• Previous Article
    Non-autonomous weakly damped plate model on time-dependent domains
  • DCDS-S Home
  • This Issue
  • Next Article
    Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $
September  2021, 14(9): 3305-3318. doi: 10.3934/dcdss.2021080

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

1. 

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China

2. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

3. 

School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China

* Corresponding author: Shenzhou Zheng

Received  March 2020 Revised  January 2021 Published  September 2021 Early access  June 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 12001160), Natural Science Foundation of Hebei Province (No. A2019205218), Science Foundation of Hebei Normal University (No. L2019B02). The second author is supported by National Natural Science Foundation of China (No. 12071021)

We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.

Citation: Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080
References:
[1]

T. AlbericoC. CapozzoliR. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.

[2]

S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.

[3]

S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages.

[4]

S.-S. ByunM. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.

[5]

S.-S. ByunJ. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.

[6]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.

[7]

L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.

[8]

M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.

[9]

J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.

[10]

H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.

[11]

H. DongN. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.

[12]

L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.

[13]

M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.

[15]

N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.

[16]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.

[17]

D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.

[18]

S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.

[19]

P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.

[20]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.

[21]

C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.

[22]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.

[23]

N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.

[24]

J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.

[25]

J. ZhangM. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.

show all references

References:
[1]

T. AlbericoC. CapozzoliR. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.

[2]

S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.

[3]

S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages.

[4]

S.-S. ByunM. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.

[5]

S.-S. ByunJ. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.

[6]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.

[7]

L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.

[8]

M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.

[9]

J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.

[10]

H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.

[11]

H. DongN. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.

[12]

L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.

[13]

M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.

[15]

N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.

[16]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.

[17]

D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.

[18]

S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.

[19]

P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.

[20]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.

[21]

C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.

[22]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.

[23]

N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.

[24]

J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.

[25]

J. ZhangM. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.

[1]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675

[2]

K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091

[3]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627

[4]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[5]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497

[6]

Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881

[7]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[8]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

[9]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279

[10]

Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785

[11]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070

[12]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[13]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[14]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763

[15]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure and Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

[16]

Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211

[17]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

[18]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

[19]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[20]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (205)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]