on the integrability properties of the right-hand sides of these equations. Key words: integrability of solutions of nonlinear elliptic equations, entropy and weak ...
Mathematical Notes, vol. 70, no. 3, 2001, pp. 337–346. Translated from Matematicheskie Zametki, vol. 70, no. 3, 2001, pp. 375–385. c Original Russian Text Copyright 2001 by A. A. Kovalevskii.
Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to L1 A. A. Kovalevskii Received September 26, 2000
Abstract—We establish a number of results on the integrability of the entropy and the weak solutions of the Dirichlet problem for nonlinear elliptic equations of second order depending on the integrability properties of the right-hand sides of these equations. Key words: integrability of solutions of nonlinear elliptic equations, entropy and weak solutions of the Dirichlet problem, Young’s inequality, Fatoux lemma.
1. INTRODUCTION AND STATEMENT OF MAIN RESULTS Suppose that Ω is a bounded open set in Rn ( n ≥ 2) and p ∈ (1, n) . Suppose that c1 , c2 are positive constants, g is a nonnegative function from Lp/(p−1) (Ω) , and for any i ∈ {1, . . . , n} ai is the Carath´eodory function on Ω × Rn . We assume that for almost all x ∈ Ω and any ξ ∈ Rn the following inequalities hold: n
|ai (x, ξ)| ≤ c1 |ξ|p−1 + g(x),
i=1
n
ai (x, ξ)ξi ≥ c2 |ξ|p .
(1) (2)
i=1
Consider the following Dirichlet problem: n ∂ ai (x, ∇u) = f − ∂x i i=1
in Ω ,
u = 0 on ∂Ω .
(3)
Questions related to the solvability and properties of solutions of problem (3) with f from L1 (Ω) or from the class of bounded measures were studied in a large number of papers (see, for example, [1– 5]). Let us cite some definitions and results relevant to the problem under consideration. In the following, we assume that f ∈ L1 (Ω) . ˚ 1,1 (Ω) for which the Definition 1. By a weak solution of problem (3) we mean a function u ∈ W following conditions are satisfied:: 1) for any i ∈ {1, . . . , n} , ai (x, ∇u) ∈ L1 (Ω) ; 2) for any function ϕ ∈ C0∞ (Ω) , n ai (x, ∇u)Di ϕ dx = f ϕ dx. Ω
i=1
0001-4346/2001/7034-0337$25.00
Ω
c 2001 Plenum Publishing Corporation
337
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A. A. KOVALEVSKII
As regards this definition, see, for example, [2]. Next, suppose that for any k > 0 Tk is a function on R such that s if |s| ≤ k, Tk (s) = k sign s if |s| > k. By T˚1,p (Ω) we denote the set of all functions u : Ω → R such that for any k > 0 we have ˚ 1,p (Ω) . Tk (u) ∈ W It is well known (see [4]) that if u ∈ T˚1,p (Ω) and i ∈ {1, . . . , n} , then there exists a unique measurable function δi u : Ω → R such that for any k > 0 , Di Tk (u) = δi u · 1{|u| 0 the following inequality holds: {|u−ϕ| 0,
(7)
i=1
then there exists a unique entropy solution of problem (3). Moreover, it was established in [4] that if u is the entropy solution of problem (3), then u is also a solution of this problem in the following sense: for any i ∈ {1, . . . , n} ai (x, δu) ∈ L1 (Ω) and for any function ϕ ∈ C0∞ (Ω) the following relation holds: n ai (x, δu)Di ϕ dx = f ϕ dx. Ω
i=1
Ω
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Moreover, there are examples showing that for p ≤ 2 − 1/n problem (3) has, generally speaking, no weak solutions. Set n(p − 1) . r= n−1 It follows from the results obtained in [4] that if u is the entropy solution of problem (3), then we have u ∈ Lλ (Ω) for any λ ∈ (0, n(p − 1)/(n − p)) and |δu| ∈ Lλ (Ω) for any λ ∈ (0, r) ; besides, if p > 2 − 1/n , then u is a weak solution of problem (3) and for any λ ∈ [1, r) we have ˚ 1,λ (Ω) . u∈W It follows, in particular, that if p > 2 − 1/n and, in addition to (1) and (2), inequality (7) ˚ 1,λ (Ω) for any holds, then there exists a weak solution of problem (3) belonging to the space W λ ∈ [1, r) . Naturally, the question arises as to whether it is possible to enhance the integrability of the entropy and the weak solutions of problem (3) under the additional conditions of integrability of the function f . Here are two relevant results. Theorem 1. Suppose that 1 n and, in addition to (1) and (2), inequality (7) holds. Let p>2−
f ln(1 + |f |) ∈ L1 (Ω). ˚ 1,r (Ω) . Then there exists a weak solution of problem (3) belonging to W Theorem 2. Suppose that 1 n and, in addition to (1) and (2), inequality (7) holds. Suppose that p>2−
1 1 , and for any k ≥ e let the following inequality be valid: meas{|u| ≥ k} ≤ M k−α (ln k)−β .
(13)
Then u ∈ Lα (Ω) . Proof. By (13), for any k ∈ N we have {ek ≤|u| 0 , set For any u ∈ T I(u, k) = Ω
|∇Tk (u)|p dx.
Lemma 3. Let u ∈ T˚1,p (Ω) , k > 0 . Then ∗
∗
∗
meas{|u| ≥ k} ≤ cpn,p k−p [I(u, k)]p
/p
.
(14)
Proof. Inequality (14) follows from (5) and the obvious inequality k
p∗
meas{|u| ≥ k} ≤
∗
Ω
|Tk (u)|p dx.
Lemma 4. Let u ∈ T˚1,p (Ω) , k, k1 > 0 . Then ∗
∗
∗
meas{|δu| ≥ k} ≤ cpn,p k1−p [I(u, k1 )]p
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/p
+ k−p I(u, k1 ).
(15)
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A. A. KOVALEVSKII
Proof. Set G = {|u| < k1 , |δu| ≥ k} . Obviously, meas{|δu| ≥ k} ≤ meas{|u| ≥ k1 } + meas G.
(16)
By (4), we have k ≤ |∇Tk1 (u)| almost everywhere on G . Then meas G ≤ k−p I(u, k1 ).
(17)
From (16), (17), and Lemma 3 we obtain (15), and the lemma is now proved. Lemma 5. Suppose that c > 0 , (n − 1)/n < σ < 1 and ψ is a nonnegative measurable function on Ω such that ψ[ln(1 + ψ)]σ ∈ L1 (Ω) . Suppose that u ∈ T˚1,p (Ω) and for any k > 0 the following inequality is valid: I(u, k) ≤ c
Ω
ψ|Tk (u)| dx.
(18)
Then u ∈ Lq (Ω) , where q = n(p − 1)/(n − p) , and |δu| ∈ Lr (Ω) . Proof. Let ci , i = 3, 4, . . . , be positive constants depending only on n , p , c , σ , meas Ω , and the L1 (Ω)-norm of the function ψ[ln(1 + ψ)]σ . Choose k ≥ e and set γ = (p − 1)/(2p) . By (18), we have ψ |Tk (u)| dx + c ψ|Tk (u)| dx. (19) I(u, k) ≤ c {ψ≤k γ }
{ψ>k γ }
For the first integral on the right-hand side of (19), using H¨ older’s inequality and inequality (5), we obtain 1/p∗ γ γ (p∗ −1)/p∗ p∗ ψ|Tk (u)| dx ≤ k |Tk (u)| dx ≤ k [meas Ω] |Tk (u)| dx {ψ≤k γ }
Ω
γ
Ω
1/p
≤ c3 k [I(u, k)]
.
(20)
For the second integral on the right-hand side of (19), we have ψ|Tk (u)| dx ≤ k ψ dx ≤ k(γ ln k)−σ ψ[ln(1 + ψ)]σ dx = c4 k(ln k)−σ . {ψ>k γ }
{ψ>k γ }
(21)
Ω
From (19)–(21), using the Young inequality, we obtain I(u, k) ≤ cc3 kγ [I(u, k)]1/p + cc4 k(ln k)−σ ≤ Therefore,
p−1 1 I(u, k) + (cc3 )p/(p−1) k1/2 + cc4 k(ln k)−σ . p p
I(u, k) ≤ c5 k1/2 + c6 k(ln k)−σ ≤ (2c5 + c6 )k(ln k)−σ .
Thus, for any k ≥ e we have
I(u, k) ≤ c7 k(ln k)−σ .
(22)
Combining this with Lemmas 3 and 2, we obtain u ∈ Lq (Ω) , where q = n(p − 1)/(n − p) . Now choose an arbitrary number k > e(n−1)/(n−p) . It is readily seen that there exists a number k1 > e such that (23) k1n−1 (ln k1 )σ = kn−p . Then
−n(p−1)/(n−p)
k1
(ln k1 )−σn/(n−p) = k−p k1 (ln k1 )−σ = k−r (ln k1 )−σn/(n−1) . MATHEMATICAL NOTES
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Obviously, ln k1 < k1 . Combining this with (23), we obtain kn−p < k1n .
(25)
I(u, k1 ) ≤ c7 k1 (ln k1 )−σ .
(26)
Further, since k1 > e , by (22) we have Using Lemma 4 and relations (24)–(26), we infer that meas{|δu| ≥ k} ≤ c8 k−r (ln k)−σn/(n−1) . Combining this with Lemmas 2 and taking into account the inequality σ > (n − 1)/n , we obtain |δu| ∈ Lr (Ω) . The lemma is proved. Lemma 6. Suppose that b > 0 , 1 < m < np/(np − n + p) and ψ is a nonnegative function on Ω , ˚1,p (Ω) and for any k > 0 the following inequality is valid: where ψ ∈ Lm (Ω) . Suppose that u ∈ T ψ|Tk (u)| dx. (27) I(u, k) ≤ b Ω
Then 1) for any λ ∈ (0, nm(p − 1)/(n − mp)) , we have u ∈ Lλ (Ω) ; 2) for any λ ∈ (0, (p − 1)m∗ ) , we have |δu| ∈ Lλ (Ω) . Proof. First of all, note that by the inequality m < np/(np − n + p) we have mp < n and (m − 1)p∗ < m . Set nm(p − 1) m−1 ∗ (m − 1)p∗ n − mp p , , m3 = , θ= . m2 = m mp n−m n − mp Obviously, 0 < m1 < 1 , 0 < m2 < 1 . Next, we have m1 ∗ = θ, (28) p 1− (1 − m2 )p m1 m3 . (29) m3 θ = p − 1 − m2 Let bi , i = 1, 2, . . . , denote positive constants depending only on n , p , b , m and the Lm (Ω)norm of the function ψ . Choose k > 0 . Using (27), H¨ older’s inequality, and inequality (5), we obtain 1/m (m−1)/m m1 1−m1 m1 m p∗ ψ |Tk (u)| dx ≤ bk ψ dx |Tk (u)| dx I(u, k) ≤ bk m1 = 1 −
Ω
Ω
≤ b1 km1 [I(u, k)]m2 .
Ω
Now we can conclude that for any k > 0 I(u, k) ≤ b2 km1 /(1−m2 ) .
(30)
Combining this with Lemma 3 and taking into account (28), for any k > 0 we obtain meas{|u| ≥ k} ≤ b3 k−θ . Then, by Lemma 1, for any λ ∈ (0, θ) we have u ∈ Lλ (Ω) . Hence assertion 1) is valid. Next, choose k > 0 and set k1 = km3 . By (30), we have I(u, k1 ) ≤ b2 km1 m3 /(1−m2 ) . Combining this with Lemma 4 and taking into account relations (28) and (29), we obtain ∗
meas{|δu| ≥ k} ≤ b4 k−(p−1)m . Hence, by Lemma 1, for any λ ∈ (0, (p − 1)m∗ ) we have |δu| ∈ Lλ (Ω) . Hence assertion 2) is also valid. The lemma is now proved. MATHEMATICAL NOTES
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Lemma 7. Suppose that b , b > 0 , 1 < m < np/(np − n + p) and ψ is a nonnegative function on Ω , where ψ ∈ Lm (Ω) . Suppose that (p − 1)m∗ ≥ 1 . Suppose that u ∈ T˚1,p (Ω) and for any k > 0 the following inequalities are valid: meas{|u| ≥ k} ≤ b k−nm(p−1)/(n−mp) , p |∇Tk (u)| dx ≤ b ψ dx.
{k−1≤|u|
