Higher-order nondivergence elliptic and parabolic equations in Sobolev spaces and Orlicz spaces. (English) Zbl 1240.35071
In the very interesting paper under review, the authors obtain global regularity estimates in Sobolev and Orlicz spaces for the strong solutions to the Cauchy problem for higher-order parabolic equations of nondivergence form
\[
u_t-\sum_{|\nu|=0}^{2m} a_\nu(x,t)D^\nu u =f(x,t)\quad \text{in}\;{\mathbb R}^n\times(0,T),
\]
where the coefficients have small BMO seminorms and satisfy
\[
(-1)^{m-1} \sum_{|\nu|=2m} a_\nu(x,t)\xi^\nu\geq\Lambda_1|\xi|^{2m},\quad \sum_{|\nu|=0}^{2m} |a_\nu(x,t)|\leq \Lambda_2.
\]
The corresponding elliptic result is derived as particular case of time-independent data.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K25 | Higher-order parabolic equations |
| 35J30 | Higher-order elliptic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | Acerbi, E.; Mingione, G., Gradient estimates for the \(p(x)\)-Laplacian system, J. Reine Angew. Math., 584, 117-148 (2005) · Zbl 1093.76003 |
| [2] | Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke Math. J., 136, 285-320 (2007) · Zbl 1113.35105 |
| [3] | Adams, R. A.; Fournier, J. J.F., Sobolev Spaces (2003), Academic Press: Academic Press New York · Zbl 0347.46040 |
| [4] | Bramanti, M.; Cerutti, M. C., \(W_p^{1, 2}\) solvability for the Cauchy Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations, 18, 1735-1763 (1993) · Zbl 0816.35045 |
| [5] | Byun, S., Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains, J. Differential Equations, 246, 9, 3518-3534 (2009) · Zbl 1170.35027 |
| [6] | Byun, S.; Ryu, S., Orlicz regularity for higher order parabolic equations in divergence form with coefficients in weak BMO, Arch. Math. (Basel), 95, 2, 179-190 (2010) · Zbl 1200.35041 |
| [7] | Byun, S.; Ryu, S., Gradient estimates for higher order elliptic equations on nonsmooth domains, J. Differential Equations, 25, 1, 243-263 (2011) · Zbl 1205.35327 |
| [8] | Byun, S.; Wang, L., Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57, 10, 1283-1310 (2004) · Zbl 1112.35053 |
| [9] | Byun, S.; Wang, L., Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176, 2, 271-301 (2005) · Zbl 1073.35046 |
| [10] | Byun, S.; Yao, F.; Zhou, S., Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal., 255, 8, 1851-1873 (2008) · Zbl 1156.35038 |
| [11] | Caffarelli, L. A.; Peral, I., On \(W^{1, p}\) estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51, 1-21 (1998) · Zbl 0906.35030 |
| [12] | Calderón, A. P.; Zygmund, A., On the existence of certain singular integrals, Acta Math., 88, 85-139 (1952) · Zbl 0047.10201 |
| [13] | Chiarenza, F.; Frasca, M.; Longo, P., Interior \(W^{2, p}\) estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat., 40, 1, 149-168 (1991) · Zbl 0772.35017 |
| [14] | Chiarenza, F.; Frasca, M.; Longo, P., \(W^{2, p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336, 2, 841-853 (1993) · Zbl 0818.35023 |
| [15] | Cianchi, A., Strong and weak type inequalities for some classical operators in Orlicz spaces, J. Lond. Math. Soc. (2), 60, 1, 187-202 (1999) · Zbl 0940.46015 |
| [16] | Di Fazio, G., \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Unione Mat. Ital. A (7), 10, 2, 409-420 (1996) · Zbl 0865.35048 |
| [17] | Dong, H., Solvability of parabolic equations in divergence form with partially BMO coefficients, J. Funct. Anal., 258, 2145-2172 (2010) · Zbl 1191.35145 |
| [18] | Dong, H.; Kim, D., Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196, 1, 25-70 (2010) · Zbl 1206.35249 |
| [19] | Dong, H.; Kim, D., On the \(L^p\)-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199, 3, 889-941 (2011) · Zbl 1228.35113 |
| [20] | Evans, L. C., Partial Differential Equations, Grad. Stud. Math., vol. 19 (1998) · Zbl 0902.35002 |
| [21] | Fabes, E., Singular integrals and partial differential equations of parabolic type, Studia Math., 28, 81-131 (1966) · Zbl 0144.35002 |
| [22] | Fefferman, C.; Stein, E. M., \(H^p\) spaces of several variables, Acta Math., 129, 3-4, 137-193 (1972) · Zbl 0257.46078 |
| [23] | Haller-Dintelmann, R.; Heck, H.; Hieber, M., \(L^p - L^q\) estimates for parabolic systems in non-divergence form with VMO coefficients, J. Lond. Math. Soc. (2), 74, 3, 717-736 (2006) · Zbl 1169.35339 |
| [24] | Han, Q., On the Schauder estimates of solutions to parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 27, 1, 1-26 (1998) · Zbl 0954.35034 |
| [25] | Jia, H.; Li, D.; Wang, L., Regularity of Orlicz spaces for the Poisson equation, Manuscripta Math., 122, 3, 265-275 (2007) · Zbl 1152.35019 |
| [26] | Jones, B. F., A class of singular integrals, Amer. J. Math., 86, 441-462 (1964) · Zbl 0123.08501 |
| [27] | Kim, D.; Krylov, N. V., Parabolic equations with measurable coefficients, Potential Anal., 26, 345-361 (2007) · Zbl 1124.35024 |
| [28] | Krylov, N. V., Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32, 1-3, 453-475 (2007) · Zbl 1114.35079 |
| [29] | Krylov, N. V., Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal., 257, 6, 1695-1712 (2009) · Zbl 1171.35352 |
| [30] | Lieberman, G. M., A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal., 201, 2, 457-479 (2003) · Zbl 1107.35030 |
| [31] | Mingione, G., Calderón-Zygmund estimates for measure data problems, C. R. Math. Acad. Sci. Paris, 344, 7, 437-442 (2007) · Zbl 1190.35088 |
| [32] | Mingione, G., Gradient estimates below the duality exponent, Math. Ann., 346, 3, 571-627 (2010) · Zbl 1193.35077 |
| [33] | Miyazaki, Y., The \(L^p\) resolvents of elliptic operators with uniformly continuous coefficients, J. Differential Equations, 188, 2, 555-568 (2003) · Zbl 1057.47052 |
| [34] | Miyazaki, Y., Higher order elliptic operators of divergence form in \(C^1\) or Lipschitz domains, J. Differential Equations, 230, 1, 174-195 (2006) · Zbl 1143.35023 |
| [35] | Palagachev, D. K., \(W^{2, p}\)-a priori estimates for the emergent Poincaré problem, J. Global Optim., 40, 305-318 (2008) · Zbl 1137.35024 |
| [36] | Palagachev, D. K.; Softova, L. G., A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst., 13, 3, 721-742 (2005) · Zbl 1091.35035 |
| [37] | Palagachev, D. K.; Softova, L. G., Precise regularity of solutions to elliptic systems with discontinuous data, Ric. Mat., 54, 2, 631-639 (2005) · Zbl 1387.35196 |
| [38] | Solonnikov, V. A., On boundary value problems for linear parabolic systems of differential equations of general form, Tr. Mat. Inst. Steklova, 83, 3-63 (1965) · Zbl 0164.12502 |
| [39] | Stein, E. M., Harmonic Analysis (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0821.42001 |
| [40] | Wang, L.; Yao, F.; Zhou, S.; Jia, H., Optimal regularity for the Poisson equation, Proc. Amer. Math. Soc., 137, 2037-2047 (2009) · Zbl 1166.35011 |
| [41] | Yao, F., Regularity theory in Orlicz spaces for the parabolic polyharmonic equations, Arch. Math. (Basel), 90, 429-439 (2008) · Zbl 1221.35186 |
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