Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 25–34

LOCAL AND GLOBAL INTEGRABILITY OF GRADIENTS IN OBSTACLE PROBLEMS

Li Gongbao^{∗} and Olli Martio

Academia Sinica, Wuhan Institute of Mathematical Sciences P.O. Box 71007, Wuhan 430071, P.R. of China University of Helsinki, Department of Mathematics

P.O. Box 4 (Hallituskatu 15), FIN–00014 University of Helsinki, Finland; olli.martio@helsinki.fi

Abstract. We establish local and global higher integralibity results for the derivatives of the solutions to obstacle problems associated with the second order degenerate elliptic partial differential equation div 𝒜 (x, ∇u(x)) = 0, where |𝒜 (x, ξ)| ≈ |ξ|^{p−1} , p > 1.

1. Introduction

In this paper we consider the obstacle problem associated with the second order degenerate elliptic equation

(1.1)  div 𝒜 (x, ∇u(x)) = 0

with |𝒜 (x, ξ)| ≤ β |ξ|^{p−1} and 𝒜 (x, ξ) · ξ ≥ α |ξ|^{p} for some 0 < α ≤ β < ∞ and p > 1, see 2.1. The prototype of equation (1.1) is the p-harmonic equation

(1.2)  div (|∇u|^{p−2} ∇u) = 0.

Suppose that Ω is a bounded open set in R^{n}, that ψ is any function in Ω with values in R ∪ {−∞, ∞}, and that θ ∈ W^{1,p}(Ω). The function ψ is an obstacle and θ determines the boundary values. Let

𝒦_{ψ,θ} = {v ∈ W^{1,p}(Ω) : v ≥ ψ a.e. and v − θ ∈ W_{0}^{1,p}(Ω)}.

A solution to the 𝒦_{ψ,θ} -obstacle problem is a function u ∈ 𝒦_{ψ,θ} such that

(1.3)  ∫_{Ω} 𝒜 (x, ∇u) · ∇(v − u) dx ≥ 0

whenever v ∈ 𝒦_{ψ,θ}.

For solutions u of equation (1.1) it is known ([GM], [Str 1–2], [I], [RZ]) that u ∈ W_{loc}^{1,q}(Ω) where q = q(p, n, α/β) > p. Our first result generalizes this to the solution of the 𝒦_{ψ,θ} -obstacle problem.

1991 Mathematics Subject Classification: Primary 35J70; Secondary 35J85.

∗ Partially supported by Youth’s Foundation NSFC.

Theorem A. Suppose that ψ ∈ W_{loc}^{1,s}(Ω), s > p. Then a solution u to the 𝒦_{ψ,θ} -obstacle problem belongs to W_{loc}^{1,q}(Ω) where q = q(p, s, n, α/β) > p.

For variational extremals the global higher integrability of the derivative ∇u has been studied by S. Granlund [G] in the case p = n. For this it seems necessary to impose a regularity condition for ∂Ω. We say that ∂Ω is p-Poincaré thick if there is γ < ∞ such that for all open cubes Q(r) ⊂ R^{n} with side length r > 0 it holds

(1.4)  ( ∫_{Q(2r)} |u|^{p} dx)^{1/p} ≤ γ( ∫_{Q(2r)} |∇u|^{pn/(p+n)} dx)^{(p+n)/pn}

whenever u ∈ W^{1,p}(Q(2r)), u = 0 a.e. on (R^{n} \ Ω) ∩ Q(2r), and Q(\frac{3}{2}r) ∩ ∁Ω ≠ ∅; here, and in the following, Q(λr), λ > 0, means a cube parallel to Q(r) with the same center as Q(r) and with side length λr. Theorem 2.3 and Corollary 2.7 below give simple sufficient conditions such that (1.4) holds for p ≥ n/(n − 1).

Theorem B. Suppose that a bounded domain Ω has a p-Poincaré thick boundary and that p ≥ n/(n − 1). Let θ and ψ belong to W^{1,s}(Ω), s > p. Then a solution u to the 𝒦_{ψ,θ} -obstacle problem belongs to W^{1,q}(Ω) where q = q(p, s, n, α/β, γ) > p and γ is the constant of (1.4).

In Section 2 the assumptions on 𝒜 together with some preliminary lemmas are presented. Section 3 is devoted to the proofs of Theorems A and B. In Remark 3.14 some variants of Theorems A and B are discussed. In particular, local and global higher integrability for the derivatives of solutions of (1.1) is a consequence of Theorems A and B, respectively. Theorems A and B also imply the corresponding results for variational obstacle problems.

The higher integrability of solutions of (1.1) were first considered by Meyers and Elcrat [ME] in 1975. See also [Str 1–2]. For obstacle problems and for differential and variational inequalities most of the regularity studies have been devoted to prove the Hölder continuity of the solutions u to the 𝒦_{ψ,θ} -obstacle problem for Hölder continuous obstacles ψ [Gi]. Michael and Ziemer [MZ] proved the continuity of u if ψ is just continuous. For p-harmonic equations (1.2) the higher regularity, i.e. the C^{1,α} -regularity, has been much studied, see [L]. For equations (1.1) the Hölder continuity and higher integrability of the derivatives are different aspects of regularity, although for p ≥ n there is an obvious connection via the Sobolev imbedding theorem.

When our work was completed, T. Kilpeläinen and P. Koskela [KK] replaced the Poincaré thickness in Theorem B by a capacitary condition on ∂Ω.

2. Equation (1.1) and preliminary results

2.1. Equation div 𝒜 (x, ∇u) = 0. We consider mappings 𝒜: R^{n} × R^{n} → R^{n} which satisfy the following assumptions for some p > 1 and 0 < α ≤ β:

(a) the mapping x ↦ 𝒜 (x, ξ) is measurable for all ξ ∈ R^{n} and the mapping ξ ↦ 𝒜 (x, ξ) is continuous for a.e. x ∈ R^{n};

for all ξ ∈ R^{n} and a.e. x ∈ R^{n}

(b) 𝒜 (x, ξ) · ξ ≥ α |ξ|^{p} and

(c) |𝒜 (x, ξ)| ≤ β |ξ|^{p−1} .

The constant p is always associated with 𝒜 as in (b) and (c).

The assumptions (a)–(c) are not strong enough to give a unique solution to the 𝒦_{ψ,θ} -obstacle problem. However, if 𝒜 satisfies the monotonicity condition

(𝒜 (x, ξ_{1}) − 𝒜 (x, ξ_{2})) · (ξ_{1} − ξ_{2}) > 0, ξ_{1} ≠ ξ_{2},

for a.e. x ∈ R^{n}, then it can be shown that the 𝒦_{ψ,θ} -obstacle problem has a unique solution provided that 𝒦_{ψ,θ} ≠ ∅. For this result see [HKM].

In [HKM] the relation between 𝒦_{ψ,θ} -obstacle problems and variational obstacle problems is explained.

2.2. A sufficient condition for (1.4). Here we show that condition (1.4) follows from a measure-theoretic property of ∁Ω; this observation is due to Granlund [G] for p = n.

2.3. Theorem. Suppose that there is μ > 0 such that each cube Q(r) with Q(\frac{3}{2}r) ∩ ∁Ω ≠ ∅ satisfies

(2.4)  m((R^{n} \ Ω) ∩ Q(2r)) ≥ μm(Q(2r)).

Then ∁Ω is p-Poincaré thick for each p ≥ n/(n − 1) and the constant γ in (1.4) depends only on n, p, and μ.

Proof. Let Q(r) be a cube with Q(r) ∩ ∂Ω = ∅ and let u ∈ W^{1,p}(Q(2r)) satisfy u = 0 on Q(2r) \ (R^{n} \ Ω). By [Mor, Theorem 3.6.5, p. 83] we have for each q > 1

(2.5)  ∫_{Q(2r)} |u|^{q} dx ≤ c_{1}(n, q, μ)r^{q} ∫_{Q(2r)} |∇u|^{q} dx.

Next let

c_{u} = \frac{1}{m(Q(2r))} ∫_{Q(2r)} u dx

be the mean value of u in Q(2r). For the following Sobolev–Poincaré inequality

(2.6) ( ∫_{Q(2r)} |u − c_{u}|^{p} dx)^{1/p} ≤ c_{2}(n, p)( ∫_{Q(2r)} |∇u|^{pn/(p+n)} dx)^{(p+n)/pn}

see [GT, p. 174]; note that p is the Sobolev conjugate exponent of q = pn/(p + n) and that q < n.

Combining (2.5) and (2.6) we obtain

( ∫Q(2r)|u|^{p} dx)^{1/p} ≤ ( ∫_{Q(2r)} |u − c_{u}|^{p})^{1/p} + m(Q(2r))^{1/p}|c_{u}|

≤ c_{2}( ∫_{Q(2r)} |∇u|^{q} dx)^{1/q} + m(Q(2r))^{−1/n}( ∫_{Q(2r)} |u|^{q} dx)^{1/q}

≤ c_{2}( ∫_{Q(2r)} |∇u|^{q} dx)^{1/q} + c′_{1}( ∫_{Q(2r)} |∇u|^{q} dx)^{1/q}

≤ γ( ∫_{Q(2r)} |∇u|^{q} dx)^{1/q}

where the Minkowski and Hölder inequality has also been used. The theorem follows.

An open set Ω is c-coplump, c ≥ 1, if for each x ∈ R^{n} \ Ω and r > 0 there is z ∈ B(x, r) such that

B(z, r/c) ∩ Ω = ∅.

If Ω is c-coplump, then Ω clearly satisfies condition (2.4) for some μ = μ(c) > 0. Hence we obtain from Lemma 2.3

2.7. Corollary. If Ω is c-coplump, then ∁Ω is p-Poincaré thick for all p ≥ n/(n − 1).

2.8. Reverse Hölder inequality. To obtain the higher integrability we use the following semilocal reverse Hölder inequality due to Giaquinta and Modica [GM, p. 164]; a new and rather simple proof for Lemma 2.9 can be derived from the work of Kinnunen [K].

2.9. Lemma. Suppose that q > 1 and that g ∈ L^{q}(Q(2r_{0})) and f ∈ L^{s}(Q(2r_{0})), s > q. If for every x ∈ Q(2r_{0}) and r < \frac{1}{2}d(x, ∂Q(2 r_{0})) we have the estimate

\overline{∫} _{Q(r)} |g|^{q} dx ≤ c[(\overline{∫} _{Q(2r)} |g| dx)^{q} +\overline{∫} _{Q(2r)} |f|^{q} dx]

for some c > 0 independent of the cube Q(r) with center at x, then g ∈ L_{loc}^{t}(Q(2 r_{0})) for some t = t(n, q, s, c) > q.

3. Proofs for Theorems A and B

3.1. Proof for Theorem A. Let u be a solution to the 𝒦_{ψ,θ} -obstacle problem and let Q(2r) ⊂ Ω be a cube. Fix a cutoff function φ ∈ C_{0}^{∞}(Q(2r)) such that 0 ≤ φ ≤ 1, |∇φ| ≤ c/r, and φ = 1 on Q(r). Consider the function

v = u − c_{u} − φ^{p}(u − c_{u} − (ψ − c_{ψ}));

here c_{u} and c_{ψ} denote the mean values of the functions u and ψ, respectively, in Q(2r), i.e.

c_{u} =\overline{∫} _{Q(2r)} u dx = \frac{1}{m(Q(2r))} ∫_{Q(2r)} u dx.

Now v ∈ 𝒦_{ψ−c_{u},θ−c_{u}}; indeed, v − (θ − c_{u}) ∈ W_{0}^{1,p}(Ω) because φ ∈ C_{0}^{∞}(Ω) and since c_{u} ≥ c_{ψ}, we obtain

\begin{aligned}v & = (1 − φ^{p})(u − c_{u}) + φ^{p}(ψ − c_{ψ}) ≥ (1 − φ^{p})(u − c_{u}) + φ^{p}(ψ − c_{u}) \\ & ≥ (1 − φ^{p})(ψ − c_{u}) + φ^{p}(ψ − c_{u}) = ψ − c_{u} \end{aligned}

a.e. in Ω. Since

∇v = (1 − φ^{p}) ∇(u − c_{u}) + φ^{p} ∇(ψ − c_{ψ}) + p φ^{p−1} ∇φ[(ψ − c_{ψ}) − (u − c_{u})]

and since u − c_{u} is a solution to the 𝒦_{ψ−c_{u},θ−c_{u}} -obstacle problem, we have

∫_{Ω} 𝒜 (x, ∇u) · ∇u dx ≤ ∫_{Ω} 𝒜 (x, ∇u) · ∇v dx

≤ ∫_{Ω}(1 − φ^{p}) 𝒜 (x, ∇u) · ∇u dx + ∫_{Ω} φ^{p} 𝒜 (x, ∇u) · ∇ψ dx

+ pβ ∫_{Ω} |∇u|^{p−1} φ^{p−1} |∇φ| (|ψ − c_{ψ}| + |u − c_{u}|) dx

where we have also used assumption (c). Using (b) and (c) again we obtain from the above inequality

(3.2)  \begin{aligned}α ∫_{Ω} φ^{p} |∇u|^{p} dx & ≤ ∫_{Ω} φ^{p} 𝒜 (x, ∇u) · ∇u dx \\ & ≤ β ∫_{Ω} φ^{p} |∇u|^{p−1} |∇ψ| dx + pβ ∫_{Ω} |∇u|^{p−1} φ^{p−1} \\ & × |∇φ| (|ψ − c_{ψ}| + |u − c_{u}|) dx. \end{aligned}

Next we use Young’s inequality

(3.3)  ab ≤ εa^{p′} + C(ε, p)b^{p},  \frac{1}{p} + \frac{1}{p′} = 1,

valid for all a, b ≥ 0, ε > 0, and p > 1. Now (3.2) yields

\begin{aligned}α ∫_{Ω} & φ^{p} |∇u|^{p} dx ≤ εβ ∫_{Ω} φ^{p} |∇u|^{p} dx + C(ε, p)β ∫_{Ω} φ^{p} |∇ψ|^{p} dx \\ & + pβε ∫_{Ω} |∇u|^{p} φ^{p} dx + 2^{p}C(ε, p)pβ ∫_{Ω} |∇φ|^{p}(|ψ − c_{ψ}|^{p} + |u − c_{u}|^{p}) dx \end{aligned}

and choosing

ε = \frac{α}{2β(1 + p)}

we obtain from the above inequality

(3.4) ∫_{Ω} φ^{p}|∇u|^{p} dx ≤ c[ ∫_{Ω} φ^{p} |∇ψ|^{p} dx + ∫_{Ω} |∇φ|^{p}(|ψ − c_{ψ}|^{p} + |u − c_{u}|^{p}) dx]

where c is a (generic) constant which depends only on n, p, and α/β. Next we estimate the last integral in (3.4) using the ordinary Poincaré inequality [GT, 7.45, p. 164]

(3.5)  ∫_{Q(2r)} |v − c_{v}|^{p} dx ≤ cr^{p} ∫_{Q(2r)} |∇v|^{p} dx

valid for all functions v ∈ W^{1,p}(Q(2r)) and the Sobolev–Poincaré inequality (2.6). Together with |∇φ| ≤ c/r these give

\begin{aligned}∫_{Ω} |∇φ|^{p} & (|ψ − c_{ψ}|^{p} + |u − c_{u}|^{p}) dx \\ & ≤ c ∫_{Q(2r)} |∇ψ|^{p} dx + \frac{c}{r^{p}} ( ∫_{Q(2r)} |∇u|^{pn/(p+n)} dx)^{(p+n)/n} \end{aligned}

and hence we obtain from (3.4) the estimate

\begin{aligned}∫_{Q(r)} |∇u|^{p} dx & ≤ ∫_{Ω} φ^{p}|∇u|^{p} dx \\ & ≤ c ∫_{Q(2r)} |∇ψ|^{p} dx + \frac{c}{r^{p}} ( ∫_{Q(2r)} |∇u|^{pn/(p+n)} dx)^{(p+n)/n}. \end{aligned}

This implies

\overline{∫} _{Q(r)} |∇u|^{p} dx ≤ c(\overline{∫} _{Q(2r)} |∇u|^{pn/(p+n)} dx)^{(p+n)/n} + c\overline{∫} _{Q(2r)} |∇ψ|^{p} dx.

Setting g ≡ |∇u|^{pn/(p+n)} , f = |∇ψ|^{pn/(p+n)} , and q = (p + n)/n we obtain from Lemma 2.9 that |∇u| ∈ L_{loc}^{t}(Ω) for some t = t(p, s, n, α/β) > p.

The Sobolev imbedding theorem [GT, p. 164] yields u ∈ L_{loc}^{np/(n−p)}(Ω) if p < n, u ∈ L_{loc}^{q}(Ω) for all q > 1 if p = n, and u ∈ L_{loc}^{∞}(Ω) if p > n. Hence u ∈ L_{loc}^{t′}(Ω), t′ = t′(p, n) > p, and choosing q = min(t, t′) > p we have proved Theorem A.

3.6. Proof for Theorem B. Since Ω is bounded, we can choose a cube Q_{0} = Q(2r_{0}) such that Ω ⊂ Q(r_{0}). Next let Q(2r) ⊂ Q_{0} . There are two possibilities:

(i) Q(\frac{3}{2}r) ⊂ Ω or (ii) Q(\frac{3}{2}r) ∩ ∁Ω ≠ ∅. In the case (i) we can follow the proof for Theorem A to obtain the estimate

\overline{∫} _{Q(r)} |∇u|^{p} dx ≤ c[(\overline{∫} _{Q( \frac{3}{2} r)} |∇u|^{np/(p+n)} dx)^{(p+n)/n} +\overline{∫} _{Q( \frac{3}{2} r)} |∇ψ|^{p} dx]

and then choosing g = |∇u|^{np/(p+n)} , f = |∇ψ|^{np/(p+n)} in Q(\frac{3}{2}r) and g = f = 0 in Q(2r) \ Q(\frac{3}{2}r) with q = (p + n)/p we arrive at the inequality

(3.7)  \overline{∫} _{Q(r)} g^{q} dx ≤ c[(\overline{∫} _{Q(2r)} g dx)^{q} +\overline{∫} _{Q(2r)} f^{q} dx]

where c = c(p, s, n, α/β) < ∞.

In the case (ii) note that replacing θ by θ_{1} = max(θ, ψ) we may assume that the boundary function θ satisfies θ ≥ ψ in Ω. Indeed, θ_{1} = (ψ − θ)^{+} + θ and since

0 ≤ (ψ − θ)^{+} ≤ (u − θ)^{+} ∈ W_{0}^{1,p}(Ω),

the function (ψ − θ)^{+} , and hence u − θ_{1} , belongs to W_{0}^{1,p}(Ω). Next let

v = u − φ^{p}(u − θ)

in Ω where φ ∈ C_{0}^{∞}(Q(2r)) is a similar cutoff function as in the proof of Theorem A. Now v ∈ 𝒦_{ψ,θ} because v − θ ∈ W_{0}^{1,p}(Ω) and u ≥ ψ, θ ≥ ψ a.e. yields

v = (1 − φ^{p})u + φ^{p}θ ≥ (1 − φ^{p})ψ + φ^{p}ψ = ψ

a.e. Since

∇v = (1 − φ^{p}) ∇ u + φ^{p} ∇ θ + pφ^{p−1}(θ − u) ∇φ,

we have the estimate

∫_{Ω} 𝒜 (x, ∇u) · ∇u dx ≤ ∫_{Ω} 𝒜 (x, ∇u) · ∇v dx
≤ ∫_{Ω}(1 − φ^{p}) 𝒜 (x, ∇u) · ∇u + β ∫_{Ω} |∇u|^{p−1}φ^{p}|∇θ| dx
+ βp ∫_{Ω} |∇u|^{p−1}φ^{p−1}|θ − u| |∇φ| dx

where assumption (c) has also been used. From this and from (b) we obtain

α ∫_{Ω} φ^{p} |∇u|^{p} dx ≤ ∫_{Ω} φ^{p}𝒜 (x, ∇u) · ∇u dx

≤ β ∫_{Ω} |∇u|^{p−1}φ^{p}|∇θ| dx + βp ∫_{Ω} |∇u|^{p−1}φ^{p−1}|θ − u| |∇φ| dx

≤ βε ∫_{Ω} φ^{p}|∇u|^{p} dx + βC(ε, β) ∫_{Ω} φ^{p}|∇θ|^{p} dx
+ βpε ∫_{Ω} φ^{p}|∇u|^{p} dx + βpC(ε, p) ∫_{Ω} |∇φ|^{p}|θ − u|^{p} dx;

here we have also used Young’s inequality (3.3) twice. Now we choose

ε = \frac{α}{2β(1 + p)}.

Then the above inequality yields

(3.8)  ∫_{Ω} φ^{p}|∇u|^{p} dx ≤ c[ ∫_{Ω} φ^{p}|∇θ|^{p} dx + ∫_{Ω} |∇φ|^{p}|θ − u|^{p} dx]

where c is a (generic) constant depending only on p, s, α/β, n, and γ.

To estimate the last integral in (3.8) we use the p-Poincaré thickness of ∂Ω. Indeed, the function θ − u can be continued as 0 to ∁Ω and hence (1.4) implies

(3.9) ∫_{Ω} |∇φ|^{p}|θ − u|^{p} dx ≤ cr^{−p}( ∫_{Q(2r)∩Ω} |∇(θ − u)|^{pn/(p+n)} dx)^{(p+n)/n};

note that ∇(θ − u) = 0 a.e. in ∁Ω. The Minkowski and Hölder inequalities yield

r^{−p}( ∫_{Q(2r)∩Ω} |∇(θ − u)|^{pn/(p+n)} dx)^{(p+n)/n}

≤ r^{−p}[( ∫_{Q(2r)∩Ω} |∇θ|^{pn/(p+n)} dx)^{(p+n)/pn}

+ ( ∫_{Q(2r)∩Ω} |∇u|^{pn/(p+n)} dx)^{(p+n)/pn}]^{p}

≤ r^{−p}[r( ∫_{Q(2r)∩Ω} |∇θ|^{p} dx)^{1/p} + ( ∫_{Q(2r)∩Ω} |∇u|^{pn/(p+n)})^{(p+n)/pn}]^{p}

≤ 2^{p}[ ∫_{Q(2r)∩Ω} |∇θ|^{p} dx + r^{−p}( ∫_{Q(2r)∩Ω} |∇u|^{pn/(p+n)} dx)^{(p+n)/n}].

From (3.8) and (3.9) we thus obtain

(3.10) \begin{aligned}_{∫Ω} & φ^{p}|∇u|^{p} dx \\ & ≤ c[ ∫_{Q(2r)∩Ω} |∇θ|^{p} dx + r^{−p}( ∫_{Q(2r)∩Ω} |∇u|^{pn/(p+n)} dx)^{(p+n)/n}]. \end{aligned}

If we now set g = |∇u|^{pn/(p+n)} and f = |∇θ|^{pn/(p+n)} in Ω ∩ Q(2r), g = f = 0 in Q(2r) \ Ω, and q = (p + n)/n, then (3.10) yields

(3.11)  \overline{∫} _{Q(r)} g^{q} dx ≤ c[\overline{∫} _{Q(2r)} f^{q} dx + (\overline{∫} _{Q(2r)} g dx)^{q}]

where c = c(p, s, n, α/β, γ) < ∞.

Lemma 2.9 together with inequalities (3.7) and (3.11) implies that |∇u| ∈ L^{t}(Ω) for some t = t(p, s, n, α/β, γ) > p.

It remains to show that u ∈ L^{δ}(Ω) for some δ = δ(n, p) > p. Continuing u − θ as 0 to R^{n} we obtain from the ordinary Sobolev imbedding theorem that for p < n, p^{∗} = pn/(n − p),

(3.12)  ( ∫_{Ω} |u − θ|^{p^{∗}} dx)^{1/p^{∗}} ≤ c( ∫_{Ω} |∇(u − θ)|^{p} dx)^{1/p} < ∞.

If now δ = min(s, p^{∗}) > p, then by the Minkowski and Hölder inequalities

(3.13)  \begin{aligned}( ∫_{Ω} |u|^{δ} dx)^{δ} & ≤ ( ∫_{Ω} |θ|^{δ} dx)^{1/δ} + ( ∫_{Ω} |u − θ|^{δ} dx)^{1/δ} \\ & ≤ ( ∫_{Ω} |θ|^{δ} dx)^{1/δ} + c_{1}( ∫_{Ω} |u − θ|^{p^{∗}} dx)^{1/p^{∗}} \end{aligned}

where c_{1} depends on diam Ω, p, and n.

Since θ ∈ L^{s}(Ω), we obtain from (3.13) that u ∈ L^{δ}(Ω). Setting q = min(t, δ) > p we see that u ∈ W^{1,q}(Ω) in the case p < n. If p ≥ n, then we can apply the above reasoning for any p^{∗} < ∞ together with Hölder’s inequality to conclude that u ∈ L^{s}(Ω) and hence u ∈ W^{1,q}(Ω) with q = min(t, s) > p in this case. The theorem follows.

3.14. Remarks. Here we present some variants of Theorems A and B.

(a) A slight modification of the proof of Theorem A shows that if u ∈ W_{loc}^{1,p}(Ω) is a solution of div 𝒜 (x, ∇u) = 0 in Ω, then u ∈ W_{loc}^{1,q}(Ω), q = q(n, p, α/β) > p. This situation has already been considered in [GM], [Str 1–2], and [I]. In fact, this situation corresponds to the case ψ = −∞.

34 Li Gongbao and Olli Martio

(b) If in Theorem B it is assumed that θ, ψ ∈ W^{1,p}(Ω) with ∇θ, ∇ψ ∈ L^{s}(Ω), s > p ≥ n/(n − 1), then it follows from the proof of Theorem B that ∇u ∈ L^{q}(Ω), q = q(n, p, s, α/β, γ) > p. Granlund proved this result for variational obstacle problems in the case p = n [G, Theorem 1.5].

(c) A simplified version of the proof for Theorem B shows that if u is a solution of ∇ · 𝒜 (x, ∇u) = 0 in Ω with u − θ ∈ W_{0}^{1,p}(Ω) and if θ ∈ W^{1,s}(Ω), s > p ≥ n/(n − 1), then u ∈ W^{1,q}(Ω), q = q(p, s, n, α/β, γ) > p.

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Received 25 September 1992