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Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $
Indiana University, Department of Mathematics, Bloomington, IN, USA |
We derive pointwise bounds for the Green's function and its derivatives for the Laplace operator on smooth bounded sets in $ {\bf R}^3 $ subject to Neumann boundary conditions. The proofs require only ordinary calculus, scaling arguments and the most basic facts of $ L^2 $-Sobolev space theory.
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[3] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[4] |
B. E. J. Dahlberg and C. E. Kenig,
Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.
doi: 10.2307/1971407. |
[5] |
O. Druet, F. Robert and J. Wei, The Lin-Ni's Problem for Mean Convex Domains, Mem. Amer. Math. Soc., 2012.
doi: 10.1090/S0065-9266-2011-00646-5. |
[6] |
L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. |
[7] |
D. Gilbarg and N. S. Trudinger,, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[8] |
D. Hoff,, Linear and Quasilinear Parabolic Systems, Mathematical Surveys and Monographs, 251, American Mathematical Society, Providence, RI, 2020. |
[9] |
D. Hoff,
Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech., 7 (2005), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[10] |
C. E. Kenig and J. Pipher,
The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math., 113 (1993), 447-509.
doi: 10.1007/BF01244315. |
[11] |
F. Robert, Construction and asymptotics for the Green Os function with Neumann boundary conditions, Informal Notes, 2010, available at https://iecl.univ-lorraine.fr/files/2021/04/NotesGreenNeumannRobert.pdf. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[3] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[4] |
B. E. J. Dahlberg and C. E. Kenig,
Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.
doi: 10.2307/1971407. |
[5] |
O. Druet, F. Robert and J. Wei, The Lin-Ni's Problem for Mean Convex Domains, Mem. Amer. Math. Soc., 2012.
doi: 10.1090/S0065-9266-2011-00646-5. |
[6] |
L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. |
[7] |
D. Gilbarg and N. S. Trudinger,, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[8] |
D. Hoff,, Linear and Quasilinear Parabolic Systems, Mathematical Surveys and Monographs, 251, American Mathematical Society, Providence, RI, 2020. |
[9] |
D. Hoff,
Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech., 7 (2005), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[10] |
C. E. Kenig and J. Pipher,
The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math., 113 (1993), 447-509.
doi: 10.1007/BF01244315. |
[11] |
F. Robert, Construction and asymptotics for the Green Os function with Neumann boundary conditions, Informal Notes, 2010, available at https://iecl.univ-lorraine.fr/files/2021/04/NotesGreenNeumannRobert.pdf. |
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