Abstract
In this chapter, we first prove the uniqueness of solutions to the Dirichlet boundary value problem (1.4) by the sub- and super-solution method. In Sect. 2.2, we use the same method to prove the stability of solutions to the corresponding parabolic initial-boundary value problem. Finally, by the same idea, we prove the uniform Lipschitz estimates for solutions to these two problems, under suitable boundary conditions.
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References
Conti, M., Terracini, S., Verzini, G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195(2), 524–560 (2005)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
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Wang, K. (2013). Uniqueness, Stability and Uniform Lipschitz Estimates. In: Free Boundary Problems and Asymptotic Behavior of Singularly Perturbed Partial Differential Equations. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33696-6_2
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DOI: https://doi.org/10.1007/978-3-642-33696-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33695-9
Online ISBN: 978-3-642-33696-6
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