) and |∇x H(x, y)| ∈ Lt (Ω × Ω) for s ∈ [1, n−2 ) n and t ∈ [1, n−1 ).

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

19

(H4) For each y ∈ Ω and φ ∈ Cc∞ (Ω),   n n ∂φ  j ∂H ij ∂H a (·, y) + b (·, y)φ + cH(·, y)φ dx = φ(y). ∂xi ∂xj j=1 ∂xj Ω i,j=1 n ) where σ is as in (C3). By Proof. Let t be a real number in [σ, n−1 (4.8) and Lemma 4.2 we see that S is a contraction from W01,t (Ω) into itself which implies that id + S is an isomorphism on W01,t (Ω). Thus for each y in Ω there is a unique H(·, y) in W01,t (Ω) for which

(4.9)

H(·, y) + S(H(·, y)) = G(·, y).

Differentiating both sides of (4.9) we have  (4.10) ∇x H(x, y) + U (z, y)∇x G(z, x)dz = ∇x G(x, y), Ω

where U (z, y) =

n  j=1

bj

∂H (z, y) + cH(z, y). ∂zj

Note that by hypothesis the function H(·, y) is independent of the n choice of t in [σ, n−1 ). Since id + S is an isomorphism from W01,t (Ω) onto itself, by (G2), (G3), we get (H1) and (H2). Moreover, in view of Lemma 4.3, Lemma 4.4 and Sobolev’s inequality, we obtain (H3). Finally, for each φ ∈ Cc∞ (Ω) and y ∈ Ω, by (G4) and (4.10) we have   n ∂φ ∂H φ(y) = aij (x, y) dx + ∂xi ∂xj Ω i,j=1   n  ∂G ∂φ U (z, y) aij (z, x) (x)dzdx + ∂x ∂x i j Ω Ω i,j=1    n ∂φ ij ∂H a (z, y) dz + U (z, y)φ(z)dz, = ∂zi ∂zj Ω i,j=1 Ω which gives (H4) and completes the proof of the theorem.



We shall see that the existence of solutions to a Dirichlet problem is a direct consequence of the above theorem. On the other hand, the following proposition is important for their uniqueness and regularity. Proposition 4.6. Let f ∈ L1 (Ω) and suppose that w ∈ W01,σ (Ω) is a weak solution to the equation Lw = f in the sense that    n n  ij ∂w ∂φ j ∂w a + b φ + cwφ dx = f φdx ∂xi ∂xj j=1 ∂xj Ω i,j=1 Ω

20 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

for all φ ∈ Cc∞ (Ω). Then we have  w(x) = −(Sw)(x) + f (y)G(x, y) dy for a.e. x in Ω. Ω

ij

ji

Proof. Since a = a and G(x, y) = G(y, x), the proposition is an immediate consequence of the following two identities:   n ∂G ∂w aij (x) (x, y) (x)dx = w(y) for a.e. y in Ω, (4.11) ∂xi ∂xj Ω i,j=1   n

(4.12)

Ω

 ∂w ∂w ∂G a (x, y) + bj G(x, y) + cwG(x, y) dx ∂xi ∂xj ∂xj i,j=1 j=1  f (x)G(x, y)dx for a.e. y in Ω. = n

ij

Ω

To prove (4.11), observing from (G7) that      |∇w(x)||∇x G(x, y)|dx dy ≤ C |∇w(x)||x − y|1−n dx dy Ω Ω Ω Ω     1−n =C |∇w(x)| |x − y| dy dx ≤ Cd(Ω) |∇w(x)|dx < +∞. Ω

Ω

Ω

This implies that the integral in (4.11) is well-defined for a.e. y in Ω. Now let {φm } ⊂ Cc∞ (Ω) be a sequence satisfying φm → w in W01,1 (Ω) and φm (y) → w(y) for a.e. y in Ω. By property (G4) we get   n ∂G ∂φm aij (x) (x, y) (x)dx = φm (y) for all y in Ω. ∂xi ∂xj Ω i,j=1 Therefore, by passing to the limit we see that (4.11) will follow if we can show that up to a subsequence we have     n n ∂G ∂φm ∂G ∂w ij a (x) (x, y) (x)dx −→ aij (x) (x, y) (x)dx ∂xi ∂xj ∂xi ∂xj Ω i,j=1 Ω i,j=1 for a.e. y in Ω. To this end, let fm (y)  be the difference of the two expressions. Then since |fm (y)| ≤ C Ω |x − y|1−n |∇φm (x) − ∇w(x)| dx for all y in Ω and by arguing as in the foregoing observation, we obtain fm → 0 in L1 (Ω) and thus (4.11) is proved. In order to prove (4.12) we first note that as the coefficients aij satisfy (C1) and

n (C2), we can extend the operator L0 = − i,j=1 Di (aij Dj ) to a neigh borhood of  Ω, say Ω. ijFor each m ∈ N and each x in Ω, define ij am (x) = Ω ρ 1 (x − z)a (z) dz, where ρ 1 is the standard mollifier. m m ∞ Then aij m (x) is in C (Ω) and satisfy (C1) and (C2) in Ω with the

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

21

same constants λ1 , λ2 and the same modulus of continuity ϕ. Also,

let Gm (x, y) denote the Green function for Lm := − ni,j=1 Di (aij m Dj ) on Ω. We remark that for each y ∈ Ω, Gm (·, y) is smooth in Ω \ {y} because Lm Gm (·, y) = 0 in Ω \ Br (y) for each Br (y) ⊂⊂ Ω. Let y ∈ Ω be fixed. By Theorem 1.1. in [13] we have for each n ) a uniform bound on ||Gm (·, y)||W 1,s (Ω) with respect to s ∈ [1, n−1 0 n m. Considering a sequence sm ↑ n−1 we find by a diagonal process and by Rellich-Kondrashov’s theorem a subsequence still denoted by

y) ∈ W01,s (Ω) for all s < n , such that {Gm (·, y)} and a function G(·, n−1 1,s n

Gm (·, y)  G(·, y) weakly in W0 (Ω), s ∈ [1, n−1 ), and Gm (x, y) →

y) pointwise for a.e. x in Ω. We are going to show that G

must G(x,

y) ≥ 0 for a.e. x in Ω be G, the Green function for L0 . First, G(x, since all Gm have this property. We also have for φ ∈ Cc∞ (Ω),   n

∂G ∂φ aij (x) (x, y) (x)dx = φ(y). ∂xi ∂xj Ω i,j=1 This is a consequence of

 n Ω

i,j=1

∂φ ∂Gm aij m (x) ∂xi (x, y) ∂xj (x)dx = φ(y) and

ij weakly in Ls (Ω) and aij m → a strongly

y) ∈ W 1,2 (Ω \ Br (y)) whenever in L∞ (Ω). We next claim that G(·, Br (y) ⊂⊂ Ω. Indeed, for such a ball we have from estimate (1.38) in [13] that  (4.13) |∇Gm (x, y)|2 dx ≤ K(n, λ1 , λ2 )r2−n .

the fact

∂Gm (·, y) ∂xi



∂G (·, y) ∂xi

Ω\Br (y)

In view of this and since Gm (x, y) ≤ K(n, λ1 , λ2 )|x − y|2−n we may

y) weakly in W 1,2 (Ω \ Br (y)). But then it also assume Gm (·, y)  G(·, follows from the weakly lower semicontinuity of the Dirichlet integral

in place of Gm . This gives the claim and that (4.13) holds with G

must be equal to G by the uniqueness of the Green function hence G (see Theorem 1.1. in [13]). Particularly, we have for any fixed y ∈ Ω, Gm (x, y) → G(x, y) for a.e. x in Ω. With these ingredients, the proof of (4.12) can be proceeded as follows: We observe that by arguing as at the beginning of this proof and using Corollary 3.5 and Theorem 3.6 one can find a set E ⊂ Ω of measure zero such that for all y ∈ Ω \ E, the functions |∇w(·)|| · −y|1−n ,

|bj (·)||∇w(·)| min {δ(·), | · −y|}| · −y|1−n ,

|c(·)||w(·)| min {δ(·), | · −y|}| · −y|1−n ,

and |f (·)|| · −y|2−n

22 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

are in L1 (Ω). Let y ∈ Ω \ E be fixed and let m ∈ N be arbitrary. Choose a sequence of functions {ϕk } ⊂ Cc∞ (Ω) such that ϕk (x) = 1 ) ∪ (Ω \ Ω 1 ), ϕk (x) = 1 if x ∈ Ω 1 \ B(y, k1 ), 0 ≤ ϕk ≤ 1 0 if x ∈ B(y, 2k 2k k and |∇ϕk | ≤ Ck, where we denote Ω = {x ∈ Ω : δ(x) ≥ }. Then the function φ(x) = Gm (x, y)ϕk (x) is in Cc∞ (Ω) and by using it as a test function in the definition of weak solutions we have     n n ∂w ∂ϕk ij ∂w ∂Gm (4.14) a (x, y)ϕk dx + aij Gm (x, y)dx ∂xi ∂xj ∂xi ∂xj Ω i,j=1 Ω i,j=1    n j ∂w b Gm (x, y)ϕk dx + cwGm (x, y)ϕk dx + ∂xj Ω j=1 Ω  f (x)Gm (x, y)ϕk (x)dx for k = 1, 2, ... = Ω

Let us denote the second integral in (4.14) by Ik . Then   |∇w(x)| |∇w(x)|δ(x) |Ik | ≤ Ck dx + Ck dx n−2 n−1 1 B(y, k1 )\B(y, 2k ) |x − y| Ω 1 \Ω 1 |x − y| 2k k   |∇w(x)| |∇w(x)| ≤C dx + C dx. n−1 n−1 1 B(y, k1 )\B(y, 2k ) |x − y| Ω 1 \Ω 1 |x − y| 2k

1−n

k

  1  ∈ L (Ω), B(y, k1 ) \ B(y, 2k ) →0 1

Hence,  Ik → 0since |∇w(·)||·−y|   and Ω 1 \ Ω 1  → 0. Using this and Lebesgue’s dominated convergence 2k k theorem we obtain from (4.14) that for all y in Ω \ E,     n n ∂w ij ∂w ∂Gm a (x, y)dx + bj Gm (x, y)dx ∂xi ∂xj ∂xj Ω i,j=1 Ω j=1   f (x)Gm (x, y)dx for m = 1, 2, ... + cwGm (x, y)dx = Ω

Ω

Then by Lebesgue’s dominated convergence theorem and the fact that Gm (x, y) → G(x, y) for a.e. x in Ω, (4.12) will follow if we can show     n n ∂w ∂G ij ∂w ∂Gm a (x, y)dx → aij (x, y)dx for a.e. y ∈ Ω. ∂xi ∂xj ∂xi ∂xj Ω i,j=1 Ω i,j=1 Thanks to (4.11), this is equivalent to proving that  n    n    ∂w ∂G ∂w ∂G   m m Jm (y) :=  aij (x, y)dx − aij (x, y)dx  m  Ω  ∂x ∂x ∂x ∂x i j i j Ω i,j=1 i,j=1

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

23

converges to 0 for

a.e. y in Ω. But this is indeed the case as for y in 1−n dx → 0. Ω \ E, Jm (y) ≤ C ni,j=1 ||aij − aij m ||L∞ (Ω) Ω |∇w(x)||x − y| The proof of the proposition is then completed.  As a combination of the foregoing results, we obtain the following existence and uniqueness theorem. We stress that our uniqueness result for this very weak solution is interesting in view of the counter-example of Serrin in [23]. In fact, when n > 2 and even when bj ≡ c ≡ 0 one can construct aij ∈ C ∞ (Rn \ {0}) explicitly satisfying (C1) in Ω := B1 (0) such that the equation L0 u = 0 admits a nonzero weak n solution w ∈ W01,s (Ω) for all 1 ≤ s < n−1 . Here, our result says that ij uniqueness for L is ensured if a are Dini-continuous and bj , c satisfy a smallness condition. Theorem 4.7. Suppose that A0 A1 A2 (1 + A0 A1 A2 ) < 1, where A0 is as Lemma 4.2. Then for each f in L1 (Ω), the function  f (y)H(x, y)dy

u(x) := Ω

n belongs to W01,s (Ω) for any 1 ≤ s < n−1 and solves the equation Lu = f weakly. Moreover, u is unique in the sense that if w ∈ W01,σ (Ω) is a weak solution to Lw = f then w ≡ u. n ), where σ is as in (C3). By TonelliProof. Let s be a number in [σ, n−1 Fubini Theorem and H¨older’s inequality we have

 

s

|f (y)||H(x, y)|dy dx     s−1 s |f (y)||H(x, y)| dy |f (y)|dy dx ≤ Ω Ω Ω  ≤ ||f ||sL1 (Ω) sup |H(x, y)|s dx < ∞. Ω

Ω

y∈Ω

Ω

 Thus the function u(x) = Ω f (y)H(x, y)dy is well-defined and belongs to Ls (Ω). Also using Tonelli-Fubini Theorem we see that u is weakly differentiable with  f (y)∇x H(x, y)dy ∇u(x) = Ω

24 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

for almost every x ∈ Ω. Thus |∇u| ∈ Ls (Ω) by repeating the above argument. Now for each φ ∈ Cc∞ (Ω),   n n  ∂u ij ∂u ∂φ a + bj φ + cuφ dx ∂xi ∂xj j=1 ∂xj Ω i,j=1   n  ∂H ∂φ f (y) aij (x, y) (x)dydx+ = ∂xi ∂xj Ω Ω i,j=1    n  j ∂H f (y) b (x, y)φ(x) + cH(x, y)φ(x) dydx = f φdy + ∂x j Ω Ω Ω j=1 by (H4) in Theorem 4.5 and Fubini Theorem. Hence u is a weak solution to the equation Lu = f in the sense stated in Proposition 4.6. Note also that in view of (4.9) and the definition of S we have  u = −Su + f (y)G(·, y)dy. Ω

Therefore, by combining with Proposition 4.6 we see that if w ∈ W01,σ (Ω) is a weak solution to the equation Lw = f then (id + S)w = (id + S)u. This implies that w = u since we have proved in Theorem  4.5 that id + S is an isomorphism on W01,σ (Ω). Our last result concerns the global regularity for solutions to the equation Lu = f . n˜ σ } < p < n. In addition Theorem 4.8. Let σ ˜ ∈ (1, 2] and max {1, n+˜ σ to (C1) and (C2), assume there exist θ > 1, γ1 , γ2 ∈ (0, 1] and a np ) nonnegative measurable function ω on Ω such that for each t ∈ [˜ σ , n−p t n there are Banach function spaces Xt , Yt for which MXt : L (R ) →   Lt (Rn ), MYt : Lt (Rn ) → Lt (Rn ) and

(4.15)

γ1 1 + t1 − (θt) 

|Q| n

sup Q:(Q)≤4d(Ω)

|| min{δ, (Q)}1−γ1 bj χΩ ||Xt ,Q < +∞,

and (4.16)

sup Q:(Q)≤4d(Ω)

γ2

|Q| n || min{δ, (Q)}1−γ2 ωχΩ ||Yt ,Q < +∞,

where we set b0 = cω −1 in (4.15). Then if u ∈ W01,˜σ (Ω) is a weak solution to the equation Lu = f with f ∈ Lp (Ω), we have u ∈ W01,r (Ω) np for all r ∈ [1, n−p ). Particularly, when p > n2 we get u ∈ C α (Ω) for any α ∈ (0, 2 − np ).

WEIGHTED SOBOLEV’S INEQUALITIES AND SINGULAR EQUATIONS

25

Proof. Note that as in Proposition 4.6 we have  (4.17) u = −Su + f (y)G(·, y)dy. Ω

On the other hand, if f is a function in Lp (Ω), 1 < p < n, then from (G7) and Young’s inequality the function x → Ω f (y)G(x, y)dy is in np ). Moreover, using (4.15), (4.16) and arguing as W 1,r (Ω) for r ∈ [1, n−p in the proof of Lemma 4.2 we see that |∇[Su]| ∈ Lθ˜σ (Ω). Thus from k (4.17) we have u ∈ W01,θ˜σ (Ω). By iteration, u belongs to W01,θ σ˜ (Ω) for np ˜ < n−p . Therefore, u lies in W01,r (Ω) integers k > 0 such that θk−1 σ np for all r ∈ [1, n−p ). The last statement follows from Morrey imbedding theorem and hence the proof is completed.  We remark that a problem similar to Theorem 4.8 has been considered in [24] and [16]. By generalizing the fundamental work of De n Giorgi and Morrey, Stampacchia proved in [24] that if |bj |2 ∈ L 2 (Ω), c ∈ Lq (Ω), q > n2 then any weak solution u ∈ W01,2 (Ω) to the equation Lu = f is H¨older continuous up to the boundary of Ω whenever f ∈ Lp (Ω), p > n2 (see also Theorem 8.29 in [12]). On the other hand, following the work of Aizenman, Simon [1] and Chiarenza, Fabes, Garofalo [7], Kurata studied in [16] a local problem, where he showed that 1,2 (Ω) to the equaif |bj |2 , c ∈ Knloc (Ω) then any weak solution u ∈ Wloc 2 loc tion Lu = 0 is continuous and |∇u| ∈ Kn (Ω). Here Knloc (Ω) is the local Kato class (see e.g., [16]) which contains all functions g ∈ Lploc (Ω), p > n2 . However, as remarked in [16] these solutions in general are not locally H¨older continuous. In Theorem 4.8 above, we consider a global problem and obtain regularity results up to the boundary for weak solutions under conditions where |bj |2 and c may not be even in L1 (Ω). Also when σ ˜ = 2 our conditions are obviously satisfied if |bj |2 , c ∈ Lq (Ω) for some q > n2 . We stress that unlike the approaches of Stampacchia and Kurata which are based on the boundedness and weak Harnack’s inequality for weak solutions of L0 , our method is quite simple relying only on some properties of the Green function for L0 and the integral representation (4.17) for weak solutions. This moreover allows us to obtain regularity results for solutions that may not a priori belong to 1,2 (Ω). We mention finally that a problem related to Theorem 4.7 Wloc and Theorem 4.8 was also considered in [2] for an elliptic operator in non-divergence form whose lower order terms could have strong singularities at the boundary of the domain. However, these terms are required to be locally bounded inside the domain and a sign condition is assumed on the lowest order term as well.

26 DUONG MINH DUC, NGUYEN CONG PHUC, AND NGUYEN VAN TRUYEN

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[22] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813–874. [23] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Pisa 18 (1964), 385–387. [24] G. Stampacchia, Le probl`eme de Dirichlet pour les ´equations elliptiques du second ordre ` a coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258. [25] E. M. Stein, Singular integrals and differentiability properties of functions, Univ. Press, Princeton, N. J., 1970. [26] K. O. Widman, Inequalities for the Green functions and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37. Department of Mathematics-Informatics, Vietnam National University at Hochiminh city, Hochiminh city, Vietnam E-mail address: [email protected] Department of Mathematics, University of Missouri-Columbia, MO 65211 E-mail address: [email protected] Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070 E-mail address: [email protected]