Differential and Integral Equations Volume 16, Number xx, Year 2003, Pages xx–xx

LINEAR ELLIPTIC SYSTEMS INVOLVING FINITE RADON MEASURES

Vicenţiu Rădulescu

Department of Mathematics, University of Craiova, RO-1100 Craiova, Romania

Michel Willem Mathématique Pure et Appliquée, Université Catholique de Louvain Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

(Submitted by: Viorel Barbu)

1. Statement of the main result

The study of elliptic boundary value problems with L^{1} or Radon measure data has been initiated in the last few decades by the pioneering works of Stampacchia [12], Brezis-Strauss [7], Brezis [5], [6].

Let Ω be a smooth bounded domain in R^{N}. Consider the problem

\{ \begin{matrix}−div (a_{i}(x)∇u_{i}) + ∑_{j=1}^{d} b_{ij}(x)u_{j} = f_{i}, in Ω, for i = 1, ⋯, d \\ u_{i} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂u_{i}}{∂ν} = g_{i}, on Γ_{N} , for i = 1, ⋯, d . \end{matrix} (1.1)

Here, ν denotes the unit normal outward vector, d ≥ 1 is an integer, and a_{i}, b_{ij} ∈ L^{∞}(Ω), for 1 ≤ i, j ≤ d. We point out that we make no symmetry assumption on the coefficients b_{ij}. We assume that {Γ_{D}, Γ_{N} } realize an open partition of the boundary ∂Ω, i.e., Γ_{D} ∩ Γ_{N} = ∅ and \overline{Γ_{D}} ∪ \overline{Γ_{N}} = ∂Ω. Moreover, we suppose that Γ_{D} has nonzero (N − 1)-Lebesgue measure, namely, meas_{N−1} (Γ_{D}) > 0. We also assume that the elliptic operator is not degenerate, i.e., there exists α > 0 such that

a_{i}(x) ≥ α  for a.e. x ∈ Ω and any i = 1, ⋯, d.  (1.2)

Set E^{1,p}(Ω) := \{u ∈ W^{1,p}(Ω); u = 0 on Γ_{D}\} and E := ⋂_{1≤p< \frac{N}{N−1}} (E^{1,p}(Ω))^{d}. We denote throughout by ∥ · ∥_{p} (resp. ∥ · ∥_{p,d}) the norm in the space L^{p}(Ω)

Accepted for publication: April 2001.

AMS Subject Classifications: ????????????

(resp. (L^{p}(Ω))^{d}). We also denote by | · |_{p} (resp. | · |_{p,d}) the norm in the space E^{1,p}(Ω) (resp. (E^{1,p}(Ω))^{d}).

We suppose that the associated bilinear form is coercive, namely there exists β > 0 such that, for every u = (u_{1}, ⋯, u_{d}) ∈ (E^{1,2}(Ω))^{d},

∫_{Ω} ( ∑_{i=1}^{d} a_{i}|∇u_{i}|^{2} + ∑_{i,j} b_{ij}u_{i}u_{j})dx ≥ β |u|_{2,d}^{2} .  (1.3)

We assume that f_{i} and g_{i} are bounded measures (finite Radon measures) on Ω, respectively Γ_{N}, that is, f_{i} ∈ M(Ω) and g_{i} ∈ M(Γ_{N}), for any i = 1, ⋯, d,.

If Γ_{N} = ∅ and f ∈ (M(Ω))^{d} Stampacchia introduced in [12] a duality method combined with a C^{0,α}-regularity argument. The purpose of this paper is to study the general elliptic system (1.1) which involves mixed boundary conditions. As in Stampacchia’s framework, our arguments are restricted to a linear setting. The proof relies on the crucial observation (see Lemma 1) that L^{1} boundedness implies the boundedness in the space E. As we shall observe in Lemma 1, this becomes true because the L^{p^{∗}}-boundedness implies E^{1,p}-boundedness, for any p < \frac{N}{N−1}.

Definition 1. A function u = (u_{1}, ⋯, u_{d}) ∈ E is said to be a solution of the problem (1.1) provided that

∫_{Ω} a_{i}∇u_{i} · ∇φ + ∫_{Ω} ( ∑_{j=1}^{d} b_{ij}u_{j})φ = ∫_{Ω} f_{i}φ + ∫_{Γ_{N}} a_{i}g_{i}φ,

for any i = 1, ⋯, d and for every φ ∈ C^{1}(\overline{Ω}) with φ = 0 on Γ_{D}.

Theorem 1. Assume that hypotheses (1.2) and (1.3) are fulfilled. Then, for any bounded measures f ∈ (M(Ω))^{d} and g ∈ (M(Γ_{N}))^{d} the problem (1.1) has at least one solution.

We point out that the celebrated non-uniqueness example constructed in Serrin [11] shows that Problem (1) may have several solutions (see also Prignet [10], p. 329).

2. Proof of Theorem 1

Let f^{n} = (f_{i}^{n})_{1≤i≤d} ∈ (L^{2}(Ω))^{d} and g^{n} = (g_{i}^{n})_{1≤i≤d} ∈ (L^{2}(Γ_{N}))^{d} be such that

f^{n} ⇀ f  weakly in the sense of measures in (M(Ω))^{d}  (2.1)

linear elliptic systems involving finite radon measures 3

g^{n} ⇀ g  weakly in the sense of measures in (M(Γ_{N}))^{d}  (2.2)
∥f^{n}∥_{1,d} ≤ ∥f∥_{(M(Ω))}_{d}  (2.3)
∥g^{n}∥_{1,d} ≤ ∥g∥_{(M(Γ_{N}))}_{d} .  (2.4)

Consider the problem

\{ \begin{matrix}−div (a_{i}∇u_{i}^{n}) + ∑_{j=1}^{d} b_{ij}u_{j}^{n} = f_{i}^{n}, in Ω, for i = 1, ⋯, d \\ u_{i}^{n} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂u_{i}^{n}}{∂ν} = g_{i}^{n}, on Γ_{N} , for i = 1, ⋯, d . \end{matrix}  (2.5)

Using the coercivity condition (1.3) and applying the Lax-Milgram Lemma we find that problem (2.5) has a unique solution u^{n} ∈ (E^{1,2}(Ω))^{d}. Proposition 1. The sequence (u^{n})_{n} is bounded in (L^{1}(Ω))^{d}. Proof of Proposition 1. We argue by contradiction and assume that ∥u^{n}∥_{1,d} → ∞. Set v_{i}^{n} = \frac{u_{i}^{n}}{∥u^{n}∥_{1,d}}, for every 1 ≤ i ≤ d and n ≥ 1. We observe that v^{n} ∈ (E^{1,2}(Ω))^{d}, ∥v^{n}∥_{1,d} = 1 and

\{ \begin{matrix}−div (a_{i}∇v_{i}^{n}) + ∑_{j=1}^{d} b_{ij}v_{j}^{n} = \frac{f_{i}^{n}}{∥u^{n}∥_{1,d}} , in Ω, for i = 1, ⋯, d \\ v_{i}^{n} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂v_{i}^{n}}{∂ν} = \frac{g_{i}^{n}}{∥u^{n}∥_{1,d}} , on Γ_{N} , for i = 1, ⋯, d . \end{matrix} (2.6)

Lemma 1. The sequence (v^{n}) is bounded in the space E. Proof of Lemma 1. Taking into account (2.3), (2.4) and the assumption ∥u^{n}∥_{1,d} → ∞ we obtain that the L^{2}-sequences r_{i}^{n} = \frac{f_{i}^{n}}{∥u^{n}∥_{1,d}} and s_{i}^{n} = \frac{g_{i}^{n}}{∥u^{n}∥_{1,d}} converge to 0 in L^{1}(Ω), respectively in L^{1}(Γ_{N}). Set

M = max_{i,j}{∥a_{i}∥_{L^{∞}(Ω)}, ∥b_{ij}∥_{L^{∞}(Ω)}}.

Fix p > 1 such that p < \frac{N}{N−1}. Set

w_{i}^{n} = [(1 + |v_{i}^{n}|)^{(Np−N−p)/(N−p)} − 1] sgn v_{i}^{n} .

By Proposition IX.5 in [4] it follows that w_{i}^{n} ∈ H_{0}^{1}(Ω). Multiplying by w_{i}^{n} in (2.6) and integrating by parts we find

− ∫_{Γ_{N}} a_{i}s_{i}^{n}w_{i}^{n} − \frac{N − (N − 1)p}{N − p} ∫_{Ω} a_{i}(1 + |v_{i}^{n}|)^{−N(2−p)/(N−p)} |∇v_{i}^{n}|^{2}

+ ∫_{Ω} ∑_{j=1}^{d} b_{ij}v_{j}^{n}w_{i}^{n} = ∫_{Ω} r_{i}^{n}w_{i}^{n} .

Thus, by (1.2) and the fact that |w_{i}^{n}| ≤ 1 we deduce that

α \frac{N − (N − 1)p}{N − p} ∫_{Ω}(1 + |v_{i}^{n}|)^{−N(2−p)/(N−p)} |∇v_{i}^{n}|^{2}  (2.7)
≤ M ∥s_{i}^{n}∥_{L^{1}(Γ_{N})} + ∥r_{i}^{n}∥_{1} + M ∥v^{n}∥_{1,d} .

Therefore,

∫_{Ω} \frac{|∇v_{i}^{n}|^{2}}{(1 + |v_{i}^{n}|)^{N(2−p)/(N−p)}} ≤ C_{1} .  (2.8)

On the other hand, by Sobolev inclusions and Hölder’s inequality,

||v_{i}^{n}||_{p^{∗}}^{p} ≤ C ∫_{Ω} |∇v_{i}^{n}|^{p}  (2.9)

≤ C( ∫_{Ω} \frac{|∇v_{i}^{n}|^{2}}{(1 + |v_{i}^{n}|)^{N(2−p)/(N−p)}} )^{p/2}( ∫_{Ω}(1 + |v_{i}^{n}|)^{\frac{Np}{N−p}} )^{(2−p)/2} ,

where C depends only on p. Relations (2.8) and (2.9) yield

||v_{i}^{n}||_{p^{∗}} ≤ C ||∇v_{i}^{n}||_{p} ≤ C_{2} ∥1 + |v_{i}^{n}|∥p^{∗}^{\frac{N(2−p)}{2(N−p)}} ≤ C_{3} (1 + ||v_{i}^{n}||p^{∗}^{\frac{N(2−p)}{2(N−p)}} ). (2.10)

We distinguish two different situations: Case 1: N ≥ 3. This implies 1 > \frac{N(2−p)}{2(N−p)}. Hence, by (2.10), the sequence (v^{n}) is bounded in (L^{p^{∗}}(Ω))^{d}, so in (L^{p}(Ω))^{d}. Returning now to (2.10) we have

∫_{Ω} |∇v_{i}^{n}|^{p} ≤ C

which shows that (v^{n}) is bounded in (E^{1,p}(Ω))^{d}, for any p < \frac{N}{N−1}. Case 2: N = 2. This implies 1 = \frac{N(2−p)}{2(N−p)}, so the above argument does not work. However, it is possible to repeat it, but for a modified sequence v^{n}. Indeed, we observe that if the constant C_{3} appearing in (2.10) is less than 1, then the boundedness of (v^{n}) in (E^{1,p}(Ω))^{d} follows with the same argument. But C_{3} depends only on C_{1}, so on the value of

M ∥s_{i}^{n}∥_{L^{1}(Γ_{N})} + ∥r_{i}^{n}∥_{1} + M ∥v^{n}∥_{1,d}.

But (r_{i}^{n}) and (s_{i}^{n}) converge to 0 in L^{1}(Ω), respectively in L^{1}(Γ_{N}). Thus, in order to get C_{3} < 1, it is sufficient to define v_{i}^{n} by v_{i}^{n} = ε \frac{u_{i}^{n}}{||u_{i}^{n}||_{1,d}}, for ε > 0 small enough. This choice is possible due to the linearity of the system (2.5). □

The key fact in the proof of the above result is the boundedness of (v^{n}) in (L^{1}(Ω))^{d} combined with the linearity of the problem (2.6). Proof of Proposition 1 continued (case N ≤ 3). Let V^{n} = (V_{1}^{n}, ⋯, V_{d}^{n}) ∈ (E^{1,2}(Ω))^{d} be the unique solution of the coercive problem

\{ \begin{matrix}−div (a_{i}∇V_{i}^{n}) + ∑_{j=1}^{d} b_{ij}v_{j}^{n} = 0, in Ω, for i = 1, ⋯, d \\ V_{i}^{n} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂V_{i}^{n}}{∂ν} = 0, on Γ_{N} , for i = 1, ⋯, d . \end{matrix}  (2.11)

It follows by Lemma 1 that the sequence (v^{n})_{n} is bounded in (L^{p^{∗}}(Ω))^{d}, for any p < \frac{N}{N−1}. Our hypothesis N ≤ 3 implies p^{∗} ≥ 2, provided that \frac{2N}{N+2} ≤ p < \frac{N}{N−1}. Hence, the sequence (v^{n})_{n} is bounded in (L^{2}(Ω))^{d}. After multiplication in (2.11) by V_{i}^{n} and integration we find

∫_{Ω} |∇V_{i}^{n}|^{2} ≤ α^{−1} ∑_{j=1}^{d} ∫_{Ω} |b_{ij}v_{j}^{n}V_{i}^{n}| ≤ α^{−1}M ∑_{j=1}^{d} ∥v_{j}^{n}∥_{2} · ∥V_{i}^{n}∥_{2} ≤ C|V^{n}|_{2,d}.

(2.12)

It follows that (V^{n})_{n} is bounded in (E^{1,2}(Ω))^{d}. On the other hand, by (2.6) and (2.11),

\{ \begin{matrix}−div (a_{i}∇(v_{i}^{n} − V_{i}^{n})) = \frac{f_{i}^{n}}{∥u^{n}∥_{1,d}} , in Ω, for i = 1, ⋯, d \\ v_{i}^{n} − V_{i}^{n} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂(v_{i}^{n} − V_{i}^{n})}{∂ν} = \frac{g_{i}^{n}}{∥u^{n}∥_{1,d}} , on Γ_{N} , for i = 1, ⋯, d . \end{matrix}  (2.13)

Observing that the sequence (v_{n}−V_{n})_{n} is bounded in (L^{1}(Ω))^{d} and arguing as in the proof of Lemma 1, we deduce that (v_{n}−V_{n})_{n} is bounded in (E^{1,p}(Ω))^{d}, for any p < \frac{N}{N−1}. So, up to a subsequence, we can assume that

v^{n} − V^{n} ⇀ 0  weakly in (E^{1,p}(Ω))^{d}, ∀p < \frac{N}{N−1}.  (2.14)

But, by Lemma 1 and passing again at a subsequence,

v^{n} ⇀ v  weakly in (E^{1,p}(Ω))^{d}, ∀p < \frac{N}{N−1}.  (2.15)

Hence, by (2.14) and (2.15),

V^{n} ⇀ v  weakly in (E^{1,p}(Ω))^{d}, ∀p < \frac{N}{N−1}.  (2.16)

But (V^{n})_{n} is bounded in (E^{1,2}(Ω))^{d}, so v ∈ (E^{1,2}(Ω))^{d}. Taking into account (2.11) we obtain that the same convergence holds in (E^{1,2}(Ω))^{d} and v ∈ (E^{1,2}(Ω))^{d}. By (2.15) and (2.16) we deduce that we can pass at the limit in (2.11) and we find

\{ \begin{matrix}−div (a_{i}∇v_{i}) + ∑_{j=1}^{d} b_{ij}v_{j} = 0, in Ω, for i = 1, ⋯, d \\ v_{i} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂v_{i}}{∂ν} = 0, on Γ_{N} , for i = 1, ⋯, d . \end{matrix}  (2.17)

By the uniqueness of the solution in (E^{1,2}(Ω))^{d} we conclude that v = 0. Consequently, (v^{n}) converges weakly to 0 in E which implies, by Rellich’s theorem that we can assume v_{n} → 0 strongly in (L^{1}(Ω))^{d} which contradicts ∥v^{n}∥_{1,d} = 1. □ Proof of Theorem 1 continued. We are now in position to conclude the proof of Theorem 1 in the case N ≤ 3. This time we argue as in the proof of Lemma 1 but with u^{n} instead of v^{n}. Indeed, since (u^{n}) ⊂ (E^{1,2}(Ω))^{d} is bounded in (L^{1}(Ω))^{d} we may repeat the same arguments as in the proof of Lemma 1 to show that (u^{n}) is bounded in E. In particular, this implies that, passing eventually at a subsequence, there exists u ∈ E such that

u^{n} ⇀ u weakly in (E^{1,p}(Ω))^{d}, ∀p < \frac{N}{N−1}.

Hence, u is solution to the problem (1.1).

In the case N ≥ 4 we shall employ several times the above arguments. For this aim we define the sequence V_{(k)}^{n} by V_{(1)}^{n} = V^{n} and, for any k ≥ 2, let V_{(k)}^{n} = (V_{1,k}^{n}, ⋯, V_{d,k}^{n}) ∈ (E^{1,2}(Ω))^{d} be the unique solution of the problem

\{ \begin{matrix}−div (a_{i}∇V_{i,k}^{n}) + ∑_{j=1}^{d} b_{ij}V_{i,k−1}^{n} = 0, in Ω, for i = 1, ⋯, d \\ V_{i,k}^{n} = 0, on Γ_{D}, for i = 1, ⋯, d \\ \frac{∂V_{i,k}^{n}}{∂ν} = 0, on Γ_{N} , for i = 1, ⋯, d . \end{matrix} (2.18)

Fix 1 ≤ p < \frac{N}{N−1}.

Lemma 2. The sequence (V_{(1)}^{n})_{n} is bounded in (E^{1,Np/(N−p)}(Ω))^{d}.

Proof of Lemma 2. We repeat the argument applied in the proof of Lemma 1, but for V_{(1)}^{n} instead of v^{n}. We already know that (v^{n}) is bounded in (L^{p}(Ω))^{d}. Multiplying in (2.11) by

w_{i}^{n} = [(1 + |V_{i}^{n}|)^{(p−1)N/(N−2p)} − 1] sgn V_{i}^{n} ∈ H_{0}^{1}(Ω)

we find

\frac{(p − 1)N}{N − 2p} α ∫_{Ω} \frac{|∇V_{i}^{n}|^{2}}{(1 + |V_{i}^{n}|)^{\frac{2N−p(N+2)}{N−2p}}}

≤ \frac{(p − 1)N}{N − 2p} ∫_{Ω} a_{i} \frac{|∇V_{i}^{n}|^{2}}{(1 + |V_{i}^{n}|)^{\frac{2N−p(N+2)}{N−2p}}} = − ∫_{Ω}(∑_{j=1}^{d} b_{ij}v_{j}^{n})w_{i}^{n}

≤ M||v^{n}||_{1,d} + M ||v^{n}||_{p,d}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{(p−1)N}{N−2p} · \frac{p}{p−1}} )^{\frac{p−1}{p}} .

Hence,

∫_{Ω} \frac{|∇V_{i}^{n}|^{2}}{(1 + |V_{i}^{n}|)^{\frac{2N−p(N+2)}{N−2p}}} ≤ C_{1} + C_{2} (∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{p−1}{p}} .  (2.19)

We observe that the hypothesis N ≥ 4 implies p < \frac{N}{N−1} ≤ \frac{2N}{N+2}, so \frac{2(N−p)}{Np} > 1. Therefore, by Sobolev inclusions and Hölder’s inequality, we obtain

∫_{Ω} |∇V_{i}^{n}|^{\frac{Np}{N−p}} ≤ ( ∫_{Ω} \frac{|∇V_{i}^{n}|^{2}}{(1 + |V_{i}^{n}|)^{\frac{2N−p(N+2)}{N−2p}}} )^{\frac{Np}{2(N−p)}} ( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{2N−p(N+2)}{2(N−p)}} .

(2.20)

By (2.19) and (2.20) we find

( ∫_{Ω} |∇V_{i}^{n}|^{\frac{Np}{N−p}} )^{\frac{N−p}{Np}}  (2.21)

≤ [C_{1} + C_{2}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{p−1}{p}} ]^{1/2}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{2N−p(N+2)}{2Np}}

≤ C_{3}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{2N−p(N+2)}{2Np}} + C_{4}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{N−2p}{2Np}} .

Our choice p < \frac{N}{N−1} < \frac{N}{2} implies \frac{Np}{N−p} < N. Therefore, by Sobolev inclusions, the space E^{1,Np/(N−p)}(Ω) is continuously embedded in L^{Np/(N−2p)}(Ω), namely

( ∫_{Ω} |V_{i}^{n}|^{\frac{Np}{N−2p}} )^{\frac{N−2p}{Np}} ≤ C( ∫_{Ω} |∇V_{i}^{n}|)^{\frac{Np}{N−p}} )^{\frac{N−p}{Np}} .  (2.22)

Thus, by (2.21) and (2.22), we deduce that

( ∫_{Ω} |V_{i}^{n}|^{\frac{Np}{N−2p}} )^{\frac{N−2p}{Np}} ≤ C( ∫_{Ω} |∇V_{i}^{n}|^{\frac{Np}{N−p}} )^{\frac{N−p}{Np}}

≤ C_{5}( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{2N−p(N+2)}{2Np}} + C_{6} ( ∫_{Ω}(1 + |V_{i}^{n}|)^{\frac{Np}{N−2p}} )^{\frac{N−2p}{2Np}}

≤ C_{7} + C_{8}||V_{i}^{n}||\frac{2N−p(N+2)}{2(N−2p)} ; \frac{Np}{N−2p} + C_{9} ||V_{i}^{n}||1/2_{\frac{Np}{N−2p}} .

Observing that \frac{2N−p(N+2)}{2(N−2p)} < 1, the above relations yield

||V_{i}^{n}||_{\frac{Np}{N−2p}} ≤ C ||∇V_{i}^{n}||_{\frac{Np}{N−p}} ≤ C_{10} + C_{11} ||V_{i}^{n}||1/2_{\frac{Np}{N−2p}} .  (2.23)

This implies that (V_{i}^{n}) is bounded in L^{Np/(N−2p)}(Ω). Then, again by (2.23), the sequence (∇V_{i}^{n}) is bounded in L^{Np/(N−p)}(Ω) which implies the boundedness of (V_{i}^{n}) in E^{1,Np/(N−p)}(Ω). □ Proof of Theorem 1 concluded. It follows by Lemma 2 that the sequence (V_{(1)}^{n}) is bounded in (L^{Np/(N−2p)}(Ω))^{d}. If \frac{Np}{N−2p} ≥ 2, then we get the boundedness of (V_{(1)}^{n}) in (L^{2}(Ω))^{d} and the proof is concluded with exactly the same arguments as in the case N ≤ 3, but for v^{n} replaced by V_{(1)}^{n}. The condition

\frac{Np}{N−2p} ≥ 2 holds true if p ≥ \frac{2N}{N+4}. Taking into account the restriction p < \frac{N}{N−1} we find either N = 4 or N = 5. If not, we will repeat the arguments done in the proof of Lemma 2. It is sufficient to point out that the proof of Lemma 2 is based on the observation that (V_{(1)}^{n}) is bounded in (E^{1,p^{∗}}(Ω))^{d}, provided that (v^{n}) is bounded in (L^{p}(Ω))^{d}. Now, with the same arguments, one can show that the boundedness of (V_{(1)}^{n}) in (L^{Np/(N−p)}(Ω))^{d} implies the bound-

edness of (V_{(2)}^{n}) in (E^{1,Np/(N−2p)}(Ω))^{d}, since \frac{Np}{N−2p} is the Sobolev conjugated

exponent of \frac{Np}{N−p}. This holds true provided that \frac{Np}{N−3p} ≥ 2 and p < \frac{N}{N−1}, namely for N = 6 or N = 7. For greater values of N the proof relies on the same principles. □

We remark that the solution obtained by approximation in the above proof is unique. Indeed, let f^{n,1}, f^{n,2} ∈ (L^{2}(Ω))^{d} and g^{n,1}, g^{n,2} ∈ (L^{2}(Γ_{N}))^{d} be such that conditions (2.1)-(2.4) are fulfilled. Denote by u^{n,1}, respectively u^{n,2} the corresponding (unique) solutions in (E^{1,2}(Ω))^{d} of the problem (2.5). Since the sequence (u^{n,1} − u^{n,2}) is bounded in (L^{1}(Ω))^{d}, it follows with the same arguments as in the above proof that

u^{n,1} − u^{n,2} ⇀ 0 weakly in (E^{1,p}(Ω))^{d}, ∀p < \frac{N}{N−1}

which implies the uniqueness of the solution obtained by approximation.

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