American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 333-339. doi: 10.3934/proc.2009.2009.333

$H^2$-solutions for some elliptic equations with nonlinear boundary conditions

Junichi Harada 1, and  Mitsuharu Ôtani 2,

1. 

Major in Pure and Applied Physics, Graduate School of Advanced Sciences and Engineering, Waseda University, Japan
2. 

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555

Received  August 2008 Revised  June 2009 Published  September 2009

The following elliptic equation with nonlinear boundary condition is considered: $-\Delta u+bu=f(x)$ in $\Omega$, $-\frac{\partial u}{\partial n}=\beta(u)-g(u)$ on $\partial\Omega$, where $b\geq0$, $f\in L^2(\Omega)$, $\beta(u)$ is a monotone increasing function on $\mathbb (R)^1$ and $g(u)$ is its small perturbation. It is shown that this problem admits a solution $u$ belonging to $H^2(\Omega)$ under suitable conditions on $\beta$ and $g$. The method of our proof relies on some approximation procedures and the classical but new arguments for $H^2$-estimates near the boundary which can work under (non-monotone) nonlinear boundary conditions.
Keywords: Nonlinear boundary condition, $H^2$-estimates.
Mathematics Subject Classification: Primary: 35A.
Citation: Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333
[1]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[2]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[3]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[4]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253
[5]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393
[6]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[7]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[8]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865
[9]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[10]

Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785
[11]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623
[12]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[13]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
[14]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075
[15]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[16]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
[17]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[18]

Michela Eleuteri. An existence result for a P.D.E. with hysteresis, convection and a nonlinear boundary condition. Conference Publications, 2007, 2007 (Special) : 344-353. doi: 10.3934/proc.2007.2007.344
[19]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[20]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878

 Impact Factor: 

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences