Abstract
Let Ω be a domain in R n with compact complement and let T be a quasilinear elliptic or degenerate elliptic operator associated with functions u ∈C 2(Ω). This paper is a study of solutions of (sgn u)Tu ≧ f (|u|, |grad u|) where f belongs to a class of functions here termed bifurcation functions. The main condition on f is that uniqueness fails for the ordinary differential equation y″ = f(y, y´) with the initial condition y(0) = y´(0) = 0. The conclusion is that u is constant for large |x| and hence, under mild supplementary hypotheses, u has compact support. Examples show that the results fail if the assumptions on f are only slightly weakened, so that the class of f is, essentially, the largest class for which the results can be stated truly.
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References
Antoncev, S. N., On the localization of solutions of nonlinear degenerate elliptic and parabolic equations, Soviet. Math. Dokl. 260, 1981, 6; translation ibid. 24, 1981, 2, 420–424.
Beénilan, Phillippe, H. Brézis & M. Crandall, A semilinear equation in LI (R“), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, 1975, 523–555.
Bensoussan, Alain, Haim Brézis & Avner Friedman, Estimates on the free boundary for quasivariational inequalities, Comm. PDE 2, 1977, 297–321.
Bensoussan, Alain, & Avner Friedman, On the support of the solution of a system of quasi-variational inequalities, J. Math. Anal. App. 65, 1978, 660–674.
Berkowitz, L. D., & Harry Pollard, A nonclassical variational problem arising from an optimal filter problem, Arch. Rational Mech. Anal. 26, 1967, 281–304.
Berkowitz, L. D., & Harry Pollard, A nonclassical variational problem arising from an optimal filter problem, II, ibid. 38, 1970, 161–172.
Bernis, F., Compactness of support for some nonlinear elliptic problems of arbitrary order in dimension n, Comm. PDE 9, 1984, 271–312.
Brézis, Haim, Solutions with compact support of variational inequalities, Uspehi Mat. Nauk 29 (1974), 103–108 = Russian Math. Surveys 29 (1974) 103–108.
Diaz, J. I., Nonlinear partial differential equations and free boundaries, Vol. I, Elliptic equations, Pitman Advanced Publishing Program, Boston, London Melbourne, Research Note 106, 1985.
Diaz, Gregoris, & Ildefonso Diaz, Finite extinction time for a class of nonlinear parabolic equations, Comm. PDE 4, 1979, 1213–1231.
Dfaz, J. Ildefonso, & M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh 89 A, 1981, 249–258.
Diaz, J. Ildefonso, & L. Veron, Compacité du support des solutions d’équations quasilinéares elliptiques ou paraboliques, C. R. Acad. Sci. Paris 297, 1983, 149–152.
Evans, Lawrence C., & Barry F. Knerr, Instantaneous shrinking of the support of non-negative solutions to certain nonlinear parabolic equations and variational inequalities, Ill. J. Math. 23, 1979, 153–166.
Herero, M. A., & J. L. Vazquez, Asymptotic behavior of solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3, 1981, 113–127.
Hestenes, Magnus, & Ray Redheffer, On the minimization of certain quadratic functionals, I, Arch. Rational Mech. and Anal. 56, 1974, 1–14.
Hestenes, W. Magnus, & Ray Redheffer, On the minimization of certain quadratic functionals, II, ibid.,15–33.
Johanssen, D. E., Solution of a linear mean square problem when process statistics are undefined, IEEE Trans. Aut. Control 11, 1966, 20–30.
Kalashnikov, A. S., The propagation of disturbances in problems of nonlinear heat conduction and absorption, USSR Comm. Math. and Math. Phys. 14, 1974, 70–85.
Meyers, Norman, & James Serrin, The exterior Dirichlet problem for second-order elliptic partial differential equations, J. Math. Mech. 9, 1960, 513–538.
Redheffer, Ray, On a nonlinear functional of Berkowitz and Pollard, Arch. Rational Mech. and Anal. 50, 1973, 1–9.
Redheffer, Ray, Nonlinear differential inequalities and functions of compact support, Trans. Amer. Math. Soc. 220, 1976, 133–157.
Redheffer, Ray, & Reinhard Redlinger, Quenching in time-delay systems: a summary and a counterexample, Siam J. Math. Anal 15, 1984, 1114–1124.
Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. Royal Soc. (London) 264, 1969, 413–496.
Wallach, Sylvan, The differential equation y’ = f(y), Am. J. Math. 70, 1948, 345–350.
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To James Serrin, in celebration of his 60th birthday
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© 1989 Springer-Verlag Berlin Heidelberg
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Redheffer, R. (1989). A Class of Quasilinear Differential Inequalities whose Solutions are Ultimately Constant. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_33
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DOI: https://doi.org/10.1007/978-3-642-83743-2_33
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