-
Previous Article
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators
- CPAA Home
- This Issue
-
Next Article
Dynamics of a nonlocal dispersal SIS epidemic model
W1, p-estimates for elliptic equations with lower order terms
Department of Mathematics, Sogang University Seoul, 121-742, Korea |
$ - {\rm{div}}{\mkern 1mu} (A\nabla u) - b \cdot \nabla u + \lambda u = f + {\rm{div}}{\mkern 1mu} F,\quad - {\rm{div}}{\mkern 1mu} ({A^t}\nabla v) + {\rm{div}}{\mkern 1mu} (vb) + \lambda v = g + {\rm{div}}{\mkern 1mu} G$ |
References:
[1] |
P. Auscher and M. Qafaoui,
Observations on W1, p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2002), 487-509.
|
[2] |
S. Byun and L. Wang,
Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.
doi: 10.1002/cpa.20037. |
[3] |
S. Byun,
Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc.,, 357 (2005), 1025-1046.
doi: 10.1090/S0002-9947-04-03624-4. |
[4] |
S. Byun and L. Wang,
The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc., 90 (2005), 245-272.
doi: 10.1112/S0024611504014960. |
[5] |
H. Dong and D. Kim,
Elliptic equations in divergence form with partially BMO coefficients, Arich. Ration. Mech. Anal., 196 (2010), 25-70.
doi: 10.1007/s00205-009-0228-7. |
[6] |
H. Dong and D. Kim,
On the Lp-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arich. Ration. Mech. Anal., 199 (2011), 889-941.
doi: 10.1007/s00205-010-0345-3. |
[7] |
H. Dong and D. Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, Recent Advances in Harmonic Analysis and Partial Differential Equations, 69–97, Contemp. Math. , 581, Amer. Math. Soc. , Providence, RI, 2012. |
[8] |
J. Droniou,
Non-coercive linear elliptic problems, Potential Anal., 17 (2002), 181-203.
doi: 10.1023/A:1015709329011. |
[9] |
J. Droniou and J. Vázquez,
Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.
doi: 10.1007/s00526-008-0189-y. |
[10] |
E. Fabes, O. Mendez and M. Mitrea,
Boundary layers on Sobolev-Besov spaces and Poisson's equation fo the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368.
doi: 10.1006/jfan.1998.3316. |
[11] |
J. Geng,
W1, p estimate for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448.
doi: 10.1016/j.aim.2012.01.004. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, 2001. |
[13] |
D. Jerison and C. Kenig,
The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.
doi: 10.1006/jfan.1995.1067. |
[14] |
H. Kim and Y. Kim,
On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290.
doi: 10.1137/14096270X. |
[15] |
G. Moscariello,
Existence and uniqueness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444.
doi: 10.1515/ACV.2011.007. |
[16] |
Z. Shen, Bounds of Riesz transform on Lp spaces for second order elliptic operators, Ann. Inst. Fourier(Grenoble), 55 (2005), 173–197. |
show all references
References:
[1] |
P. Auscher and M. Qafaoui,
Observations on W1, p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2002), 487-509.
|
[2] |
S. Byun and L. Wang,
Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.
doi: 10.1002/cpa.20037. |
[3] |
S. Byun,
Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc.,, 357 (2005), 1025-1046.
doi: 10.1090/S0002-9947-04-03624-4. |
[4] |
S. Byun and L. Wang,
The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc., 90 (2005), 245-272.
doi: 10.1112/S0024611504014960. |
[5] |
H. Dong and D. Kim,
Elliptic equations in divergence form with partially BMO coefficients, Arich. Ration. Mech. Anal., 196 (2010), 25-70.
doi: 10.1007/s00205-009-0228-7. |
[6] |
H. Dong and D. Kim,
On the Lp-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arich. Ration. Mech. Anal., 199 (2011), 889-941.
doi: 10.1007/s00205-010-0345-3. |
[7] |
H. Dong and D. Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, Recent Advances in Harmonic Analysis and Partial Differential Equations, 69–97, Contemp. Math. , 581, Amer. Math. Soc. , Providence, RI, 2012. |
[8] |
J. Droniou,
Non-coercive linear elliptic problems, Potential Anal., 17 (2002), 181-203.
doi: 10.1023/A:1015709329011. |
[9] |
J. Droniou and J. Vázquez,
Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.
doi: 10.1007/s00526-008-0189-y. |
[10] |
E. Fabes, O. Mendez and M. Mitrea,
Boundary layers on Sobolev-Besov spaces and Poisson's equation fo the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368.
doi: 10.1006/jfan.1998.3316. |
[11] |
J. Geng,
W1, p estimate for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448.
doi: 10.1016/j.aim.2012.01.004. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, 2001. |
[13] |
D. Jerison and C. Kenig,
The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.
doi: 10.1006/jfan.1995.1067. |
[14] |
H. Kim and Y. Kim,
On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290.
doi: 10.1137/14096270X. |
[15] |
G. Moscariello,
Existence and uniqueness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444.
doi: 10.1515/ACV.2011.007. |
[16] |
Z. Shen, Bounds of Riesz transform on Lp spaces for second order elliptic operators, Ann. Inst. Fourier(Grenoble), 55 (2005), 173–197. |
[1] |
N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119 |
[2] |
Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012 |
[3] |
Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure and Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163 |
[4] |
Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 |
[5] |
Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 |
[6] |
Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263 |
[7] |
Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 |
[8] |
John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276 |
[9] |
John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337 |
[10] |
John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291 |
[11] |
John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89 |
[12] |
John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269 |
[13] |
Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure and Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587 |
[14] |
Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989 |
[15] |
Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975 |
[16] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 |
[17] |
Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507 |
[18] |
Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations and Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 |
[19] |
Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure and Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785 |
[20] |
Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]