Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

Proposition 2.1

If a sequence converges weakly in W∘1,q(ν,Ω) , then it converges strongly in L¹(Ω).

Further, let for every k > 0 T_k : ℝ → ℝ be the function such that

By analogy with known results for nonweighted Sobolev spaces (see for instance [25]) we have: if u ∈ W∘1,q(ν,Ω) and k > 0, then T_k(u) ∈ W∘1,q(ν,Ω) and for every i ∈ {1,…,n}

We denote by T∘1,q(ν,Ω) the set of all functions u : Ω → ℝ such that for every k > 0, T_k(u) ∈ W∘1,q(ν,Ω) .

Clearly,

For every u : Ω → ℝ and for every x ∈Ω we set

Definition 2.2

Let u ∈ T∘1,q(ν,Ω) , and i ∈{1,…,n}. Then δ_iu : Ω → ℝ is the function such that for every x ∈ Ω δ_iu(x) = D_iT_k(u,x)(u)(x).

Definition 2.3

If u ∈ T∘1,q(ν,Ω) , then δ u : Ω → ℝⁿ is the mapping such that for every x ∈Ω and for every i ∈{1,…,n} (δ u(x))_i = δ_iu(x).

Now we give several propositions which will be used in the next sections.

Proposition 2.4

Let u ∈ T∘1,q(ν,Ω) . Then for every k > 0 we have D_i T_k(u) = δ_iu⋅ 1_{{|u| < k}} a.e. in Ω, i = 1,…,n.

If u ∈ W∘1,q(ν,Ω) , then for every i ∈{1,…,n} δ_iu = D_iu a.e. in Ω.

Proposition 2.5

Let u∈T∘1,q(ν,Ω)andw∈W∘1,q(ν,Ω)∩L∞(Ω).Thenu−w∈T∘1,q(ν,Ω), and for every i ∈{1,…,n} and for every k > 0 we have

Proposition 2.6

There exists a positive constant c₀ depending on n, q, and ∥ 1/ν_i∥_{L^{1/(qi − 1)}(Ω)}, i = 1,…,n, such that for every function u ∈ W∘1,q(ν,Ω)

Proposition 3.1

The following assertions hold:

1. if u,w ∈ W∘1,q(ν,Ω) and i ∈ {1,…,n}, then a_i(x,∇ u)D_iw ∈ L¹(Ω);

2. if u∈T∘1,q(ν,Ω),w∈W∘1,q(ν,Ω)∩L∞(Ω), k > 0 and i ∈ {1,…,n}, then a_i(x,δ u)D_i T_k(u − w) ∈ L¹(Ω).

Let F : Ω × ℝ → ℝ be a Carathéodory function. We consider the following Dirichlet problem:

Definition 3.2

An entropy solution of problem (7), (8) is a function u ∈ T∘1,q(ν,Ω) such that:

for every function w ∈ W∘1,q(ν,Ω) ∩ L^∞(Ω) and for every k ⩾ 1

Note that the left-hand integral in (10) is finite. It follows from assertion b) of Proposition 3.1. The right-hand integral in (10) is also finite. It follows from the boundedness of the function T_k and inclusion (9).

Theorem 4.1

Suppose the following conditions are satisfied:

1. for a.e. x ∈ Ω the function F(x,⋅) is nonincreasing on ℝ;

2. for any s ∈ ℝ the function F (., s) belongs to L¹(Ω).

Then there exists an entropy solution of the Dirichlet problem (7), (8).

Proof

According to the approach from [1], we will consider a sequence of the approximating problems for the equations with smooth right-hand sides. Then we will obtain special estimates of the solutions of these problems. Finally, we will pass to the limit. The proof is in 9 steps.

Step 1. We set f = F (⋅, 0). Let for every l ∈ ℕ, F_l : Ω × ℝ → ℝ be the function such that

By virtue of condition 1), we have:

Further, in view of condition 2), we have f ∈ L¹(Ω). Hence there exists {f_l} ⊂ C0∞(Ω) such that:

Using the inequalities (4)–(6), property (11), and well-known results on the solvability of the equations with monotone operators (see for instance [26]), we obtain: if l ∈ ℕ, then there exists the function u_l ∈ W∘1,q(ν,Ω) such that for every function w ∈ W∘1,q(ν,Ω)

It means that the function u_l ∈ W∘1,q(ν,Ω) is a generalized solution of the Dirichlet problem:

We denote by c_i, i = 3,4,…, the positive constants depending only on n, q, c₁, c₂, ∥g₁∥_L¹(Ω), ∥g₂∥_L¹(Ω), ∥f∥_L¹(Ω), ∥F(⋅,−1)∥_L¹(Ω), ∥F (⋅, 1)∥_L¹(Ω), ∥1/ν_i∥_{L^{1/(qi − 1)}(Ω)}, i = 1,…,n, and meas Ω.

Let us show that for every k ⩾ 1 and l ∈ ℕ the following inequalities hold:

In fact, let k ⩾ 1 and l ∈ ℕ. As u_l ∈ W∘1,q(ν,Ω) we have T_k(u_l) ∈ W∘1,q(ν,Ω) . In view of (14) and (13) we obtain

Using (2) and (5) in the left-hand side of this inequality, we get

Assertion (11) and properties of the function T_k imply that

The estimate (15) follows from (18) and (17). Finally, the inequality (16) follows from (19) and (17).

Step 2. Now we show that for every k ⩾ 1 and l ∈ ℕ

In fact, let k ⩾ 1 and l ∈ ℕ. We have |T_k(u_l)| = k on {|u_l| ⩾ k}; then

Using Proposition 2.6, (2) and (15), we obtain

The inequality (20) follows from the latter estimate and (22).

Next, we fix i ∈ {1,…,n}, and set

We have

From (20) it follows that

Moreover, in view of the set’s G definition and (15) we get

The inequality (21) follows from the latter estimate and (23), (24).

Step 3. Assertions (2) and (15) imply that for every k ⩾ 1 the sequence {T_k(u_l)} is bounded in W∘1,q(ν,Ω) . As the space W∘1,q(ν,Ω) is reflexive, then there exist an increasing sequence {l_h} ⊂ ℕ, and sequence {z_k} ⊂ W∘1,q(ν,Ω) such that for every k ∈ ℕ we have a weak convergence T_k(u_lh) → z_k in W∘1,q(ν,Ω) . Without loss of generality it can be assumed that

Step 4. Let us show that the sequence {u_l} is fundamental on measure.

Indeed, let k ⩾ 1, l,j ∈ ℕ. We fix t > 0, and set G′ = {|u_l| < k, |u_j| < k,|u_l − u_j| ⩾ t}. It is clear that

As t ⩽ |T_k(u_l) − T_k(u_j)| on G′, we obtain

This inequality, (20) and (26) imply that for every k ⩾ 1, and l,j ∈ ℕ

Let ε > 0. We fix k ∈ ℕ such that

Taking into account (25) and Proposition 2.1, we infer a strong convergence T_k(u_l) → z_k in L¹(Ω). Then there exists N ∈ ℕ such that for every l,j ∈ ℕ, l,j ⩾ N

From this inequality, (27), and (28) we deduce that for every l,j ∈ ℕ, l,j ⩾ N

This means that the sequence {u_l} is fundamental on measure.

Step 5. Now we show that for every i ∈ {1,…,n} the sequence {νi1/qiDiul} is fundamental on measure.

For every t > 0 and l,j ∈ ℕ we put

Besides, for every t > 0, h,k ⩾ 1, and l,j ∈ ℕ we set

Using (21), we establish that for every t > 0, h ⩾ n, k ⩾ 1, and l,j ∈ ℕ

Further, we get one estimate for some integrals over E_t,h,k(l, j). So we introduce now auxiliary functions and sets.

Let for every x ∈ Ω Φ_x : ℝⁿ × ℝⁿ → ℝ be a function such that for every pair (ξ,ξ′) ∈ ℝⁿ × ℝⁿ

Recall that a_i,i = 1,…,n, are Carathéodory functions, and inequality (6) holds for almost every x ∈ Ω and every ξ,ξ′ ∈ ℝⁿ, ξ ≠ ξ′. Then there exists a set E ⊂ Ω, meas E = 0, such that:

1. for every x ∈ Ω ∖ E the function Φ_x is continuous on ℝⁿ × ℝⁿ;

2. for every x ∈ Ω ∖ E and ξ,ξ′ ∈ ℝ,ⁿξ ≠ ξ′, we have Φ_x(ξ,ξ′) > 0.

Put for every t > 0, h > t, and x ∈ Ω

As ν_i > 0 a.e. in Ω,i = 1,n, then there exists a set E~ ⊂ Ω, meas E~ = 0, such that the set G_t,h(x) is nonempty for every t > 0,h > t, and x ∈ Ω ∖ E~ .

Let for every t > 0 and h > t μ_t,h : Ω → ℝ be a function such that

For every t > 0 and h > t we have μ_t,h > 0 a.e. in Ω, and μ_t,h ∈ L¹(Ω).

Let t > 0, h ⩾ t + 1, k ⩾ 1, and l,j ∈ ℕ. We fix x ∈ E_t,h,k(l, j) ∖ (E ∪ E~ ), and set ξ = ∇ u_l(x), ξ′ = ∇ u_j(x).

As (ξ; ξ′) ∈ G_t,h(x), then μ_t,h(x) ⩽ Φ_x(ξ,ξ′). This inequality and function’s Φ_x definition imply that

Then, taking into account (6) and (2), we obtain

In view of (14) we have

From these equalities and (30) it follows that

Using (16) and conditions 1), 2), we find that for every l,j ∈ ℕ

From the latter estimate and (31) we deduce that for every t > 0,h ⩾ t + 1,k ⩾ 1, and l,j ∈ ℕ the following inequality holds:

The sequence {u_l} is fundamental on measure. Then there exists an increasing sequence {n_k} ⊂ ℕ such that for every k ∈ ℕ and l,j ∈ ℕ, l,j ⩾ n_k, we have

Let t > 0 and ε > 0. We fix h ⩾ t + n such that

Put for every k ∈ ℕ

Let us show that α_k → 0. Assume the converse. Then there exist τ > 0, an increasing sequence {k_s} ⊂ ℕ, and sequences {l_s},{j_s} ⊂ ℕ such that for every s ∈ ℕ we have l_s,j_s ⩾ n_ks and

Let G_s = E_t,h,ks(l_s,j_s), s ∈ ℕ.

In view of (32) and (13) for every s ∈ ℕ we get

It follows that

From this assertion, taking into account μ_t,h ∈ L¹(Ω) and μ_t,h > 0 a.e. in Ω, we infer that meas G_s → 0. This fact is in contradiction to (35). Hence, we conclude that α_k → 0.

Finally, we fix k ∈ ℕ such that the inequalities hold:

Let l,j ∈ ℕ,l,j ⩾ n_k. From (29), (33), (34) and (36) it follows that N_t(l,j) ⩽ ε.

This means that for every i ∈ {1,…,n} the sequence {νi1/qiDiul} is fundamental on measure.

Step 6. From results of the Steps 4 and 5, and F. Riesz’s theorem we get the following facts: there exist measurable functions u : Ω → ℝ and v⁽ⁱ⁾ : Ω → ℝ,i = 1,…,n, such that the sequence {u_l} converges to u on measure, and for every i ∈ {1,…,n} the sequence {νi1/qiDiul} converges to v⁽ⁱ⁾ on measure. As is generally known, we can extract subsequences converging almost everywhere in Ω to the corresponding functions. We may assume without loss of generality that

From (37), (25) and Proposition 2.1 we deduce that for every k ∈ ℕ

Let us show that u ∈ T∘1,q (ν, Ω). Indeed, let k > 0. Take h ∈ ℕ, h > k. In view of (39) we have T_h(u)∈ W∘1,q (ν, Ω). Hence, by inclusion (3) we obtain T_k(T_h(u)) ∈ W∘1,q (ν, Ω). This fact and the equality T_k(u) = T_k(T_h(u)) imply that T_k(u)∈ W∘1,q (ν, Ω). Therefore, u ∈ T∘1,q (ν, Ω).

Step 7. Now we show that

In fact, let i ∈ {1, …, n}. In view of (37) there exists a set E′ ⊂ Ω, meas E′ = 0, such that

and in view of (38) there exists a set E″ ⊂ Ω, meas E″ = 0, such that

Fix k ∈ ℕ. By (2) we have: if l ∈ ℕ, then there exists a set E^(l) ⊂ Ω, meas E^(l) = 0, such that

We denote by Ê a union of sets E′, E″ and E^(l), l ∈ N. Clearly, meas Ê = 0. Let x ∈ {|u| < k} \ Ê. In view of (42) there exists l₀ ∈ ℕ such that for every l ∈ ℕ, l ⩾ l₀, we have |u_l(x)| < k. Let l ∈ ℕ, l ⩾ l₀. Then x ∈ {|u_l| < k}\ E^(l) and according to (44) we get νi1/qi(x)DiTk(ul)(x)=νi1/qi(x)Diul(x). From this equality and (43) we deduce that νi1/qiDiTk(ul)(x)→v(i)(x). Thus,

Besides, in view of (2) and (15) for every l ∈ ℕ

Using Fatou’s lemma, from (45) and (46) we infer that the function |v⁽ⁱ⁾|^q_i is summable in {|u| < k}.

Further, let φ : Ω → ℝ be a measurable function such that |φ| ⩽ 1 in Ω, and let ε > 0. As the function |v⁽ⁱ⁾| is summable on {|u| < k}, then there exists ε₁ ∈ (0, ε) such that for every measurable set G ⊂ {|u| < k}, meas G ⩽ ε₁, we have

Moreover, in view of (45) and Egorov’s theorem, there exists a measurable set Ω′ ⊂ {|u| < k} such that

From (47) and (48) we infer that

and from (49) we deduce that there exists l₁ ∈ ℕ such that for every l ∈ ℕ, l ⩾ l₁,

Let l ∈ ℕ, l ⩾ l₁. Using (50), (51), Hölder’s inequality, (48), and (46), we get

Since ε is an arbitrary constant, from the latter estimate it follows that

On the other hand, let F : W∘1,q (ν, Ω) → ℝ be a functional such that for every function v ∈ W∘1,q (ν, Ω)

It is easy to see that F ∈ ( W∘1,q (ν, Ω))^*. Hence, by virtue of (40), we have

This fact and functional’s F definition imply that

From (52) and (53) we deduce that

In turn, from this equality and Proposition 2.1 we infer that

Since k ∈ ℕ is an arbitrary number, from the latter assertion it follows that

Taking into account that ν_i > 0 a.e. in Ω, from (38) and (54) we obtain that D_i u_l →δ_iu a.e. in Ω. Thus, (41) is proved.

Assertion (41) along with the fact that a_i, i = 1, …, n, are Carathéodory functions implies that

Step 8. Let us show that the following assertions are fulfilled: