Abstract
We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.
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Acknowledgments
S. Kim is partially supported by National Research Foundation of Korea (NRF) Grant No. NRF-2016R1D1A1B03931680 and No. NRF-20151009350.
S. Hwang is partially supported by the Yonsei University Research Fund (Post Doc. Researcher Supporting Program) of 2017 (project no.: 2017-12-0031).
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Appendix
Appendix
The following lemma is well known to experts and essentially due to Campanato. We provide its proof for the reader’s convenience.
Lemma A.1
Let\({\Omega } \in \mathbb {R}^{n}\)bea domain satisfying the following condition: there exists a constantA0 ∈ (0, 1] such thatfor everyx ∈Ω and 0 < r < diam Ω, wehave
Suppose that a function\(u \in L_{\text {loc}}^{1}(\overline {\Omega })\)is of Dini mean oscillation in Ω, then there exists a uniformly continuous functionu∗on Ω such thatu∗ = ua.e. in Ω.
Proof
In the proof we shall denote
By taking the average over \({\Omega }(x, \frac 12 r)\) to the triangle inequality
and using \(\lvert {{\Omega }(x, r)}\rvert / \lvert {{\Omega }(x,\frac 12 r)}\rvert \le 2^{n} /A_{0}\), we get
By telescoping, we get
where in the last step we used the fact that \(\omega (t) \simeq \omega \left (\frac {1}{2^{j}}r\right )\) when \(t \in \left (\frac {1}{2^{j + 1}}r, \frac {1}{2^{j}}r\right ]\); see [4]. Note that the last inequality also implies that
Now, we define the function u∗ on Ω by setting \(u^{\ast }(x)=\lim _{r\to 0} \bar u_{x,r}\). By the Lebesgue differentiation theorem, we have u = u∗ a.e. By letting k →∞ in (A.2), we obtain
For any x, y in Ω, let r = |x − y|, \(z=\frac 12 (x+y)\), and use Eq. A.4 to get
By taking the average over \({\Omega }(z,\frac 12 r)\) to the triangle inequality
and noting that \({\Omega }(z, \frac 12 r) \subset {\Omega }(x,r)\cap {\Omega }(y,r)\), we get
Combining together and using Eq. A.3, we conclude that
Therefore, we see that u∗ is uniformly continuous with it the modulus of continuity dominated by the function \(\displaystyle \rho (r):={\int }_0^r \frac {\omega (t)}{t},dt\). □
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Hwang, S., Kim, S. Green’s Function for Second Order Elliptic Equations in Non-divergence Form. Potential Anal 52, 27–39 (2020). https://doi.org/10.1007/s11118-018-9729-z
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DOI: https://doi.org/10.1007/s11118-018-9729-z