On Carlson′s Type Removability Test for the Degenerate Quasilinear Elliptic Equations

Definition 1.1. A function $$ is called a solution of (1.1) if the integral identity

is fulfilled for any test function $$.

Definition 1.2. Let E ⊂ ⊂D be a compact subset of the bounded domain D ⊂ ℝⁿ. One will say that the set E is removable for the class of C^α(D) of solutions of (1.1) if any solution of (1.1) in D∖E from the space $$ belongs to the space $$ throughout the domain D and is extendable inside the compactum E as solution.

Definition 1.3. Let E ⊂ ℝⁿ be a bounded closed subset, h : ℝ → (0, ∞) a continuous function, and h(0) = 0,   μ some Radon measure. A finite system of balls $$, the radii of which do not exceed δ > 0, covers the set E, that is, E ⊂ ⋃_νB_ν. Assume that $$, where the lower bound is taken with respect to all the mentioned balls.

Assume that

In the case μ = dx,   h(t) = t^{−p+(p−1)α}, the number $$ is a Hausdorff measure of order n − p + (p − 1)α of the set E. We will sometimes denote it by mes_{n−p+(p−1)α}(E). Denote by $$ the term $$ for h(t) = t^{−p+(p−1)α}, dμ = ω dx.

Lemma 1.4 (see [12].)Assume that a function u ∈ L¹(D) satisfies the inequality

for any ball B(x, r) ⊂ D, where α ∈ (0,1). Then, u ∈ C^α(D) and for any D^′ ⊂ ⊂D the estimate where C = C(n, α, D^′, D), is satisfied.

Lemma 1.5. Let ϕ(t), ω(t) be nonnegative nondecreasing functions on [0, R]. Assume that s > 0 is such that

for all r > 0 and 0 < λ < 1. Suppose that for any 0 < ρ < r < R, with A, B, α, β nonnegative constants and β < α. Then, for any γ ∈ (β, α), there exists a constant ε₀ = ε₀(A, α, β, γ, s) such that if ε < ε₀, then one has, far all 0 < ρ < r < R, where c = c(β, α, A, s, γ) > 0.

Proof. For τ ∈ (0,1) and r < R, we have

Choose τ < 1 in such a way that 2Aτ^α = τ^γ and assume ε₀τ^−α−s ≤ 1. Then, we get, for every r < R, and therefore, for all integers k > 0, Choosing k such that τ^k+2r < ρ ≤ τ^k+1r, the last inequality gives This proves Lemma 1.5.

Lemma 1.6. Let 1 ≤ p ≤ 2. Let x, y ∈ ℝⁿ be arbitrary points. Then, the estimate

is valid.

Proof. Let us introduce the vector function

acting from [0,1] into ℝⁿ. Applying the methods of differential calculus for the vector function, we obtain

The set of points {l(θ) ∈ ℝⁿ : l(θ) = θx + (1 − θ)y;   0 ≤ θ ≤ 1} in ℝⁿ forms a segment of the straight line that connects the point x with the point y. We denote this segment by [x, y]. Let |dl| be a length element of this segment. It is obvious that |dl| = |x − y|dθ. Therefore, for the above integral expression, we have the estimate

To proceed with the estimation of this expression, we introduce into consideration the triangle, the base of which is the segment [x, y] and the vertex lies at the point 0. Now, the integration in the preceding estimate will be carried out with respect to the base of the triangle. It is not difficult to verify that the above integral expression takes a maximal value when the point 0 lies in the middle of the segment [x, y], which means that for it we have the estimate

To show that this is true, let us choose a new coordinate system, where the x_n-axis is directed along the segment [x, y]. Let (u₁, u₂, …, u_n) be the coordinates of the the point 0 in the new coordinate system. Then, the preceding integral expression is equal to

Theorem 1.7. Let D ⊂ ℝⁿ be a bounded domain, E ⊂ ⊂D be a compact subset. Let 2 < p < ∞ and ω be a positive, locally integrable function satisfying condition (1.3) or 1 < p < 2 and let any of the following conditions be fulfilled for the function ω:

• (1)

  the function ω is integrable along any finite smooth n − 1-dimensional surface and condition (1.9′) is fulfilled for it;

• (2)

  for any x ∈ D and sufficiently small ρ > 0, the condition $$ is fulfilled for some s > n − p + 1, where the constant C > 0 does not depend on x.

Then, for a compact set E to be removable in the class C^α(D), 0 < α ≤ 1 of solutions of (1.1) in D∖E, $$, it is sufficient that condition (1.15) be fulfilled.

Theorem 1.8. Let ω ∈ A_p, E ⊂ ⊂D be a compact subset of the domain D. Let 0 < α < κ be some number. In that case, if $$, then the set E is not removable in the class of $$ solutions of (1.1) which belong to C^α(D).

Corollary 1.9. Let 0 < α < κ, 2 ≤ p < ∞, ω ∈ A_p, or 1 < p < 2 and any of the following conditions be fulfilled:

• (1)

  the function ω satisfies condition (1.9′) and is integrable along any finite smooth n − 1-dimensional surface;

• (2)

  for any x ∈ D and sufficiently small ρ > 0, the condition $$ is fulfilled by some s > n − p + 1, where the constant C > 0 does not depend on x.

Then, for the compact set E to be removable for the class of $$ solutions of (1.1) belonging to C^α(D), it is necessary and sufficient that condition (1.15) be fulfilled.

Corollary 1.10. Let 0 < α ≤ 1, 1 < p < ∞. Then, for the compact set E to be removable in the class $$ of solutions of the equation

belonging to the class C^α(D) throughout the domain D, it is necessary and sufficient that condition (1.13) be fulfilled.

Lemma 2.1. Let D be a bounded domain. Let 2 ≤ p < ∞ and the function ω : ℝⁿ → [0, ∞] satisfy condition (1.3) or 1 < p < 2, and let any of the following conditions be fulfilled for the function ω:

• (1)

  condition (1.9′) is fulfilled and ω is integrable along any finite smooth n − 1-dimensional surface;

• (2)

  for any x ∈ D and sufficiently small ρ > 0, the condition $$, where the constant C > 0 does not depend on x, is fulfilled for some s > n − p + 1.

Assume that f : D → ℝ is a sufficiently smooth function (one can also assume the condition f(x) ∈ C^β(D), where β ≥ min {p^′, 1}). Then, for any ε > 0, there exist a finite number of balls {B_ν}, ν = 1,2, …, N, such that

Proof. We will follow the same reasoning as that used in proving Lemma  1  in [9] (see also [3], Lemma 2.1). The set O_f = {x ∈ D : ∇f(x) = 0} is divided into two parts $$; here, $$ is the set of points where ∇²f(x) ≠ 0, and $$ is the set of points where ∇²f(x) = 0. Let 1 < p < 2. Then, for the set $$, our reasoning is as follows. By virtue of the implicit function theorem, the set $$ lies on a countable quantity of smooth n − 1-dimensional surfaces {S_j};   j = 1,2, …. Let x ∈ S_j be a fixed point on the j-th surface. For sufficiently small r > 0, we have

where C₁ = sup _D |∇²f| and ω(S(x, r)) is integral omega over the n − 1 dimensional surface S(x, r). By virtue of Fubini’s formula, Let E be the set of points t ∈ (r_x, 2r_x), for which the following condition is fulfilled: Then whence for the one-dimensional Lebesgue measure of the set E we obtain the estimate μ₁(E) ≤ r_x/2. Hence, by virtue of the doubling condition, there exists a point t_x ∈ (r_x, 2r_x), for which Then, for sufficiently small t_j > 0, for any x ∈ S_j, it can be assumed that there exists a number ρ_x ∈ (t_j, 2t_j), for which Therefore, For the surface S_j, from the system of balls {B(x, ρ_x); x ∈ S_j}, we can extract, by virtue of Besicovtich theorem [16], a subcovering {B(x_ν, ρ_ν);   x_ν ∈ S_j, ν ∈ ℕ} with finite intersections: Therefore and by construction, for x_ν ∈ S_j,   ν = 1,2, …, we have ρ_ν ∈ (t_j, 2t_j). Thus, Here, we have used the sufficient smallness of t_j, condition (1.9′), and the inequality $$. Choosing now t_j, $$, where ε > 0 is an arbitrary number, we obtain $$, whence, after summing the inequalities over all surfaces S_j,   j ∈ ℕ, we find

In the case of the second condition 1 < p < 2, using $$, we immediately pass from the inequality (2.8) to (2.10) as

Due to the latter inequality, an estimate analogous to (2.10) will have the form After choosing t_j sufficiently small and taking the condition s > n − p + 1 into account, we can make the right-hand part smaller than ε/2^j.

In the case p ≥ 2, the whole reasoning of [9] is applicable. Note that only instead of the inequality (2.10) we will have

where $$ is the t_j neighborhood of the surface S^j. After choosing a sufficiently small t_j, we can make the right-hand part of this inequality smaller than ε/2^j. This is possible because the n-dimensional Lebesgue measure of the surface S^j is equal to zero.

Now, it remains to obtain the covering for the set of points $$. Let 1 < p < 2. Let us decompose $$, where $$ is the set of points $$, for which ∇³f ≠ 0. Here, we repeat the reasoning for $$. As above, the set $$ is divided into two parts. In one part, we have ∇⁴f(x) ≠ 0, to which we apply the same reasoning as for $$. The second part of $$, where ∇⁴f(x) = 0, is again divided into two parts. At the k-th step, when k(p − 1) ≥ 1 and t > 0 is sufficiently small, this process yields the estimate

where η > 0 is arbitrary.

Note that, in the case p ≥ 2, the foregoing estimate gives the desired results immediately at the first step (k = 1).

Remark 2.2. It is not difficult to verify that under the assumptions of Lemma 2.1, instead of the condition of sufficient smoothness it sufficed to assume that f(x) ∈ C^β(D), where β ≥ min {p^′, 1}.

Lemma 2.3. Let 2 ≤ p < ∞ and the function ω : ℝⁿ → [0, ∞] satisfy condition (1.3) or 1 < p < 2 and let any of the following conditions be fulfilled for the function ω:

• (1)

  condition (1.9′) is fulfilled and ω is integrable along any finite smooth n − 1-dimensional surface;

• (2)

  for any x ∈ D and sufficiently small ρ > 0, the condition $$ is fulfilled by some s > n − p + 1, where the constant C > 0 does not depend on x.

Let D be some domain lying in the spherical layer $$ and having limit points on the surfaces of the spheres S(x₀, 2r) and S(x₀, r). Let $$ be the quadratic form, the coefficients of which are well defined and continuously differentiable in the domain D and for which the inequalities

are fulfilled for any x ∈ D,   η ∈ ℝⁿ for some λ ∈ (0,1). Assume that f : D → ℝ is a sufficiently smooth function.

Then, there exists a piecewise-smooth surface Σ, separating, in the domain D, the surfaces of the spheres S(x₀, r) and S(x₀, 2r) and being such that

where $$ is conormal derivative on Σ, n = (n₁, n₂, …, n_n) is unit orthogonal vector to the surface Σ, and the constant K depends on p,  λ, and the dimension n.

Proof. It suffices to consider the case r = 1. Indeed, after the change of variables x = ry, the function f : D → R transforms to the function $$, where $$. Also, $$. $$ lies in the spherical layer B(0,2)∖B(0,1). It suffices to show that $$. Indeed, let a sufficiently small element of the surface Σ satisfy the equation φ(x) = 0 in coordinates x. Then, after the change of variables, this equation takes the form $$, where $$. In other words, the normals of the surfaces Σ and $$ are related by $$. Therefore,

Applying these equalities, from the estimate we obtain (2.17).

Let us now prove (2.19). Following the notation and reasoning of [15] (see also [9]), we assume that ε = ω(D)(oscf)^p−1. For this ε, we find the corresponding balls Q₁, Q₂, …, Q_N of Lemma 2.1 and remove them from the domain D. Assume that $$ and intersect D^* with the closed layer (1 + 1/4) ≤ |x| ≤ (1 + 3/4). Denote this intersection by D^′. On the closed set D^′, we have ∇f ≠ 0. Let us choose some δ-neighborhood $$ with δ < 1/4 so small that in $$ we would have |∇f| > α > 0. We consider on $$ the system of equations

In $$, there are no stationary points of the system (2.20), and at every point $$ the direction of the field forms with the direction of the gradient an angle different from the straight angle. Let l(x) be the vector of the field at the point x. Then, using $$, we obtain From this inequality, it follows that in $$ there are no closed trajectories and all the trajectories have the uniformly bounded length.

Let some surface S be tangential, at each of its points, to the field direction. Then,

since the integrand is identically zero. We will use this fact in constructing the needed surface Σ. The base of Σ consists of ruled surfaces, while the generatrices are the trajectories of the system (2.20). Note that they will add nothing to the integral in which we are interested. These surfaces will have the form of fine tubes which will cover the entire D^′. Let us insert partitions into some of the tubes. The integral over these partitions will not any longer be equal to zero, but we can make it infinitesimal. The construction of tubes practically repeats that given in [15, pages 129–132].

Denote by E the intersection of D^′ with the sphere $$. Let N be the set of points x ∈ E, where the direction of the field of the system (2.20) is tangential to the sphere $$. At the points x ∈ N, we have ∂f/∂v = 0, where ∂/∂v is the derivative with respect to the conormal to the sphere $$. Cover N by a set G, open on the sphere $$ and being such that

Put E^′ = E∖G. At the points x ∈ E^′, the direction of the field is transversal to the sphere. Cover E′ on the sphere $$ by a finite number of uncovered domains with piecewise-smooth boundaries. We will call them cells. We will choose their diameters so small that at the points of the cells the field would be transversal to the sphere and the bundle of trajectories passing through each of the cells would diverge by δ/2n at most. The surface with trajectories lying inside the ball |x| < (1 + 3/4) and passing through the cell boundary will be called a tube. Thus, we obtain a finite number of tubes. We will call a tube a through tube if, without intersecting this tube, we can connect by a broken line a point of its corresponding cell with a point of the sphere $$ within the limits of the intersection of D′ with a spherical layer 1 + 1/4 − δ < |x| < 1 + 3/4. Such through tubes are denoted by T₁, T₂, …, T_s. If every through tube is partitioned, then the spheres $$ and $$ are separated in D by the set-theoretic sum of nonthrough tubes, partitions T₁, T₂, …, T_s, the spheres S₁, S₂, …, S_N, and the set G on the sphere $$.

Let us now take care to choose partitions in such a way that the integral ∫ω|∇f|^p−2|∂f/∂v|ds over them would have the value which we need. Denote by U_i the domain bounded by T_i. Choose any trajectory on this tube. Denote it by L_i. The length μ₁L_i of the curve L_i satisfies the inequality μ₁L_i > 1/2. Introduce, on L_i, the parameter l which is the length of the arc counted from $$. Denote by σ_i(l) the section U_i with a hypersurface which is orthogonal, at the point l, to the trajectory L_i. Let the diameter at the beginning of the tube be so small that $$. Then, the set H of points l ∈ L_i, where $$, satisfies the inequality μ₁L_i < 1/4. Thus, for E = L_i∖H, the inequality μ₁L_i > 1/4 is valid and

At the points of the curve L_i, the derivative ∂f/∂l preserves the sign and therefore Hence, using μ₁L_i > 1/4 and the mean value theorem, we see that there exists a point l₀ ∈ E such that $$. On the other hand, since, by virtue of (2.21), $$, we have $$. This together with (2.24) gives the estimate Let us now choose a cell diameter so small that This can be done since the derivatives ∂f/∂x_k,   k = 1,2, …, n, are uniformly continuous. Therefore, Denote by Σ the set-theoretic sum of all nonthrough tubes, all σ_i(l₀), all spheres S_i, and the set G on the sphere $$. Then, from Lemma 2.1 and (2.22)–(2.28), we obtain

Lemma 2.3 is proved.

Proof of Theorem 1.7 (Approximation). Let $$ and u ∈ C^α(D); let $$ be a solution of (1.1). Denote by u^j a mean value of the function u with smooth kernel ρ with finite support, $$. Then, it is obvious that u^(j) ∈ C^∞(D), j = 1,2, …. Moreover, u^j → u uniformly in any subdomain $$. Also, for any open set E′⊃E contained in G, u^(j) → u in the norm of the space $$ (see [2, 16]). Since, by condition (1.15), we have mes_nE = 0, it can be assumed that mes_nE^′ < η, where η > 0 is an arbitrary number.

Let ε > 0 be an arbitrary number. Cover the set E by a finite system of balls $$, $$ such that diam B_ν < δ,

Assume that the number δ = δ(ε, η) is so small that the set $$ lies in E^′.

For every ν, there exists, by virtue of Lemma 2.3 and inequality (1.31), a piecewise-smooth surface $$, separating the surfaces of the spheres ∂(2B_ν) and ∂(4B_ν), such that

Denote by $$ the interiority of the surface $$. Then, $$. Assume that $$. Let $$ for some ν. Then, for ν, the inequality (2.31) implies the estimate

It is obvious that the set $$ is a strictly interior subdomain of the domain D∖E. Thus, we have the identity

for any $$. From this, by virtue of the convergence $$ as j → ∞ and the fact that the $$ satisfies (2.16) by some λ > 0, we find This follows from the fact that the integrand is a system of equi-integrable functions: for any subset g ⊂ D∖Γ^′, we have as mes_ng → 0. Here and in the sequel, speaking in general, we denote by δ_j different sequences tending to zero as j → ∞.

Theorem 1.7 is proved.

Proof of Theorem 1.8. Let $$ for some compact set E ⊂ D. Let us use the recent results for a Frosman type lemma with measure [11, 17] and follow the reasoning of the original paper [3]. We come to the following conclusion. There exists a Radon measure μ with a support on the set E, such that μ(E) > 0 and for any ball B = B(x, r) we have

Let $$ be a solution of the equation

in the domain D = B(0, R), where B is a sufficiently large ball. The solution of (2.50) is understood in the sense as follows: the integral identity is fulfilled for any test function $$. Such a solution exists by virtue of $$. Let us show the latter inclusion.

By virtue of ω ∈ A_p for q > p and the fact that q is sufficiently close to p, inequality (2.49) implies for 0 < ρ < r:

where $$. This inequality defines the constant in the Adams inequality as Then, by virtue of (37′), we have Whence, by virtue of Hölder inequality, we obtain Taking into account inequalities (2.49), (37′′) and the A_p-condition, for any function $$, we have whence it follows that $$.

Let us, following ideas of [10], show that u(x) ∈ C^α(D). Let $$ be a solution of the equation

with the condition $$. Then, for it, we have the integral identity where v = u − h. The integrand in the left-hand part of (2.56) is positive. Therefore, for ρ < r/2, we have By virtue of Young’s inequality and conditions (1.2), (1.3), from (2.57), we find From (2.55), we obtain whence by virtue of Hölder inequality, we find For the first summand of (2.58), we have the following estimates. According to [14], there exists a positive number κ = κ(n, p, λ, C_p)∈(0,1) such that for the solution of (1.1) the inequality where C = C(n, p, C_p, λ, κ), is fulfilled for any r₁ < r₂. If we take into account the Caccioppoli type estimate (see [14]) then, by virtue of Moser’s inequality, we obtain From (2.60), we derive the estimate Indeed, where $$ is the lower bound of the function h in the ball B(x₀, r). Inequality (2.63) is proved.

Using the estimate (2.63) in (2.58), by virtue of (41′), we have for 0 < ρ < r/2

Now, let us derive an estimate for the last summand in (2.65). To this end, we use inequality (38′) to obtain

where for the constant we have the estimate By virtue of (2.65) and (2.67), we find whence, by virtue of Young’s inequality, we obtain Assuming that 0 < α < κ, from (2.69) and Lemma 1.5, we obtain the estimate This inequality implies whence, by virtue of the Poincaré inequality, we obtain where (u)_ρ is average of the function u with respect to the ball B(x₀, ρ).

By (2.72) and Campanato’s Lemma 1.4, we find u ∈ C^α.

Theorem 1.8 is proved.