# American Institute of Mathematical Sciences

• Previous Article
Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions
• DCDS-B Home
• This Issue
• Next Article
Averaging of ordinary differential equations with slowly varying averages
September  2010, 14(2): 327-351. doi: 10.3934/dcdsb.2010.14.327

## Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation

Received  September 2009 Revised  January 2010 Published  June 2010

We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations $\Delta u+f(x,u)=0$ in $\Omega$ ε with nonlinear boundary conditions of type $\frac{\partial u}{\partial n}+g(x,u)=0$, when the boundary of the domain varies very rapidly. We show that if the oscillations are very rapid, in the sense that, roughly speaking, its period is much smaller than its amplitude and the function $g$ is of a dissipative type, that is, it satisfies $g(x,u)u\geq b|u|^{d+1}$, then the boundary condition in the limit problem is $u=0$, that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in $H^1$ and $C^0$ norms and the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. Moreover, if a solution of the limit problem is hyperbolic (non degenerate) and some extra conditions in $g$ are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.
Citation: José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
 [1] Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713 [2] Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 [3] 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 [4] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [5] Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030 [6] Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593 [7] 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 [8] Giuseppe Da Prato, Alessandra Lunardi. On a class of elliptic and parabolic equations in convex domains without boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 933-953. doi: 10.3934/dcds.2008.22.933 [9] Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations & Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61 [10] Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493 [11] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [12] José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4217-4246. doi: 10.3934/dcdsb.2019079 [13] Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761 [14] Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007 [15] Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559 [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] Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144 [18] Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot, Ahmed Medeghri. Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 523-538. doi: 10.3934/dcdss.2011.4.523 [19] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [20] Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637

2018 Impact Factor: 1.008