Abstract
In this chapter we establish a maximum principle type result that provides pointwise control on minimal solutions. In contrast to the usual maximum principle, it does not hold for solutions in general, not even for local minimizers in the scalar case. We obtain it as a corollary of a replacement lemma modeled after Lemmas 2.4 and 2.5.
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References
Alikakos, N.D., Fusco, G.: A maximum principle for systems with variational structure and an application to standing waves. J. Eur. Math. Soc. 17(7), 1547–1567 (2015)
Antonopoulos, P., Smyrnelis, P.: A maximum principle for the system Δu −∇W(u) = 0. C. R. Acad. Sci. Paris Ser. I 354, 595–600 (2016)
Ball, J.M., Crooks, E.C.M.: Local minimizers and planar interfaces in a phase-transition model with interfacial energy. Calc. Var. Partial Differ. Equ. 40(3), 501–538 (2011)
Bates, S.M.: Toward a precise smoothness hypothesis in Sard’s theorem. Proc. Am. Math. Soc. 117(1), 279–283 (1993)
Boccardo, L., Ferone, V., Fusco, N., Orsina, L.: Regularity of minimizing sequences for functionals of the Calculus of Variations via the Ekeland principle. Differ. Integral Equ. 12(1), 119–135 (1999)
Casten, R.G., Holland, C.J.: Instability results for reaction-diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27, 266–273 (1978)
Czarnecki, A., Kulczychi, M., Lubawski, W.: On the connectedness of boundary and complement for domains. Ann. Pol. Math. 103, 189–191 (2011)
de Pascale, L.: The Morse-Sard theorem in Sobolev spaces. Indiana Univ. Math. J. 50(3), 1371–1386 (2001)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Evans, L.C.: A strong maximum principle for parabolic systems in a convex set with arbitrary boundary. Proc. Am. Math. Soc. 138(9), 3179–3185 (2010)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Kohn, R.V., Sternberg, P.: Local minimisers and singular perturbations. Proc. R. Soc. Edinb. Sect. A 111(1–2), 69–84 (1989)
Lions, P.L.: The concentration-compactness principle in the Calculus of Variations. The locally compact case, Part 1. Ann. I. H. Poincaré Anal. Nonlinear 1, 109–145 (1984)
Matano, H.: Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. 15(2), 401–454 (1979)
Modica, L., Mortola, S.: Un esempio di Γ-convergenza. Boll. Unione Mat. Ital. Sez B 14, 285–299 (1977)
Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. A Series of Modern Surveys in Mathematics, vol. 34, 4th edn. Springer, Berlin (2008)
Villegas, S.: Nonexistence of nonconstant global minimizers with limit at ∞ of semilinear elliptic equations in all of \({\mathbb R}^N\). Commun. Pure Appl. Anal. 10(6), 1817–1821 (2011)
Weinberger, H.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. di Matem. Ser. VI 8, 295–310 (1975)
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Alikakos, N.D., Fusco, G., Smyrnelis, P. (2018). The Cut-Off Lemma and a Maximum Principle. In: Elliptic Systems of Phase Transition Type. Progress in Nonlinear Differential Equations and Their Applications, vol 91. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90572-3_4
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DOI: https://doi.org/10.1007/978-3-319-90572-3_4
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