American Institute of Mathematical Sciences

Advanced
November  2008, 21(4): 1047-1069. doi: 10.3934/dcds.2008.21.1047

Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach

Mariane Bourgoing 1,

1. 

Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Fédération de Recherche Denis Poisson (FR 2964), Université François Rabelais, Tours. Parc de Grandmont, 37200 Tours

Received  June 2007 Revised  January 2008 Published  May 2008

In this article, we continue the study of viscosity solutions for second-order fully nonlinear parabolic equations, having a $L^1$ dependence in time, associated with nonlinear Neumann boundary conditions, which started in a previous paper (cf [2]). First, we obtain the existence of continuous viscosity solutions by adapting Perron's method and using the comparison results obtained in [2]. Then, we apply these existence and comparison results to the study of the level-set approach for front propagations problems when the normal velocity has a $L^1$-dependence in time.
Keywords: level-set approach, Fully nonlinear second-order parabolic equations, $L^1$ dependence in time, viscosity solutions., existence, Neumann boundary conditions.
Mathematics Subject Classification: Primary: 35K60, 49L25 ; Secondary: 35B0.
Citation: Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[1]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[2]

Anca Croitoru, Gabriela Tănase. On a nonlocal and nonlinear second-order anisotropic reaction-diffusion model with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022155
[3]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273
[4]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053
[5]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596
[6]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[7]

Wen Cheng, Anna L. Mazzucato, Victor Nistor. Approximate solutions to second-order parabolic equations: Evolution systems and discretization. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022158
[8]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[9]

Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3983-3999. doi: 10.3934/dcdsb.2019228
[10]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control and Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[11]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
[12]

Rosaria Di Nardo. Nonlinear parabolic equations with a lower order term and $L^1$ data. Communications on Pure and Applied Analysis, 2010, 9 (4) : 929-942. doi: 10.3934/cpaa.2010.9.929
[13]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[14]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155
[15]

P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[16]

Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[17]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems and Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070
[18]

Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems and Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137
[19]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[20]

Debdip Ganguly, Yehuda Pinchover, Prasun Roychowdhury. Stochastic completeness and $ L^1 $-Liouville property for second-order elliptic operators. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022138

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy