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Hessian Estimates for Non-divergence form Elliptic Equations Arising from Composite Materials

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Abstract

In this paper, we prove that any W2,1 strong solution to second-order non-divergence form elliptic equations is locally \(W^{2,\infty }\) and piecewise C2 when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be in C1,Dini. We also derive global weak-type (1,1) estimates with respect to A1 Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients

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Acknowledgements

This work was completed while the second author was visiting Brown University. She would like to thank the Division of Applied Mathematics at Brown University for the hospitality and the stimulating environment. The authors would like to thank Prof. Seick Kim and Yanyan Li for helpful discussions.

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Correspondence to Hongjie Dong.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

H. Dong was partially supported by the NSF under agreement DMS-1600593.

L. Xu was partially supported by the China Scholarship Council (No. 201706040139).

Appendix

Appendix

In the Appendix, we give the \(W_{w}^{2,p}\)-estimate and solvability for the non-divergence form elliptic equation in C1,1 domains with the zero Dirichlet boundary condition. Consider

$$ \left\{\begin{array}{ll} \lambda u-Lu=f&\quad\text{in}~\mathcal{D},\\ u=0&\quad\text{on}~\partial\mathcal{D}, \end{array}\right. $$
(A.1)

where λ ≥ 0, \(\mathcal {D}\in C^{1,1}\), and Lu := aijDiju + biDiu + cu. Let \(p\in (1,\infty )\) and w be an Ap weight. Denote

$$ \mathring{W}_{w}^{2,p}(\mathcal{D}):=\{u\in W_{w}^{2,p}(\mathcal{D}): u=0~\text{on}~\partial\mathcal{D}\}. $$

Now we impose the regularity assumptions on aij. Let \(\gamma _{0}=\gamma _{0}(n,p,\delta ,[w]_{A_{p}})\in (0,1)\) be a sufficiently small constant to be specified. There exists a constant r0 ∈ (0, 1) such that aij satisfy (2.4) in the interior of \(\mathcal {D}\) and are VMO near the boundary: for any \(x_{0}\in \partial \mathcal {D}\) and r ∈ (0, r0], we have

In addition, bi and c are bounded by a constant Λ. Then we have the following

Theorem A.1

Let \(p\in (1,\infty )\), wAp, and L1 ≤ 0. There exists a sufficient small constant \(\gamma _{0}=\gamma _{0}(n,p,\delta ,[w]_{A_{p}})\in (0,1)\) such that under the above conditions, for any λ ≥ 0 and \(f\in {L_{w}^{p}}(\mathcal {D})\), there exists a unique \(u\in W_{w}^{2,p}(\mathcal {D})\) satisfying (A.1). Furthermore, there exists a constant \(C=C(n,p,\delta ,{\Lambda },\mathcal {D},[w]_{A_{p}},r_{0})\) such that

$$ \|u\|_{W_{w}^{2,p}(\mathcal{D})}\leq C\|f\|_{{L_{w}^{p}}(\mathcal{D})}. $$

First we note that when \(\lambda \ge \lambda _{1}(n,p,\delta ,\lambda , \mathcal {D}, [w]_{A_{p}},r_{0})\), the theorem follows from the proofs of [13, Theorems 6.3 and 6.4] combined with the argument in [10, Theorem 2.5] and [23, Sections 8.5]. To deal with the case when λ ∈ [0, λ1), we need the following lemmas. The first one is a local regularity of solutions in weighted Sobolev spaces.

Lemma A.2

Let \(1<p\leq q<\infty \), \(z\in \overline {\mathcal {D}}\). Denote \(\mathcal {D}_{r}:=\mathcal {D}\cap B_{r}(z)\). Then if

$$ \xi u\in\mathring{W}_{w}^{2,p}(\mathcal{D}_{2R})\quad\forall~\xi\in C_{0}^{\infty}(B_{2R}(z)),\quad Lu\in {L_{w}^{q}}(\mathcal{D}_{2R}), $$
(A.2)

we have

$$ \xi u\in\mathring{W}_{w}^{2,q}(\mathcal{D}_{2R})\quad\forall~\xi\in C_{0}^{\infty}(B_{2R}(z)). $$
(A.3)

Furthermore, there exists a constant \(C=C(R,p,q,n,\delta ,{\Lambda },[w]_{A_{p}},r_{0})\) such that if (A.2) holds, then

$$ \|u\|_{W_{w}^{2,q}(\mathcal{D}_{R})}\leq C\big(\|Lu\|_{{L_{w}^{q}}(\mathcal{D}_{2R})}+\|u\|_{{L_{w}^{p}}(\mathcal{D}_{2R})}\big). $$
(A.4)

Proof

We follow the proof of [23, Theorem 11.2.3] when w = 1. For q = p, (A.3) is obvious and (A.4) is obtained by using the method in the proof of [23, Theorem 9.4.1]. For q > p, we define

$$ \begin{array}{@{}rcl@{}} \alpha=\frac{n}{n-1}\quad\text{for}~n\geq2;\quad p(j)=\alpha^{j}p,\quad j=0,1,\dots,k-1,\quad p(k)=q, \end{array} $$

where k − 1 is the last j such that p(j) < q. Take λ sufficiently large that λL, as an operator acting from \(\mathring {W}_{w}^{2,p(j)}(\mathcal {D})\) onto \(L_{w}^{p(j)}(\mathcal {D})\) for \(j=0,1,\dots ,k\), is invertible. Denote

$$ f=Lu,\quad g=(L-\lambda)(\xi u)=\xi f+2a^{ij}D_{i}uD_{j}\xi+u(L-c-\lambda)\xi. $$

By weighted Sobolev embedding theorem, see [18, Theorem 1.3], we have

$$ \zeta u\in W_{w}^{1,p(1)}(\mathcal{D}) $$

for any \(\zeta \in C_{0}^{\infty }(B_{2R}(z))\). Hence, \(g\in L_{w}^{p(1)}(\mathcal {D})\). By the choice of λ, the equation

$$ (L-\lambda)v=g $$

has a solution in \(\mathring {W}_{w}^{2,p(1)}(\mathcal {D})\subset \mathring {W}_{w}^{2,p}(\mathcal {D})\) which is unique in \(\mathring {W}_{w}^{2,p}(\mathcal {D})\). We thus obtain that for j = 1,

$$ v=\xi u\in \mathring{W}_{w}^{2,p(1)}(\mathcal{D}),\quad\forall~\xi\in C_{0}^{\infty}(B_{2R}(z)). $$
(A.5)

If p(1) < q, then by repeating this argument with p(1) in place of p, we get (A.5) for j = 2. In this way we conclude (A.3).

Next we prove (A.4). By the choice of λ, for j ≥ 1 and any \(\xi ,\eta \in C_{0}^{\infty }(B_{2R}(z))\) such that η = 1 on the support of ξ, we have

$$ \begin{array}{@{}rcl@{}} \|\xi u\|_{W_{w}^{2,p(j)}(\mathcal{D})}&\leq& C\|\xi f+2a^{ij}D_{i}uD_{j}\xi+u(L-c-\lambda)\xi\|_{L_{w}^{p(j)}(\mathcal{D})}\\ &\leq &C\big(\|f\|_{{L_{w}^{q}}(\mathcal{D}_{2R})}+\|\eta u\|_{W_{w}^{1,p(j)}(\mathcal{D})}\big)\\ &\leq &C\big(\|f\|_{{L_{w}^{q}}(\mathcal{D}_{2R})}+\|\eta u\|_{W_{w}^{2,p(j-1)}(\mathcal{D})}\big), \end{array} $$

where we used the weighted Sobolev embedding theorem in the last inequality. By iterating the above inequality, we obtain that for any \(\xi \in C_{0}^{\infty }(B_{3R/2}(z))\), there is an \(\eta \in C_{0}^{\infty }(B_{7R/4}(z))\) such that

$$ \|\xi u\|_{W_{w}^{2,q}(\mathcal{D})}\leq C\big(\|f\|_{{L_{w}^{q}}(\mathcal{D}_{2R})}+\|\eta u\|_{W_{w}^{2,p}(\mathcal{D})}\big). $$

Finally, recalling the conclusion for the case when p = q, we have

$$ \|\eta u\|_{W_{w}^{2,p}(\mathcal{D})}\leq C\|u\|_{W_{w}^{2,p}(\mathcal{D}_{7R/4})}\leq C\big(\|f\|_{{L_{w}^{p}}(\mathcal{D}_{7R/2})}+\| u\|_{{L_{w}^{p}}(\mathcal{D}_{7R/2})}\big). $$

This yields (A.4) and the lemma is proved. □

Next we recall the resolvent operator of LλI by

$$ \mathcal{R}_{\lambda}: {L_{w}^{p}}(\mathcal{D})\rightarrow \mathring{W}_{w}^{2,p}(\mathcal{D}). $$

Then \(\mathcal {R}_{\lambda }\) is a bounded operator for λλ1. The following properties of \(\mathcal {R}_{\lambda }\) for λ large play an important role in proving Lemma A.3 below.

Lemma A.3

Let the coefficients of L be infinitely differentiable, L1 ≤ 0, and λλ1. Then

  1. (1)

    For any bounded f and any γ ∈ (0, 1), we have \(\mathcal {R}_{\lambda }f\in C^{1+\gamma }(\mathcal {D})\), \(\mathcal {R}_{\lambda }f=0\) on \(\partial \mathcal {D}\), and in \(\mathcal {D}\),

    $$ |\mathcal{R}_{\lambda}f(x)|\leq\mathcal{R}_{\lambda}|f|(x)\leq\lambda^{-1}\sup_{x\in\mathcal{D}}|f(x)|. $$
    (A.6)
  2. (2)

    There exists an integer \(m_{0}=m_{0}(n,p,\delta ,{\Lambda },\mathcal {D},[w]_{A_{p}},r_{0})\), such that for any \(f\in {L_{w}^{p}}(\mathcal {D})\), we have

    $$ \sup_{x\in\mathcal{D}}|\mathcal{R}_{\lambda_{1}}^{m_{0}}f(x)|\leq C\|f\|_{{L_{w}^{p}}(\mathcal{D})}, $$
    (A.7)

    where \(C=C(n,p,\delta ,{\Lambda },\mathcal {D},[w]_{A_{p}},r_{0})>0\).

Proof

For \(f\in L^{\infty }(\mathcal {D})\), (A.6) is proved in [23, Theorem 11.2.1(3)]. To prove (A.7), we set α = n/(n − 1), p(j) = αjp, and

$$ u_{0}=f,\quad u_{j}=\mathcal{R}_{\lambda_{1}}^{j}f,\quad j\geq1. $$

Notice that for j ≥ 1, we have

$$ \lambda_{1}u_{j+1}-Lu_{j+1}=u_{j}. $$

Therefore, by using the solvability and estimates for λ large, we have \(u_{j+1}\in \mathring {W}_{w}^{2,p}(\mathcal {D})\) and

$$ \|u_{j+1}\|_{W_{w}^{2,p}(\mathcal{D})}\leq C\|u_{j}\|_{{L_{w}^{p}}(\mathcal{D})}\leq C\|u_{j}\|_{L_{w}^{p(j)}(\mathcal{D})}. $$

By using Lemma A.2, we get

$$ \|u_{j+1}\|_{W_{w}^{2,p(j)}(\mathcal{D})}\leq C\big(\|u_{j}\|_{L_{w}^{p(j)}(\mathcal{D})}+\|u_{j+1}\|_{{L_{w}^{p}}(\mathcal{D})}\big). $$

Hence,

$$ \|u_{j+1}\|_{W_{w}^{2,p(j)}(\mathcal{D})}\leq C\|u_{j}\|_{L_{w}^{p(j)}(\mathcal{D})}. $$
(A.8)

By the weighted embedding theorem, we have

$$ \|u_{j+1}\|_{L_{w}^{p(j+1)}(\mathcal{D})}\leq C\|u_{j}\|_{L_{w}^{p(j)}(\mathcal{D})}. $$

Iterating the above inequality yields that for j ≥ 0, we obtain

$$ \|u_{j}\|_{L_{w}^{p(j)}(\mathcal{D})}\leq C\|u_{0}\|_{L_{w}^{p(0)}(\mathcal{D})}=C\|f\|_{{L_{w}^{p}}(\mathcal{D})}. $$
(A.9)

It follow from Hölder’s inequality and the definition of Ap weights that \(u_{j+1}\in W^{2,p(j)/p}(\mathcal {D})\). Then we fix a j = j(n, p) by choosing p(j) > np/2. For such j, we conclude from (A.8) and (A.9) that

$$ \sup_{x\in\mathcal{D}}|u_{j+1}(x)|\leq C\|u_{j+1}\|_{W^{2,p(j)/p}(\mathcal{D})}\leq C\|u_{j+1}\|_{W_{w}^{2,p(j)}(\mathcal{D})}\leq C\|f\|_{{L_{w}^{p}}(\mathcal{D})}, $$

which shows that (A.7) holds with m0 = j + 1. The lemma is proved. □

Next we show the solvability when λ𝜖0 for a positive constant 𝜖0 > 0.

Lemma A.4

Let \(p\in (1,\infty )\), wAp, 𝜖0 > 0, and L1 ≤ 0. Under the above conditions, for any λ𝜖0 and \(u\in \mathring {W}_{w}^{2,p}(\mathcal {D})\), we have

$$ \|u\|_{W_{w}^{2,p}(\mathcal{D})}\leq C\|\lambda u-Lu\|_{{L_{w}^{p}}(\mathcal{D})}, $$
(A.10)

where C > 0 depends on \(n,p,\delta ,{\Lambda },\mathcal {D},\epsilon _{0},[w]_{A_{p}}\), and r0.

Proof

We define f := λuLu. As noted after Theorem A.1, it suffices to prove the case when 𝜖0λ < λ1. We follow the idea in [23, Section 11.3]. Here we list the main differences. By approximations, we may assume that the coefficients are smooth. Similar to the proof of [23, Theorem 8.5.6], we have

$$ \|u\|_{W_{w}^{2,p}(\mathcal{D})}\leq C\big(\|f\|_{{L_{w}^{p}}(\mathcal{D})}+\|u\|_{{L_{w}^{p}}(\mathcal{D})}\big). $$

Therefore, it suffices to prove for ε0λ < λ1, we have

$$ \|u\|_{{L_{w}^{p}}(\mathcal{D})}\leq C\|f\|_{{L_{w}^{p}}(\mathcal{D})}. $$

Since

$$ \lambda_{1}u-Lu=(\lambda_{1}-\lambda)u+f,\quad u=(\lambda_{1}-\lambda)\mathcal{R}_{\lambda_{1}}u+\mathcal{R}_{\lambda_{1}}f, $$

by induction on m, we have

$$ u=\big((\lambda_{1}-\lambda)\mathcal{R}_{\lambda_{1}}\big)^{m}u+\sum\limits_{i=0}^{m-1}\big((\lambda_{1}-\lambda)\mathcal{R}_{\lambda_{1}}\big)^{i}\mathcal{R}_{\lambda_{1}}f, $$
(A.11)

where m ≥ 1 is any integer. We next introduce constants C1 and Mm such that

$$ \|\mathcal{R}_{\lambda_{1}}g\|_{{L_{w}^{p}}(\mathcal{D})}\leq C_{1}\|g\|_{{L_{w}^{p}}(\mathcal{D})}\quad\forall~g\in {L_{w}^{p}}(\mathcal{D}),\quad M_{m}=\sum\limits_{i=0}^{m-1}(\lambda_{1}-\epsilon_{0})^{i}C_{1}^{i+1}. $$
(A.12)

By using (A.11), 0 < λ1λλ1𝜖0, (A.12), (A.6), and (A.7), we obtain for m > m0,

$$ \begin{array}{@{}rcl@{}} \|u\|_{{L_{w}^{p}}(\mathcal{D})}&\leq& w(\mathcal{D})^{1/p}(\lambda_{1}-\epsilon_{0})^{m}\sup_{x\in\mathcal{D}}|\mathcal{R}_{\lambda_{1}}^{m-m_{0}}\mathcal{R}_{\lambda_{1}}^{m_{0}}u(x)|+M_{m}\|f\|_{{L_{w}^{p}}(\mathcal{D})}\\ &\leq& w(\mathcal{D})^{1/p}\lambda_{1}^{m_{0}}(1-\epsilon_{0}/\lambda_{1})^{m}\sup_{x\in\mathcal{D}}|\mathcal{R}_{\lambda_{1}}^{m_{0}}u(x)|+M_{m}\|f\|_{{L_{w}^{p}}(\mathcal{D})}\\ &\leq& C_{2}w(\mathcal{D})^{1/p}\lambda_{1}^{m_{0}}(1-\epsilon_{0}/\lambda_{1})^{m}\|u\|_{{L_{w}^{p}}(\mathcal{D})}+M_{m}\|f\|_{{L_{w}^{p}}(\mathcal{D})}. \end{array} $$

Fixing m > m0 such that

$$ C_{2}w(\mathcal{D})^{1/p}\lambda_{1}^{m_{0}}(1-\epsilon_{0}/\lambda_{1})^{m}\leq1/2, $$

we get (A.10). The lemma is proved. □

We now give

Proof of Theorem A.1.

By using the method of continuity, we only need to prove that for any \(u\in \mathring {W}_{w}^{2,p}(\mathcal {D})\) and λ ≥ 0, we have (A.10). To this end, without loss of generality we may assume that \(\mathcal {D}\subset B_{2R_{0}}\), where \(R_{0}=\text {diam}~{\mathcal {D}}\). We take the global barrier v0 from [23, Lemma 11.1.2]:

$$ v_{0}(x)=\cosh(4c_{0}R_{0})-\cosh(c_{0}|x|) $$

satisfies Lv0 ≤− 1, and v0 > 0 in \(B_{4R_{0}}\) and v0 = 0 on \(\partial B_{4R_{0}}\), where c0 > 0 is a constant to be chosen. Next we introduce a new operator \(L^{\prime }\) by

$$ L^{\prime}u=v_{0}^{-1}L(v_{0}u). $$

Notice that in \(\mathcal {D}\subset B_{2R_{0}}\), according to the construction of v0, we have \(L^{\prime }1\leq -\delta ^{\prime }\) for a constant \(\delta ^{\prime }>0\) depending on n, δ,Λ, and R0, provided that c0 = c0(d, δ,Λ) is sufficiently large. By using Lemma A.3 applied to \(L^{\prime \prime }:=L^{\prime }+\delta ^{\prime }\), we get

$$ \begin{array}{@{}rcl@{}} \|u\|_{W_{w}^{2,p}(\mathcal{D})}&\leq& C\|uv_{0}^{-1}\|_{W_{w}^{2,p}(\mathcal{D})}\\ &\leq& C\|(\lambda+\delta^{\prime})uv_{0}^{-1}-L^{\prime\prime}(uv_{0}^{-1})\|_{{L_{w}^{p}}(\mathcal{D})}\\ &=& C\|(\lambda u-v_{0}L^{\prime}(uv_{0}^{-1}))v_{0}^{-1}\|_{{L_{w}^{p}}(\mathcal{D})}\\ &=&C\|(\lambda u-Lu)v_{0}^{-1}\|_{{L_{w}^{p}}(\mathcal{D})}\leq C\|(\lambda -L)u\|_{{L_{w}^{p}}(\mathcal{D})}. \end{array} $$

Hence, we finish the proof of the theorem. □

Finally we give the solvability of the adjoint operator of L defined by

$$ L^{*}u:=D_{ij}(a^{ij}u)-D_{i}(b^{i}u)+cu. $$

By using a similar argument in the proof of Lemma 2.8, from Theorem A.1, we have

Corollary A.5

Let \(p\in (1,\infty )\), wAp, and L1 ≤ 0. Assume that \(g=(g^{ij})_{i,j=1}^{n}\in {L_{w}^{p}}(\mathcal {D})\). The coefficients aij, bi, and c satisfy the same conditions as imposed in Theorem A.1. Then for any λ ≥ 0,

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} L^{*}u-\lambda u=\operatorname{div}^{2}g&\quad\text{in}~\mathcal{D},\\ u=\frac{g\nu\cdot\nu}{A\nu\cdot\nu}&\quad\text{on}~\partial\mathcal{D} \end{array}\right. \end{array} $$

admits a unique adjoint solution \(u\in {L_{w}^{p}}(\mathcal {D})\). Moreover, the following estimate holds

$$ \|u\|_{{L_{w}^{p}}(\mathcal{D})}\leq C\|g\|_{{L_{w}^{p}}(\mathcal{D})}, $$
(A.13)

where \(C=C(n,p,\delta ,{\Lambda },\mathcal {D},r_{0},[w]_{A_{p}})>0\).

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Dong, H., Xu, L. Hessian Estimates for Non-divergence form Elliptic Equations Arising from Composite Materials. Potential Anal 54, 409–449 (2021). https://doi.org/10.1007/s11118-020-09832-w

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