By Conti M., Terracini S., Verzini G.
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Extra resources for An optimal partition problem related to nonlinear eigenvalues
Math. 77 (1955), 329–354  P. Korman, A. Leung On the existence and uniqueness of positive steady states in Lotka– Volterra ecological models with diffusion, Appl. Anal. C. J. McKenna, On steady state solutions of a system of reaction–diffusion equations from biology, Nonlinear Anal. TMA 6 (1982), 523–530  C. Miranda, Un’osservazione sul teorema di Brouwer, Boll. I. Serie II, Anno II, 1 (1940), 5–7  Z. Nehari, Characteristic values associated with a class of nonlinear second order differential equations, Acta Math.
2 it suffices to show that, for i = j, u0i and u0j have disjoint support: this is consequence of property (iii) and of the strong convergence of U ε to its weak limit. e. x ∈ Ω and proves supp(u0i ) ∩ supp(u0j ) = ∅ for every i = j. Moreover by (iv) we also obtain J(U 0 ) = c0 , finally proving the assertion. As a final remark, note that, if we pass to the limit in (17) as ε → 0, by exploiting the strong convergence of the solutions U ε to the k–uple U 0 and (iii), we realize that −∆u0i (x) = f (u0i (x)) x ∈ supp(u0i ) ∀i.
27) From this the strong convergence of each uεi to its limit easily follows. Indeed, if we assume for contradiction that lim Ω |∇uεi |2 > Ω |∇u0i |2 , then all the above inequalities become strict, leading a contradiction in (27). Thus (ii) is proved. Let us now prove assertion (iii). To this aim we deduce by (25) and λ0i = 1, the relation 0 2 0 0 Ω |∇ui | = Ω f (ui )ui . Back to (26) we obtain that lim ε−1 ε→0 Ω Hui (U ε )uεi = 0. Now (27) becomes lim ε−1 ε→0 Ω H(U ε ) = 0 as desired. Furthermore, we deduce by (iii) that lim cε = J(U 0 ) ≤ c0 ε→0 where the last inequality follows by the definition of c0 and (i).
An optimal partition problem related to nonlinear eigenvalues by Conti M., Terracini S., Verzini G.