Abstract
In this chapter we study some properties of solutions to the problem (1.5). We first give an interior Hausdorff measure estimate of the free boundary of these solutions. Then we prove the uniqueness of the solution to (1.5), which then can be identified as the minimizer of a functional. Finally, we prove the uniqueness of solutions to the corresponding parabolic problem of (1.5).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Caffarelli, L.A., Lin, F.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21(3), 847–862 (2008)
Caffarelli, L.A., Lin, F.: Nonlocal heat flows preserving the L 2 energy. Discrete Contin. Dyn. Syst., Ser. A 23(1–2), 49–64 (2009)
Caffarelli, L.A., Karakhanyan, A.L., Lin, F.: The geometry of solutions to a segregation problem for non-divergence systems. J. Fixed Point Theory Appl. 5(2), 319–351 (2009)
Conti, M., Terracini, S., Verzini, G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198(1), 160–196 (2003)
Conti, M., Terracini, S., Verzini, G.: A variational problem for the spatial segregation of reaction diffusion systems. Indiana Univ. Math. J. 54(3), 779–815 (2005)
Han, Q., Lin, F.: Nodal sets of solutions of elliptic differential equations. Books available on Han’s homepage
Lin, F., Yang, X.: Geometric Measure Theory: An Introduction. Science Press/International Press, Beijing/Boston (2002)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wang, K. (2013). Uniqueness in the Singular Limit. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-33696-6_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33695-9
Online ISBN: 978-3-642-33696-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)