American Institute of Mathematical Sciences

Advanced
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

Mei Ming ,

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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[9]

P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985.  Google Scholar

[10]

P. Grisvard, Singularities in Boundary Value Problems, Research notes in applied mathematics, Springer-Verlag, 1992.  Google Scholar

[11]

V. A. Kondrat'ev, Boundary Value Problems for Elliptic Equations in Conical Regions, , Soviet Math. Dokl., 1963. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Ya. B. Lopatinskiy, On one type of singular integral equations, Teoret. i Prikl. Mat. (Lvov), 2 (1963), 53–57 (in Russsian).  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[9]

P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985.  Google Scholar

[10]

P. Grisvard, Singularities in Boundary Value Problems, Research notes in applied mathematics, Springer-Verlag, 1992.  Google Scholar

[11]

V. A. Kondrat'ev, Boundary Value Problems for Elliptic Equations in Conical Regions, , Soviet Math. Dokl., 1963. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Ya. B. Lopatinskiy, On one type of singular integral equations, Teoret. i Prikl. Mat. (Lvov), 2 (1963), 53–57 (in Russsian).  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[1]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[2]

Yu-Hao Liang, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1075-1105. doi: 10.3934/dcds.2020071
[3]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[4]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[5]

Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
[6]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[7]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
[8]

Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238
[9]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[10]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[11]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
[12]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[13]

Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041
[14]

Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469
[15]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
[16]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181
[17]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315
[18]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[19]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[20]

Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (77)
  • HTML views (97)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences