Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-\Delta_p u=\mu$ with $p\in (1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in \big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22], respectively. In this paper, we improve the results in [22] and address the open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.

Here ffl E stands for the integral average over a measurable set E and is the truncated first-order Riesz potential.Later, the case when p ∈ 3n−2 2n−1 , 2 − 1 n was treated in [22], in which the authors obtained a pointwise gradient bound involving the Wolff potential under stronger assumptions on A and ω.Namely, under the conditions (1.2)-(1.4)and further assuming that where C = C(n, p, λ, α 0 , ω, γ) and is a truncated nonlinear Wolff potential.We recall that in general, the truncated Wolff potential is defined as Our first main result is stated as follows.
For the more singular case when p ∈ 1, 3n−2 2n−1 , which was open, we obtain the following Lipschitz estimate.
We also obtain a modulus of continuity estimate of ∇u in Theorem 4.3, which directly implies the following sufficient condition for the continuity of ∇u.
Recall the Lorentz space L n,1 is the collection of measurable functions f such that Theorem 1.4 has the following corollary.
A further, actually immediate, corollary of Theorem 1.4 concerns measures with certain density properties.
Another interesting consequence of Theorem 4.3 is the following gradient Hölder continuity result.
Corollary 1.7 (Gradient Hölder continuity via Riesz potential).Let p ∈ (1, 2) and u ∈ W 1,p loc (Ω) be a solution to (1.1).Then under the assumptions (1.2)-(1.5),there exists a constant α ∈ (0, 1) depending only on n, p, and λ, such that if ω(r) ≤ Cr β whenever r > 0 and I ρ 1 (|µ|)(x) ≤ Cρ β whenever B ρ (x) ⊂⊂ Ω, for some constants C > 0 and β ∈ (0, α), then u ∈ C 1,β loc (Ω).Remark 1.8.We should stress that the constant α in Corollary 1.7 is the natural Hölder exponent of the gradients of solutions to corresponding homogeneous equations with x-independent nonlinearities (cf.Lemma 2.2).Therefore, our result in Corollary 1.7 provides the best possible Hölder exponent for the gradient of the solution.The previous Corollary is an improvement of the gradient Hölder regularity result by Liebermann [18,Theorem 5.3], who proved u ∈ C 1,β1  loc for some for some β ∈ (0, 1).It is easily seen that the last condition implies We also obtain up-to-boundary gradient estimates for the p-Laplace equations with measure data in domains with C 1,Dini boundaries.Definition 1.9.Let Ω be a domain in R n .We say that Ω has C 1,Dini boundary if there exists a constant R 0 ∈ (0, 1] and a non-decreasing function ω 0 : [0, 1] → [0, 1] satisfying the Dini condition such that the following holds: for any x 0 = (x 01 , x ′ 0 ) ∈ ∂Ω, there exists a C 1,Dini function (i.e., C 1 function whose first derivatives are uniformly Dini continuous) χ : R n−1 → R and a coordinate system depending on x 0 such that sup and that in the new coordinate system, we have holds for any x ∈ Ω and R ∈ (0, 1].
As a corollary, we also obtain the global Lipschitz estimate when Ω is bounded.
Corollary 1.12.Let Ω ⊂ R n be a bounded domain.Under the assumptions of Theorem 1.11, there exists a constant A global modulus of continuity estimate is also established in Theorem 5.10 under the same conditions, which implies that corresponding up-to-boundary gradient continuity results as in Theorem 1.4 and Corollaries 1.5-1.7 also hold for the p-Laplace equations with measure data.
Let us give a brief description of the proofs.We first apply an iteration argument to get an L γ0 -mean oscillation estimate of the gradients of solutions to the homogeneous equation with x-independent nonlinearities −div(A 0 (∇v)) = 0 in Section 2, where γ 0 ∈ (0, 1).Our proofs of the interior gradient estimates are then based on a comparison estimate between the original solution u of (1.1) and the solution to the homogeneous equation −div(A(x, ∇w)) = 0 in a ball B R with the boundary condition u = w on ∂B R .The outcome is the inequality which holds for some constant γ 0 ∈ (0, 1).The details can be found in Lemma 3.2.For the case when p ∈ 3n−2 2n−1 , 2 , we can choose γ 0 = 2 − p, which is the same integral exponent on the right hand side.We then borrow an idea in [6] by estimating the L γ0 -mean oscillation to adapt the iteration scheme used, for instance, in [9].However, for the case when p ∈ 1, 3n−2 2n−1 , we are only able to prove the comparison estimate (1.20) for some γ 0 < 2 − p and that is the reason why we only obtain Lipschitz estimates instead of pointwise gradient estimates in this case.
For the gradient estimates up to the boundary, we use the technique of flattening the boundary and generalize the interior oscillation estimates to half balls.We adapt an idea in [3] to establish the global L γ0 -mean oscillation estimates by a delicate combination of the interior estimates and the estimates near a flat boundary.To this end, we also apply an odd extension argument to derive an L γ0 -mean oscillation estimate on half balls for homogeneous equations with x-independent nonlinearities.This argument only works for equations in diagonalized form, such as the p-Laplace equation, so the global estimates for general equations remain open.As a partial result in this direction, we refer the reader to [22] for a weighted pointwise boundary estimate under the condition that ∂Ω is sufficiently flat in the sense of Reifenberg.We also refer the reader to [2,16] for boundary regularity results for quasi-linear equations with sufficiently regular right-hand side.
The rest of the paper is organized as follows.In the next section, we derive an L γ0 -mean oscillation estimate of solutions to the homogeneous equation with x-independent nonlinearities.In Section 3, we give the proof of Theorem 1.1.Section 4 is devoted to the Lipschitz estimate and the interior modulus of continuity estimate of the gradient of solutions as well as its corollaries.Finally, in Section 5 we consider the corresponding boundary estimates.

An oscillation estimate
This section is devoted to the proof of the following interior oscillation estimate for solutions to the homogeneous equation where A 0 = A 0 (ξ) is a vector field independent of x satisfying conditions (1.2) and (1.3) for some s ≥ 0, λ ≥ 1, and p > 1.In this section, we denote the integral average over B R (x) by (•) BR(x) .
Corollary 2.3.Under the conditions of Lemma 2.2, there exist constants C > 1 and α ∈ (0, 1) depending only on n, p, and λ, such that for every B R (x 0 ) ⊂ Ω, we have and for any r ∈ [R/2, R), (2.4) Proof.Without loss of generality, we assume x 0 = 0.For any x ∈ B R/2 and r ≤ R/2, by Lemma 2.2 we have By Campanato's characterization of Hölder continuous functions, we obtain (2.3).Now for any r > R/2 and z ∈ B r , using (2.3) and the triangle inequality, we have (2.5) . We can divide the line segment connecting x and y into N equal segments using x 1 , ..., x N −1 , x 0 = x, and x N = y, such that Then by the triangle inequality and (2.5), we have which directly implies (2.4).The corollary is proved.Now we are ready to give the proof of Theorem 2.1.
We prove the claim by using Corollary 2.3 and iteration.For any R/2 < r < R ≤ dist(x 0 , ∂Ω), using (2.4) and the triangle inequality, we get , where we used Young's inequality with exponents p/γ 0 and p/(p − γ 0 ) in the last line.
The theorem is proved.

Interior pointwise gradient estimates
In order to prove the interior pointwise gradient estimates, we follow the outline of arguments given in [22] while replacing their oscillation estimates with our new oscillation estimate in Section 2. We also borrow an idea in [6] by estimating the L γ0 -mean oscillations of solutions, where γ 0 ∈ (0, 1).
Let u ∈ W 1,p loc (Ω) be a solution to (1.1) and B 2r (x 0 ) ⊂⊂ Ω.We consider the unique solution w ∈ u + W 1,p 0 (B 2r (x 0 )) to the equation We first recall an interior reverse Hölder inequality that can be found in [11,Theorem 6.7].
We also have the following comparison result, which generalizes and refines similar results in [9,20,21].Lemma 3.2.Let w be a solution to (3.1) and assume that p ∈ (1, 2).Then for any where C is a constant depending only on n, p, λ, and γ 0 .
We now let v ∈ w + W 1,p 0 (B r (x 0 )) be the unique solution to By testing (3.1) and (3.9) with v − w, we obtain an estimate for the difference ∇v − ∇w: For a ball B ρ (x) ⊂⊂ Ω and a function f ∈ W 1,p loc (Ω), there exists .
We denote q x,ρ = q x,ρ (u) and by taking the average over z ∈ B ρ (x) and then taking the γ 0 -th root, we obtain . Therefore, from the definition of φ and the fact that 0 < γ 0 < 1, we obtain that lim ρ→0 q x,ρ = ∇u(x) (3.12) holds for any Lebesgue point x ∈ Ω of the vector-valued function ∇u.
For an integer j ≥ 0, set r j = ε j R, B j = B 2rj (x 0 ), Let j 0 and m be positive integers to be specified later such that j 0 ≤ m.Summing the above inequality over j = j 0 , j 0 + 1, . . ., m, we obtain Since by taking the average over x ∈ B rj+1 (x 0 ) and then taking γ 0 -th root, we obtain Then, by iterating, we get which together with (3.17) implies By the definition of φ j0 , we have by taking the average over x ∈ B rj 0 (x 0 ) and taking the γ 0 -th root, we obtain Therefore, (3.18) implies that By (1.5) and the comparison principle for Riemann integrals, there exists j 0 = j 0 (n, p, ε, C, ω) > 1 sufficiently large such that 4 where C is the constant in (3.19).Note that by the comparison principle for Riemann integrals, and since p < 2 we also have To prove (1.12) at x = x 0 , it is sufficient to show that To this end, we consider the following possibilities.Case 2: If T j < |∇u(x 0 )|, ∀j 0 ≤ j ≤ j 1 , and |∇u(x 0 )| ≤ T j1+1 , then since γ 0 = 2 − p < 1, we have where the last inequality follows from the definitions of φ j1 and q j1 .Now applying (3.19) with m = j 1 − 1 and using (3.21) and (3.22), from (3.24) we get where , and C ′′ is a constant depending on n, p, and λ.Hence using (3.20) and Young's inequality, we find Here we used (3.20) in the last inequality.Letting m → ∞ and using (3.12), we get Then using Young's inequality, we deduce (3.23).The proof is completed.

4.
Interior Lipschitz estimate and modulus of continuity estimate of the gradient In this section, we give the proof of the Lipschitz estimate in Theorem 1.3 and derive an interior modulus of continuity estimate of ∇u under the same conditions.We first adapt the argument in [6] to obtain some decay estimates from Proposition 3.3.Let α ∈ (0, 1) be the same constant as in Theorem 2.1, α 1 ∈ (0, α), R ∈ (0, 1] and B R (x 0 ) ⊂ Ω. Choose ε = ε(n, p, λ, γ 0 , α, α 1 ) > 0 sufficiently small such that where C is the constant in (3.13).
Lemma 4.1.Let B 2r (x) ⊂⊂ B R (x 0 ) ⊂ Ω with r ≤ R/4.There exists a constant C depending only on ε, n, p, λ, γ 0 , and α 1 , such that for any ρ ∈ (0, r], we have To prove Lemma 4.1, we need the following technical lemma. Lemma 4.2.Let B R (x 0 ) ⊂ Ω.Then there exist constants c 1 and c 2 depending on ε, n, p, and α 1 , such that for any fixed x ∈ B R (x 0 ) and any Proof.We will only show the proof for g since the other cases are similar.For fixed x ∈ B R/4 (x 0 ), we set and observe that by (4.4), Suppose that 0 < εt ≤ s ≤ t.It is easy to see from the definitions of g and G that G(x, s) ≤ ε 1−n G(x, t) and therefore g(x, s) ≤ ε 1−n g(x, t).Also the fact that 0 < εs ≤ εt ≤ s implies that g(x, εt) ≤ ε 1−n g(x, s).On the other hand, The lemma is proved.Now we are ready to prove Lemma 4.1.
Proof of Lemma 4.1.We first prove Assertion (i).For given ρ ∈ (0, r], let j be an integer such that ε j+1 < ρ/r ≤ ε j .Then by (4.3) with ε −j ρ in place of r, we get Therefore, Assertion (i) holds.Now applying (4.5) with ε j ρ in place of ρ and summing in j, we get Hence, by using Lemma 4.2 and the comparison principle for Riemann integrals, we can easily get (4.6).The lemma is proved.
Recall the definition of q x,ρ from Section 3. Since we have by taking the average over z ∈ B ερ (x) and then taking the γ 0 -th root, we obtain |q x,ερ − q x,ρ | ≤ Cφ(x, ερ) + Cφ(x, ρ).
Then, by iterating, we get Therefore, by using (3.12), we obtain that holds for any Lebesgue point x ∈ Ω of the vector-valued function ∇u.Now we are ready to prove the interior Lipschitz estimate.
Proof of Theorem 1.3.We prove the theorem around a given point x = x 0 assuming that B R (x 0 ) ⊂ Ω with R ∈ (0, 1] and We first derive an a priori estimate for the case when u ∈ C 1 and then use approximation to prove the general case. Step 1: The case when u ∈ C 1 B R (x 0 ) .

+ s
Multiplying the above inequality by 3 −nk/γ0 , and summing the terms with respect to k = k 0 , k 0 + 1, • • • , we obtain that where each summation is finite.By subtracting from both sides of the above inequality, we get the following L ∞ -estimate for ∇u: We can simplify the terms in (4.11) to get (4.12) Indeed, by the definition of g in (4.4), we have The first term above is equal to The second term is equal to Therefore, We can similarly get Recalling the fact that γ 0 ≤ 2−p (cf.Lemma 3.2), using (4.13), (4.14), and Hölder's inequality, from (4.11) we obtain (4.12).
Step 2: The general case.We take r 1 ∈ (0, R), r 2 = (R + r 1 )/2, and a sequence of standard mollifiers {ϕ k } such that for any positive integer k, Then we mollify µ and A by setting We note that A k is well defined and satisfies the growth, ellipticity and continuity assumptions . By the corollary after [12, Theorem 1], (4.8) implies µ ∈ W −1,p ′ (B r2 (x 0 )), where p ′ = p/(p − 1), and therefore Next we let u k ∈ u + W 1,p 0 (B r2 (x 0 )) be the unique solution to −div(A k (x, ∇u k )) =µ k in B r2 (x 0 ), u k =u on ∂B r2 (x 0 ).(4.16) Choosing u k − u as a test function in (4.16), we obtain (4.17) Using the fundamental theorem of calculus, (1.2), (1.3), and Young's inequality with exponents p and p/(p − 1), we have On the other hand, using (1.2) and Young's inequality, we obtain . Therefore, (4.17) implies that where C is a constant not depending on k.Now we recall a well-known inequality where c = c(n, p) > 1 is a positive constant and the mapping V (•) is defined as , for some positive constant c 0 = c 0 (n, p, λ).We note that the above inequality also holds for A k .Then choosing (u k − u)1 Br 2 (x0) as a test function in (1.1) and (4.16), we have . From the definition of A k , by using the Minkowski inequality and (1.4), we obtain which together with (4.15) and (4.18), yields By (4.19), we have and therefore using Hölder's inequality with exponents 2/p and 2/(2 − p), we obtain that which implies that Thus there exists a subsequence {k j } such that ∇u kj → ∇u almost everywhere in B r2 (x 0 ).Since A k and µ k are smooth in x, by the classical regularity theory (see, for instance, [1], [23]), we know that u k ∈ C 1,α loc (B r2 (x 0 )).Therefore the Lipschitz estimate (4.12) from Step 1 holds for u k in B r1/2 (x 0 ).Namely, Note that by direct computation, for any t > 0, it follows that Therefore, for sufficiently large k, by the Fubini-Tonelli theorem we have Thus by taking k = k j ր ∞ and then r 1 ր R, we obtain the Lipschitz estimate (1.13) around x = x 0 .
Proof of Theorem 4.3.For any x, y ∈ B R/4 (x 0 ) being Lebesgue points of ∇u, by the triangle inequality, we have We set ρ = |x − y|, take the average over z ∈ B(x, ρ) ∩ B(y, ρ), and then take the γ 0 -th root to get Here we used (4.7) in the second inequality.If ρ < R/8, by using (4.23), (4.6) with R/8 in place of r, and the fact that Clearly, (4.24) still holds when ρ ≥ R/8.We can simplify the terms in (4.24) as follows.For any y 0 ∈ B R/4 (x 0 ) and ρ ∈ (0, R/2), by the definition of g in (4.4), we have Recalling the definition of Ĩ1 from (4.20), the first term above is equal to The second term is equal to Let K be the positive integer such that We can similarly get the following estimate Using (1.13), (4.25), and (4.26), from (4.24) we obtain (4.22).The theorem is proved.
Proof of Theorem 1.4.Since the set of Lebesgue points of ∇u is dense in Ω, it suffices to show the right-hand side of (4.22) converges to zero when ρ → 0. In fact, we have which must converge to 0 by using (1.14) and the dominated convergence theorem.We can similarly prove the convergence of other terms in (4.22).
Proof of Corollary 1.6.This is an immediate corollary of Theorem 1.4 since the assumption (1.14) is verified by (1.16) and (1.17).

Global gradient estimates for the p-Laplacian equations
This section is devoted to the proof of the global pointwise gradient estimate in Theorem 1.10, the Lipschitz estimate in Theorem 1.11, Corollary 1.12, as well as the derivation of a global modulus of continuity estimate of ∇u stated in Theorem 5.10 for the following (possibly nondegenerate) p-Laplace equation with Dirichlet boundary condition: where a(•) satisfies (1.5), (1.7), and (1.8), and Ω has a C 1,Dini boundary characterized by R 0 and ω 0 as in Definition 1.9.
First, we derive a gradient estimate around any point x 0 ∈ ∂Ω.Without loss of generality, we assume that x 0 = 0 ∈ ∂Ω.Then we can choose a local coordinate around x 0 = 0 and a function χ as in Definition 1.9 such that χ(0 ′ ) = 0. Let Note that the determinants of the Jacobian of Γ(•) and Λ(•) are equal to 1. Since Ω has C 1,Dini boundary, from the proof of [3, Lemma 2.2], there exists Therefore, there exist constants c 1 (n) and c 2 (n) depending only on n, such that for any x ∈ Ω and 0 < r ≤ R 1 , Now we use the technique of flattening the boundary.We denote u 1 (y) = u(Γ(y)), a 1 (y) = a(Γ(y)), and µ + .
Lemma 5.1.Let w be a solution to (5.9).There exists a constant θ 1 > p depending only on n, p, and λ, such that for any t > 0, the estimate On the other hand, using (5.6) and Young's inequality, we have |∇w| q dx 1/q (5.13) for any q such that max 1, np n+p ≤ q < p.Thus by combining (5.12) and (5.13), we have Similarly, the following interior version of estimate holds for all B 2ρ (y 0 ) ⊂⊂ B + 2r .Therefore, by a standard covering argument and Gehring's lemma, we get (5.10).
Thus we can apply Theorem 2.1 to v to get inf .
Since v is an odd function in y 1 , by the triangle inequality, there exists θ ρ ∈ R such that .
By the triangle inequality again, it is easily seen that inf . Then (5.15) is a direct consequence of the three inequalities above.
We now define .
Next we define where the functions g and h are defined in (4.2).
Now we are ready to prove Lemma 5.7.
We use an approximation argument with the aid of the regularized distance introduced by Lieberman [15].Here we refer to a modified version in [7].Let d(•) be the regularized distance defined in [7, Lemma 5.1] (ψ(•) in that paper) and Ω k = {x ∈ Ω : d(x) > 1/k}.Then from [7, Lemma 5.1], we know that Ω k has a smooth boundary and the C 1,Dini -properties of ∂Ω k are the same as those of ∂Ω up to some constant independent of k.We take a sequence of standard mollifiers {ϕ k } and mollify µ and a by setting We know that µ ∈ W −1,p ′ (Ω R (x 0 )) and therefore µ k − µ W −1,p ′ (ΩR(x0)) → 0.
Next we let u k ∈ uζ k + W 1,p 0 (Ω k ∩ B R (x 0 )) be the unique solution to  x0)  .
By extracting a subsequence and taking the limit as k → ∞, we obtain (5.33).

( 3 . 15 )
Thus from (3.14) and (3.15), we have Now we can apply Lemma 3.2 to bound the second term on the right-hand side of (3.16) to conclude the proof.Now we are ready to prove Theorem 1.1.