# American Institute of Mathematical Sciences

May  2017, 16(3): 799-822. doi: 10.3934/cpaa.2017038

## W1, p-estimates for elliptic equations with lower order terms

 Department of Mathematics, Sogang University Seoul, 121-742, Korea

* Corresponding author

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No. 2013R1A1A2007290).

We consider the Neumann and Dirichlet problems for second-order linear elliptic equations
 $- {\rm{div}}{\mkern 1mu} (A\nabla u) - b \cdot \nabla u + \lambda u = f + {\rm{div}}{\mkern 1mu} F,\quad - {\rm{div}}{\mkern 1mu} ({A^t}\nabla v) + {\rm{div}}{\mkern 1mu} (vb) + \lambda v = g + {\rm{div}}{\mkern 1mu} G$
in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, where $A: \mathbb{R}^n \to \mathbb{R}^{n^2}$, $b: \Omega \to \mathbb{R}^n$ and $\lambda \geq 0$ are given. Some $W^{1, 2}$-estimates have been already known, provided that $A \in L^\infty(\Omega)^{n^2}$ and $b \in L^r(\Omega)^n$, where $n \leq r < \infty$ if $n \geq 3$ and $2 < r < \infty$ if $n=2$. Under more regularity assumptions on $A$ and $\Omega$, we establish the existence and uniqueness of weak solutions satisfying $W^{1, p}$-estimates. Our $W^{1, p}$-estimates are uniform on $\lambda \geq 0$ for the case of the Dirichlet problems. For the Neumann problems, the $W^{1, p}$-estimates are uniform with respect to $\lambda \geq 0$ if $f$ and $g$ satisfy some compatibility conditions. These uniform estimates allow us to obtain strong stability results in $W^{1, p}$ with respect to $\lambda$ for the Neumann and Dirichlet problems.
Citation: Byungsoo Kang, Hyunseok Kim. W1, p-estimates for elliptic equations with lower order terms. Communications on Pure and Applied Analysis, 2017, 16 (3) : 799-822. doi: 10.3934/cpaa.2017038
##### 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.

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##### 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.
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