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Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients. (English) Zbl 1444.35043

The interesting paper under review deals with regularity issues regarding strong solutions to the oblique derivative problem \[ \begin{cases} -a^{ij}D_{ij}u+b^iD_iu+cu=f & \text{in}\ \Omega,\\ \beta^iD_iu+\beta_0u=g & \text{on}\ \partial\Omega \end{cases} \] over a bounded domain \(\Omega\subset\mathbb{R}^d\) with \(d\geq2.\) The second-order differential operator is assumed to be uniformly elliptic and the vector field \((\beta^1,\ldots,\beta^d)\) is strictly oblique to \(\partial\Omega.\)
The central result of the paper asserts that any strong solution is twice continuously differentiable up to the boundary assuming \(\partial\Omega\) can be locally represented by a \(C^1\)-smooth function with Dini continuous first derivatives and the mean oscillation of the coefficients \(a^{ij}\) satisfy the Dini condition. An interesting application to fully nonlinear equations with linear oblique boundary conditions is given as well.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35J15 Second-order elliptic equations
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[1] Dong, Hongjie, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205, 1, 119-149 (2012) · Zbl 1258.35040 · doi:10.1007/s00205-012-0501-z
[2] Dong, Hongjie; Escauriaza, Luis; Kim, Seick, On \(C^1, C^2\), and weak type-\((1,1)\) estimates for linear elliptic operators: part II, Math. Ann., 370, 1-2, 447-489 (2018) · Zbl 1406.35071 · doi:10.1007/s00208-017-1603-6
[3] Dong, Hongjie; Kim, Seick, On \(C^1, C^2\), and weak type-\((1,1)\) estimates for linear elliptic operators, Comm. Partial Differential Equations, 42, 3, 417-435 (2017) · Zbl 1373.35117 · doi:10.1080/03605302.2017.1278773
[4] Dong, Hongjie; Krylov, N. V., Second-order elliptic and parabolic equations with \(B(\mathbb{R}^2, \text{VMO})\) coefficients, Trans. Amer. Math. Soc., 362, 12, 6477-6494 (2010) · Zbl 1207.35174 · doi:10.1090/S0002-9947-2010-05215-8
[5] DLK Hongjie Dong, Jihoon Lee, and Seick Kim, On conormal and oblique derivative problem for elliptic equations with dini mean oscillation coefficients, to appear in Indiana Univ. Math. J., arXiv:1801.09836. · Zbl 1459.35119
[6] DL Hongjie Dong and Zongyuan Li, On the \(W^2_p\) estimate for oblique derivative problem in Lipschitz domains, 2018. arXiv:1808.02124.
[7] Escauriaza, Luis; Montaner, Santiago, Some remarks on the \(L^p\) regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28, 1, 49-63 (2017) · Zbl 1367.35052 · doi:10.4171/RLM/751
[8] Di Fazio, G.; Palagachev, D. K., Oblique derivative problem for elliptic equations in non-divergence form with VMO coefficients, Comment. Math. Univ. Carolin., 37, 3, 537-556 (1996) · Zbl 0881.35028
[9] Di Fazio, Giuseppe; Palagachev, Dian K., Oblique derivative problem for quasilinear elliptic equations with VMO coefficients, Bull. Austral. Math. Soc., 53, 3, 501-513 (1996) · Zbl 0879.35056 · doi:10.1017/S0004972700017275
[10] Giaquinta, Mariano; Martinazzi, Luca, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 11, xiv+366 pp. (2012), Edizioni della Normale, Pisa · Zbl 1262.35001 · doi:10.1007/978-88-7642-443-4
[11] Ishii, Hitoshi, Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs, Duke Math. J., 62, 3, 633-661 (1991) · Zbl 0733.35020 · doi:10.1215/S0012-7094-91-06228-9
[12] Kovats, Jay, Fully nonlinear elliptic equations and the Dini condition, Comm. Partial Differential Equations, 22, 11-12, 1911-1927 (1997) · Zbl 0899.35036 · doi:10.1080/03605309708821325
[13] Krylov, N. V., Lectures on elliptic and parabolic equations in H\"{o}lder spaces, Graduate Studies in Mathematics 12, xii+164 pp. (1996), American Mathematical Society, Providence, RI · Zbl 0865.35001 · doi:10.1090/gsm/012
[14] Li, Yanyan, On the \(C^1\) regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients, Chin. Ann. Math. Ser. B, 38, 2, 489-496 (2017) · Zbl 1367.35055 · doi:10.1007/s11401-017-1079-4
[15] Li, Dongsheng; Zhang, Kai, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228, 3, 923-967 (2018) · Zbl 1390.35084 · doi:10.1007/s00205-017-1209-x
[16] Lieberman, Gary M., Regularized distance and its applications, Pacific J. Math., 117, 2, 329-352 (1985) · Zbl 0535.35028
[17] Lieberman, Gary M., H\"{o}lder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148, 77-99 (1987) · Zbl 0658.35050 · doi:10.1007/BF01774284
[18] Lieberman, Gary M., Oblique derivative problems in Lipschitz domains. I. Continuous boundary data, Boll. Un. Mat. Ital. B (7), 1, 4, 1185-1210 (1987) · Zbl 0637.35028
[19] Lieberman, Gary M., Higher regularity for nonlinear oblique derivative problems in Lipschitz domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1, 1, 111-151 (2002) · Zbl 1170.35423
[20] Lieberman, Gary M., Oblique derivative problems for elliptic equations, xvi+509 pp. (2013), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1273.35006 · doi:10.1142/8679
[21] Lin, Fang-Hua, Second derivative \(L^p\)-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc., 96, 3, 447-451 (1986) · Zbl 0599.35039 · doi:10.2307/2046592
[22] Lions, P.-L.; Trudinger, N. S., Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation, Math. Z., 191, 1, 1-15 (1986) · Zbl 0593.35046 · doi:10.1007/BF01163605
[23] Maugeri, Antonino; Palagachev, Dian K., Boundary value problem with an oblique derivative for uniformly elliptic operators with discontinuous coefficients, Forum Math., 10, 4, 393-405 (1998) · Zbl 0908.35030 · doi:10.1515/form.10.4.393
[24] Milakis, Emmanouil; Silvestre, Luis E., Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations, 31, 7-9, 1227-1252 (2006) · Zbl 1241.35093 · doi:10.1080/03605300600634999
[25] Safonov, M. V., On the oblique derivative problem for second order elliptic equations, Comm. Partial Differential Equations, 20, 7-8, 1349-1367 (1995) · Zbl 0841.35035 · doi:10.1080/03605309508821135
[26] Sa-notes M. V. Safonov, On the boundary value problems for fully nonlinear elliptic equations of second order, Mathematics Research Report No. MRR 049-94, The Australian National University, Canberra, 1994. http://www-users.math.umn.edu/ safon002/NOTES/BVP_94/BVP.pdf
[27] Spanne, Sven, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 19, 593-608 (1965) · Zbl 0199.44303
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