Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T06:31:43.963Z Has data issue: false hasContentIssue false
  • English
  • Français

EXISTENCE AND UNIQUENESS OF WEAK AND CLASSICAL SOLUTIONS FOR A FOURTH-ORDER SEMILINEAR BOUNDARY VALUE PROBLEM

Part of: Thin bodies, structures Elliptic equations and systems

Published online by Cambridge University Press:  19 August 2019

CRISTIAN-PAUL DANET Open the ORCID record for CRISTIAN-PAUL DANET [Opens in a new window]
CRISTIAN-PAUL DANET*
Affiliation:
Department of Applied Mathematics, University of Craiova, Al. I. Cuza St., 13, 200585 Craiova, Romania email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.

Keywords

fourth-ordermaximum principleplate theoryPaneitz–Branson operator

MSC classification

Primary: 35J40: Boundary value problems for higher-order elliptic equations
Secondary: 35J35: Variational methods for higher-order elliptic equations 74K20: Plates
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 3 , July 2019 , pp. 305 - 319
Copyright
© 2019 Australian Mathematical Society 

References

Adams, R. A. and Fournier, J., Sobolev spaces, 2nd edn, Volume 140 of Pure and Applied Mathematics Series (Academic Press, Boston, MA, 2003); https://www.elsevier.com/books/sobolev-spaces/adams/978-0-12-044143-3.Google Scholar
Bonheure, D., “Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity”, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 319–340; doi:10.1016/j.anihpc.2003.03.001.Google Scholar
Danet, C.-P., “Uniqueness in some higher order elliptic boundary value problems in $n$ dimensional domains”, Electron. J. Qual. Theory Differ. Equ. 54 (2011) 1–12; doi:10.14232/ejqtde.2011.1.54.Google Scholar
Danet, C.-P., The classical maximum principle. Some of its extensions and applications (Lambert Academic Publishing, Saarbrcken, Germany, 2013); https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-40556-3/the-classical-maximum-principle-some-extensions-and-applications.Google Scholar
Danet, C.-P., “On a hinged plate equation of nonconstant thickness”, Differ. Equ. Appl. 10 (2018) 235–238; doi:10.7153/dea-2018-10-16.Google Scholar
Danet, C.-P. and Mareno, A., “Maximum principles for a class of linear equations of even order”, Math. Inequal. Appl. 16 (2013) 809–822; doi:10.7153/mia-16-61.Google Scholar
Djadli, Z., Malchiodi, A. and Ahmedou, M., “Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2002) 387–434; https://www.academia.edu/2086163/Prescribing_a_fourth_order_conformal_invariant_on_the_standard_sphere_Part_II_blow-up_analysis_and_applications.Google Scholar
Gazolla, F., Grunau, H.-C. and Sweers, G., Polyharmonic boundary value problems (Springer, Berlin, Heidelberg, 2010); https://www.springer.com/gp/book/9783642122446.Google Scholar
Gelfand, I. M., “Some problems in the theory of quasilinear equations”, Amer. Math. Soc. Transl. 29(2) (1963) 295–381; doi:10.1007/978-3-642-61705-8.Google Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer, Berlin, Heidelberg, 2001); https://www.springer.com/gp/book/9783540411604.Google Scholar
Goyal, V. B., “Liouville-type results for fourth order elliptic equations”, Proc. Roy. Soc. Edinburgh 103A (1986) 209–213; doi:10.1017/S0308210500018862.Google Scholar
Goyal, V. B. and Schaefer, P. W., “On a subharmonic functional in fourth order nonlinear elliptic problems”, J. Math. Anal. Appl. 83 (1981) 20–25; doi:10.1016/0022-247X(81)90243-2.Google Scholar
Hunt, G. W., Bolt, H. M. and Thompson, J. M. T., “Structural localization phenomena and the dynamical phase-space analogy”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 425 (1989) 245–267; doi:10.1098/rspa.1989.0105.Google Scholar
Karageorgis, P. and Stalker, J., “A lower bound for the amplitude of traveling waves of suspension bridges”, Nonlinear Anal. 75 (2012) 5212–5214; doi:10.1016/j.na.2012.04.037.Google Scholar
Kawohl, B. and Sweers, G., “On the differential equation $u_{xxxx}+u_{yyyy}=f$ for an anisotropic stiff material”, SIAM J. Math. Anal. 37 (2006) 1828–1853; doi:10.1137/050624704.Google Scholar
Komkov, V., “Certain estimates for solutions of nonlinear elliptic differential equations applicable to the theory of thin plates”, SIAM J. Appl. Math. 28 (1975) 24–34; doi:10.1137/0128003.Google Scholar
Ladyzhenskaya, O. A., The mathematical theory of viscous incompressible flow (Gordon and Breach Science Publishers, New York, 1969); https://www.worldcat.org/title/mathematical-theory-of-viscous-incompressible-flows/oclc/639875838.Google Scholar
Lazer, A. C. and McKenna, P. J., “On traveling waves in a suspension bridge model as the wave speed goes to zero”, Nonlinear Anal. 74 (2011) 3998–4001; doi:10.1016/j.na.2011.03.024.Google Scholar
Mareno, A., “Maximum principles for some higher-order semilinear elliptic equations”, Glasg. Math. J. 53 (2010) 313–320; doi:10.1017/S001708951000073X.Google Scholar
Mareno, A., “A maximum principle result for a general fourth order semilinear elliptic equation”, J. Appl. Math. Phys. 4 (2016) 1682–1686; doi:10.4236/jamp.2016.48176.Google Scholar
Miranda, C., “Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche di due variabili”, Giorn. Mat. Battaglini 78 (1948) 97–118.Google Scholar
Payne, L. E., “Some remarks on maximum principles”, J. Anal. Math. 30 (1976) 421–433; doi:10.1007/BF02786729.Google Scholar
Peletier, L. A. and Troy, W. C., Spatial patterns. Higher order models in physics and mechanics, Volume 45 of Progress in Nonlinear Differential Equations and their Applications (Birkhäuser Boston Inc., Boston, MA, 2001); https://www.springer.com/gp/book/9780817641108.Google Scholar
Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Prentice-Hall Inc., Englewood Cliffs, NJ, 1967); https://www.springer.com/gp/book/9780387960685.Google Scholar
Schaefer, P. W., “On a maximum principle for a class of fourth-order semilinear elliptic equations”, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 319–323; doi:10.1017/S0308210500025233.Google Scholar
Sperb, R. P., Maximum principles and their applications (Academic Press, New York, 1981); https://www.elsevier.com/books/maximum-principles-and-their-applications/sperb/978-0-12-656880-6.Google Scholar
Tseng, S. and Lin, C.-S., “On a subharmonic functional of some even order elliptic problems”, J. Math. Anal. Appl. 207 (1997) 127–157; doi:10.1006/jmaa.1997.5272.Google Scholar
Zhang, H. and Zhang, W., “Maximum principles and bounds in a class of fourth-order uniformly elliptic equations”, J. Phys. A: Math. Gen. 35 (2002) 9245–9250; doi:10.1088/0305-4470/35/43/318.Google Scholar