August  2022, 15(4): 535-550. doi: 10.3934/krm.2021037

Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $

Indiana University, Department of Mathematics, Bloomington, IN, USA

Bob Glassey and I often discussed the pedagogy of applied analysis, agreeing in particular that elementary facts should have elementary proofs. This work is offered in that spirit and in his memory

Received  June 2021 Revised  October 2021 Published  August 2022 Early access  November 2021

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.

Citation: David Hoff. Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $. Kinetic and Related Models, 2022, 15 (4) : 535-550. doi: 10.3934/krm.2021037
References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.

[2]

S. AgmonA. 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. AgmonA. 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. AgmonA. 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. AgmonA. 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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