doi: 10.3934/krm.2021037
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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

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 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 & Related Models, doi: 10.3934/krm.2021037
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[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.  Google Scholar

[8]

D. Hoff,, Linear and Quasilinear Parabolic Systems, Mathematical Surveys and Monographs, 251, American Mathematical Society, Providence, RI, 2020.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[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.  Google Scholar

[8]

D. Hoff,, Linear and Quasilinear Parabolic Systems, Mathematical Surveys and Monographs, 251, American Mathematical Society, Providence, RI, 2020.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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