Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain

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October 2019, 39(10): 6039-6067. doi: 10.3934/dcds.2019264

Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain

School of Mathematics, Sun Yat-sen University, No.135 Xingangxi Road, Haizhu District, Guangzhou 510275, China

^* Corresponding author: Mei Ming

Received January 2019 Revised March 2019 Published July 2019

Fund Project: The author is supported by NSFC grant 11401598.

Based on the $ H^2 $ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $ \omega\in(0,\pi/2) $. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $ \omega\in(0,\pi) $.

Keywords: Weighted elliptic estimates, corner domain, mixed boundary problem, Dirichlet-Neumann operator.

Mathematics Subject Classification: Primary: 35Q31, 35J25; Secondary: 35Q35.

Citation: Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

References:

[1]	J. Banasiak and G. F. Roach, On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary, Journal of differential equations, 79 (1989), 111-131. doi: 10.1016/0022-0396(89)90116-2.
[2]	M. Sh. Birman and G. E. Skvortsov, On the quadratic integrability of the highest derivatives of the Dirichlet problem in a domain with piecewis smooth boundary, Izv. Vyssh. Uchebn. Zaved. Mat., 1962 (1962), 11–21 (in Russsian).
[3]	M. Borsuk and V. A. Kondrat'ev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North-Holland Mathematical Library, 69. Elsevier Science B.V., Amsterdam, 2006. doi: 10.1016/S0924-6509(06)80026-7.
[4]	M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems, Ⅰ, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 109-155. doi: 10.1017/S0308210500021272.
[5]	M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems Ⅱ., Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 157-184. doi: 10.1017/S0308210500021272.
[6]	M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341. Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0086682.
[7]	M. Dauge, S. Nicaise, M. Bourlard and M. S. Lubuma, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques Ⅰ: résultats généraux pour le problème de Dirichlet, Mathematical Modelling and Numerical Analysis, 24 (1990), 27-52. doi: 10.1051/m2an/1990240100271.
[8]	G. I. Eskin, General boundary values problems for equations of principle type in a plane domain with angular points, Uspekhi Mat. Nauk, 18 (1963), 241–242 (in Russian).
[9]	P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985.
[10]	P. Grisvard, Singularities in Boundary Value Problems, Research notes in applied mathematics, Springer-Verlag, 1992.
[11]	V. A. Kondrat'ev, Boundary Value Problems for Elliptic Equations in Conical Regions, , Soviet Math. Dokl., 1963.
[12]	V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.
[13]	V. A. Kondart'ev and O. A. Oleinik, Boundary value problems for partial differential equations in nonsmooth domains, Russian Math. Surveys, 38 (1983), 3-76.
[14]	V. A. Kozlov, V. G. Mazya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, 52, American Mathematical Society, Providence, RI, 1997.
[15]	D. Lannes, Well-posedness of the water-wave equations, Journal of the American Math. Society, 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.
[16]	Ya. B. Lopatinskiy, On one type of singular integral equations, Teoret. i Prikl. Mat. (Lvov), 2 (1963), 53–57 (in Russsian).
[17]	V. G. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, 162, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162.
[18]	V. G. Maz'ya, The solvability of the Dirichlet problem for a region with a smooth irregular boundary, Vestnik Leningrad. Univ., 19 (1964), 163–165 (in Russian).
[19]	V. G. Maz'ya, The behavior near the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergence form, Mat. Zametki, 2 (1967), 209–220 (in Russian).
[20]	V. G. Maz'ya and B. A. Plamenevskiy, On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr., 76 (1977), 29-60. doi: 10.1002/mana.19770760103.
[21]	V. G. Maz'ya and B. A. Plamenevskiy, $L^p$ estimates of solutions of elliptic boundary value problems in a domains with edges, Trans. Moscow Math. Soc., 1 (1980), 49-97.
[22]	V. G. Maz'ya and B. A. Plamenevskiy, Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone, Journal of Soviet Mathematics, 9 (1978), 750-764. doi: 10.1007/BF01085326.
[23]	V. G. Maz'ya and J. Rossmann, On a problem of Babu$\breve {\rm{s}}$ka (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points), Math. Nachr., 155 (1992), 199-220. doi: 10.1002/mana.19921550115.
[24]	M. Ming and C. Wang, Elliptic estimates for Dirichlet-Neumann operator on a corner domain., Asymptotic Analysis, 104 (2017), 103-166. doi: 10.3233/ASY-171427.
[25]	M. Ming and C. Wang, Water waves problem with surface tension in a corner domain Ⅰ: A priori estimates with constrained contact angle, preprint, arXiv: 1709.00180.

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References:

[1]	J. Banasiak and G. F. Roach, On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary, Journal of differential equations, 79 (1989), 111-131. doi: 10.1016/0022-0396(89)90116-2.
[2]	M. Sh. Birman and G. E. Skvortsov, On the quadratic integrability of the highest derivatives of the Dirichlet problem in a domain with piecewis smooth boundary, Izv. Vyssh. Uchebn. Zaved. Mat., 1962 (1962), 11–21 (in Russsian).
[3]	M. Borsuk and V. A. Kondrat'ev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North-Holland Mathematical Library, 69. Elsevier Science B.V., Amsterdam, 2006. doi: 10.1016/S0924-6509(06)80026-7.
[4]	M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems, Ⅰ, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 109-155. doi: 10.1017/S0308210500021272.
[5]	M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems Ⅱ., Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 157-184. doi: 10.1017/S0308210500021272.
[6]	M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341. Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0086682.
[7]	M. Dauge, S. Nicaise, M. Bourlard and M. S. Lubuma, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques Ⅰ: résultats généraux pour le problème de Dirichlet, Mathematical Modelling and Numerical Analysis, 24 (1990), 27-52. doi: 10.1051/m2an/1990240100271.
[8]	G. I. Eskin, General boundary values problems for equations of principle type in a plane domain with angular points, Uspekhi Mat. Nauk, 18 (1963), 241–242 (in Russian).
[9]	P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985.
[10]	P. Grisvard, Singularities in Boundary Value Problems, Research notes in applied mathematics, Springer-Verlag, 1992.
[11]	V. A. Kondrat'ev, Boundary Value Problems for Elliptic Equations in Conical Regions, , Soviet Math. Dokl., 1963.
[12]	V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.
[13]	V. A. Kondart'ev and O. A. Oleinik, Boundary value problems for partial differential equations in nonsmooth domains, Russian Math. Surveys, 38 (1983), 3-76.
[14]	V. A. Kozlov, V. G. Mazya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, 52, American Mathematical Society, Providence, RI, 1997.
[15]	D. Lannes, Well-posedness of the water-wave equations, Journal of the American Math. Society, 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.
[16]	Ya. B. Lopatinskiy, On one type of singular integral equations, Teoret. i Prikl. Mat. (Lvov), 2 (1963), 53–57 (in Russsian).
[17]	V. G. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, 162, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162.
[18]	V. G. Maz'ya, The solvability of the Dirichlet problem for a region with a smooth irregular boundary, Vestnik Leningrad. Univ., 19 (1964), 163–165 (in Russian).
[19]	V. G. Maz'ya, The behavior near the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergence form, Mat. Zametki, 2 (1967), 209–220 (in Russian).
[20]	V. G. Maz'ya and B. A. Plamenevskiy, On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr., 76 (1977), 29-60. doi: 10.1002/mana.19770760103.
[21]	V. G. Maz'ya and B. A. Plamenevskiy, $L^p$ estimates of solutions of elliptic boundary value problems in a domains with edges, Trans. Moscow Math. Soc., 1 (1980), 49-97.
[22]	V. G. Maz'ya and B. A. Plamenevskiy, Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone, Journal of Soviet Mathematics, 9 (1978), 750-764. doi: 10.1007/BF01085326.
[23]	V. G. Maz'ya and J. Rossmann, On a problem of Babu$\breve {\rm{s}}$ka (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points), Math. Nachr., 155 (1992), 199-220. doi: 10.1002/mana.19921550115.
[24]	M. Ming and C. Wang, Elliptic estimates for Dirichlet-Neumann operator on a corner domain., Asymptotic Analysis, 104 (2017), 103-166. doi: 10.3233/ASY-171427.
[25]	M. Ming and C. Wang, Water waves problem with surface tension in a corner domain Ⅰ: A priori estimates with constrained contact angle, preprint, arXiv: 1709.00180.

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