American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space
  • DCDS Home
  • This Issue
  • Next Article
    Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model
January  2008, 21(1): 21-39. doi: 10.3934/dcds.2008.21.21

Varying domains: Stability of the Dirichlet and the Poisson problem

Wolfgang Arendt 1, and  Daniel Daners 2,

1. 

Institute of Applied Analysis, University of Ulm, D-89069 Ulm, Germany
2. 

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Received  March 2007 Revised  October 2007 Published  February 2008

For $\Omega$ a bounded open set in $\R^N$ we consider the space $H^1_0(\bar{\Omega})=${$u_{|_{\Omega}}: u \in H^1(\R^N):$ $u(x)=0$ a.e. outside $\bar{\Omega}$}. The set $\Omega$ is called stable if $H^1_0(\Omega)=H^1_0(\bar{\Omega})$. Stability of $\Omega$ can be characterised by the convergence of the solutions of the Poisson equation

$ -\Delta u_n = f$ in $D(\Omega_n)^$´, $ u_n \in H^1_0(\Omega_n)$

and also the Dirichlet Problem with respect to $\Omega_n$ if $\Omega_n$ converges to $\Omega$ in a sense to be made precise. We give diverse results in this direction, all with purely analytical tools not referring to abstract potential theory as in Hedberg's survey article [Expo. Math. 11 (1993), 193--259]. The most complete picture is obtained when $\Omega$ is supposed to be Dirichlet regular. However, stability does not imply Dirichlet regularity as Lebesgue's cusp shows.

Keywords: harmonic function., Dirichlet problem, stability, Poisson problem.
Mathematics Subject Classification: Primary: 35J05, Secondary: 31B0.
Citation: Wolfgang Arendt, Daniel Daners. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 21-39. doi: 10.3934/dcds.2008.21.21
[1]

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
[2]

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
[3]

C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267
[4]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[5]

E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731
[6]

V. V. Motreanu. Uniqueness results for a Dirichlet problem with variable exponent. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1399-1410. doi: 10.3934/cpaa.2010.9.1399
[7]

Isabeau Birindelli, Francoise Demengel. The dirichlet problem for singluar fully nonlinear operators. Conference Publications, 2007, 2007 (Special) : 110-121. doi: 10.3934/proc.2007.2007.110
[8]

Giuseppe Maria Coclite, Mario Michele Coclite. On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4923-4944. doi: 10.3934/dcds.2013.33.4923
[9]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[10]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[11]

Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[12]

Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics & Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004
[13]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[14]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[15]

Pierpaolo Esposito, Maristella Petralla. Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1935-1957. doi: 10.3934/cpaa.2012.11.1935
[16]

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
[17]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[18]

Salvatore A. Marano, Sunra J. N. Mosconi. Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 279-291. doi: 10.3934/dcdss.2018015
[19]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[20]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences