Computer-assisted proofs for radially symmetric solutions of PDEs

Previous Article
Balanced model order reduction for linear random dynamical systems driven by Lévy noise
JCD Home
This Issue
Next Article
The snapback repellers for chaos in multi-dimensional maps

June & December 2018, 5(1&2): 61-80. doi: 10.3934/jcd.2018003

Computer-assisted proofs for radially symmetric solutions of PDEs

István Balázs ^1, , Jan Bouwe van den Berg ^2, , Julien Courtois ^3, , János Dudás ^4, , Jean-Philippe Lessard ^5,, , Anett Vörös-Kiss ^4, , JF Williams ^6, and Xi Yuan Yin ^5,

1.	MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720
2.	VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
3.	Université de Montréal, Département de Mathématiques et de Statistique, Pavillon André-Aisenstadt, 2920 chemin de la Tour, Montreal, QC, H3T 1J4, Canada
4.	Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720
5.	McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
6.	Simon Fraser University, Department of Mathematics, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

* Corresponding author: Jean-Philippe Lessard

Fund Project: The first, fourth and sixth authors were supported by the Hungarian Scientific Research Fund (NKFIH-OTKA), Grant No. K109782. The second author was supported in part by NWO-Vici grant 639.033.109. The fifth and the seventh authors were supported by NSERC

We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

Keywords: Radially symmetric solutions, geometric elliptic PDEs on manifolds, computer-assisted proofs, Taylor series, contraction mapping theorem.

Mathematics Subject Classification: 58J05, 58C40, 65G40, 41A58, 58J20, 65M99, 65G40, 35K57.

Citation: István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003

References:

[1]	G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051. doi: 10.1007/s00205-010-0309-7. Google Scholar
[2]	G. Arioli and H. Koch, Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133. doi: 10.1137/10078298X. Google Scholar
[3]	G. Arioli and H. Koch, Validated numerical solutions for some semilinear elliptic equations on the disk, 2017. Preprint.Google Scholar
[4]	I. Balázs, J.B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, MATLAB code for "Computer-assisted proofs for radially symmetric solutions of PDEs", 2017, http://www.math.vu.nl/~janbouwe/code/radialpdes/.Google Scholar
[5]	A. N. Baltagiannis and K. E. Papadakis, Periodic solutions in the Sun—Jupiter—Trojan Asteroid—Spacecraft system, Planetary and Space Science, 75 (2013), 148-157. Google Scholar
[6]	J. F. Barros and E. S. G. Leandro, The set of degenerate central configurations in the planar restricted four-body problem, SIAM J. Math. Anal., 43 (2011), 634-661. doi: 10.1137/100789701. Google Scholar
[7]	J. F. Barros and E. S. G. Leandro, Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM J. Math. Anal., 46 (2014), 1185-1203. doi: 10.1137/130911342. Google Scholar
[8]	B. Breuer, J. Horák, P. J. McKenna and M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq., 224 (2006), 60-97. doi: 10.1016/j.jde.2005.07.016. Google Scholar
[9]	J. Burgos-García and M. Gidea, Hill's approximation in a restricted four-body problem, Celestial Mech. Dynam. Astronom., 122 (2015), 117-141. doi: 10.1007/s10569-015-9612-9. Google Scholar
[10]	CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/.Google Scholar
[11]	A. Castro and E. M. Fischer, Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729. doi: 10.4153/CMB-2015-056-7. Google Scholar
[12]	L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187. Google Scholar
[13]	L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414. doi: 10.1307/mmj/1028999194. Google Scholar
[14]	J.-L. Figueras, M. Gameiro, J.-P. Lessard and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1070-1088. doi: 10.1137/16M1073777. Google Scholar
[15]	A. Hungria, J.-P. Lessard and J. D. Mireles James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459. doi: 10.1090/mcom/3046. Google Scholar
[16]	H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604. doi: 10.1137/S0036144595284180. Google Scholar
[17]	O. E. Lanford, Ⅲ. A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434. doi: 10.1090/S0273-0979-1982-15008-X. Google Scholar
[18]	E. S. G. Leandro, On the central configurations of the planar restricted four-body problem, J. Diff. Eq., 226 (2006), 323-351. doi: 10.1016/j.jde.2005.10.015. Google Scholar
[19]	P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101. Google Scholar
[20]	S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Phys. D, 239 (2010), 1581-1592. doi: 10.1016/j.physd.2010.04.004. Google Scholar
[21]	R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716. Google Scholar
[22]	M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356. doi: 10.1081/NFA-100105107. Google Scholar
[23]	S. M. Rump, INTLAB - INTerval LABoratory, In Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. http://www.ti3.tu-harburg.de/rump/.Google Scholar
[24]	S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449. doi: 10.1017/S096249291000005X. Google Scholar
[25]	A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc., 165 (2003), ⅷ+86 pp. doi: 10.1090/memo/0786. Google Scholar
[26]	C. Simó, Relative equilibrium solutions in the four-body problem, Celestial Mech., 18 (1978), 165-184. doi: 10.1007/BF01228714. Google Scholar
[27]	J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 1977.Google Scholar
[28]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117. doi: 10.1007/s002080010018. Google Scholar
[29]	W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. Google Scholar
[30]	J. B. van den Berg, A. Deschênes, J.-P. Lessard and J. D. Mireles James, Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14 (2015), 942-979. doi: 10.1137/140984506. Google Scholar
[31]	J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061. doi: 10.1090/noti1276. Google Scholar
[32]	J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584. doi: 10.1090/S0025-5718-10-02325-2. Google Scholar
[33]	M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189-R239. doi: 10.1088/1361-6544/aabe4b. Google Scholar
[34]	N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013. doi: 10.1137/S0036142996304498. Google Scholar
[35]	P. Zgliczyński, Rigorous numerics for dissipative partial differential equations. Ⅱ. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math., 4 (2004), 157-185. doi: 10.1007/s10208-002-0080-8. Google Scholar
[36]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s002080010010. Google Scholar

show all references

References:

[1]	G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051. doi: 10.1007/s00205-010-0309-7. Google Scholar
[2]	G. Arioli and H. Koch, Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133. doi: 10.1137/10078298X. Google Scholar
[3]	G. Arioli and H. Koch, Validated numerical solutions for some semilinear elliptic equations on the disk, 2017. Preprint.Google Scholar
[4]	I. Balázs, J.B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, MATLAB code for "Computer-assisted proofs for radially symmetric solutions of PDEs", 2017, http://www.math.vu.nl/~janbouwe/code/radialpdes/.Google Scholar
[5]	A. N. Baltagiannis and K. E. Papadakis, Periodic solutions in the Sun—Jupiter—Trojan Asteroid—Spacecraft system, Planetary and Space Science, 75 (2013), 148-157. Google Scholar
[6]	J. F. Barros and E. S. G. Leandro, The set of degenerate central configurations in the planar restricted four-body problem, SIAM J. Math. Anal., 43 (2011), 634-661. doi: 10.1137/100789701. Google Scholar
[7]	J. F. Barros and E. S. G. Leandro, Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM J. Math. Anal., 46 (2014), 1185-1203. doi: 10.1137/130911342. Google Scholar
[8]	B. Breuer, J. Horák, P. J. McKenna and M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq., 224 (2006), 60-97. doi: 10.1016/j.jde.2005.07.016. Google Scholar
[9]	J. Burgos-García and M. Gidea, Hill's approximation in a restricted four-body problem, Celestial Mech. Dynam. Astronom., 122 (2015), 117-141. doi: 10.1007/s10569-015-9612-9. Google Scholar
[10]	CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/.Google Scholar
[11]	A. Castro and E. M. Fischer, Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729. doi: 10.4153/CMB-2015-056-7. Google Scholar
[12]	L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187. Google Scholar
[13]	L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414. doi: 10.1307/mmj/1028999194. Google Scholar
[14]	J.-L. Figueras, M. Gameiro, J.-P. Lessard and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1070-1088. doi: 10.1137/16M1073777. Google Scholar
[15]	A. Hungria, J.-P. Lessard and J. D. Mireles James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459. doi: 10.1090/mcom/3046. Google Scholar
[16]	H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604. doi: 10.1137/S0036144595284180. Google Scholar
[17]	O. E. Lanford, Ⅲ. A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434. doi: 10.1090/S0273-0979-1982-15008-X. Google Scholar
[18]	E. S. G. Leandro, On the central configurations of the planar restricted four-body problem, J. Diff. Eq., 226 (2006), 323-351. doi: 10.1016/j.jde.2005.10.015. Google Scholar
[19]	P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101. Google Scholar
[20]	S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Phys. D, 239 (2010), 1581-1592. doi: 10.1016/j.physd.2010.04.004. Google Scholar
[21]	R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716. Google Scholar
[22]	M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356. doi: 10.1081/NFA-100105107. Google Scholar
[23]	S. M. Rump, INTLAB - INTerval LABoratory, In Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. http://www.ti3.tu-harburg.de/rump/.Google Scholar
[24]	S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449. doi: 10.1017/S096249291000005X. Google Scholar
[25]	A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc., 165 (2003), ⅷ+86 pp. doi: 10.1090/memo/0786. Google Scholar
[26]	C. Simó, Relative equilibrium solutions in the four-body problem, Celestial Mech., 18 (1978), 165-184. doi: 10.1007/BF01228714. Google Scholar
[27]	J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 1977.Google Scholar
[28]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117. doi: 10.1007/s002080010018. Google Scholar
[29]	W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. Google Scholar
[30]	J. B. van den Berg, A. Deschênes, J.-P. Lessard and J. D. Mireles James, Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14 (2015), 942-979. doi: 10.1137/140984506. Google Scholar
[31]	J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061. doi: 10.1090/noti1276. Google Scholar
[32]	J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584. doi: 10.1090/S0025-5718-10-02325-2. Google Scholar
[33]	M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189-R239. doi: 10.1088/1361-6544/aabe4b. Google Scholar
[34]	N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013. doi: 10.1137/S0036142996304498. Google Scholar
[35]	P. Zgliczyński, Rigorous numerics for dissipative partial differential equations. Ⅱ. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math., 4 (2004), 157-185. doi: 10.1007/s10208-002-0080-8. Google Scholar
[36]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s002080010010. Google Scholar

Figure 1. (Left) Ten relative equilibria of (CR4BP) with equal masses. (Right) Eight relative equilibria of (CR4BP) with masses $m_1 = 0.9987451087$, $m_2 = 0.0010170039$ and $m_3 = 0.0002378873$. In both plots, some level sets of the effective potential $\Omega$ are depicted.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. (Left) The first solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11). Since $r_{\min}<10^{-8}$, the true solution lies with the line-width by Theorem 2.1.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. The second solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. (Left) The third solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Six solutions of (22) for $\lambda \in \{118.2,120,250,350,450,500\}$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. (Left) A stationary solution of the Swift-Hohenberg equation (20) on the unit ball in $\mathbb{R}^3$ at $\lambda = 500$. (Right) The corresponding graph of $u(s) = u(\sqrt{x^2+y^2+z^2})$.

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164
[2]	A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721
[3]	Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045
[4]	Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677
[5]	Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
[6]	Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297
[7]	Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989
[8]	Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure & Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679
[9]	Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[10]	Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335
[11]	Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
[12]	Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427
[13]	Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587
[14]	Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
[15]	Thomas I. Vogel. Comments on radially symmetric liquid bridges with inflected profiles. Conference Publications, 2005, 2005 (Special) : 862-867. doi: 10.3934/proc.2005.2005.862
[16]	Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[17]	Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517
[18]	Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115
[19]	Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[20]	Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences