By Nicholas D. Alikakos (auth.), Antonin Chambolle, Matteo Novaga, Enrico Valdinoci (eds.)

This ebook is the end result of a convention held on the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. the purpose of the convention was once to debate contemporary effects on nonlinear partial differential equations, and extra particularly geometric evolutions and reaction-diffusion equations. specific consciousness was once paid to self-similar strategies, resembling solitons and vacationing waves, asymptotic behaviour, formation of singularities and qualitative homes of suggestions. those difficulties come up in lots of versions from Physics, Biology, snapshot Processing and utilized arithmetic in most cases, and feature attracted loads of realization in contemporary years.

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12 In [7] some necessary conditions are given in the case m = 2, such as the impossibility of cusps or re-entrant corners in ∂(K 1 K 2 ). 6), the star-shaped combination of the star-shaped anisotropies φ1 , . . , φm : V → [0, +∞) is defined as: m m φ j := j=1 j=1 −1/2 1 φ 2j . 1 On the hessian of the combined anisotropy Let : (R N )∗ → [0, +∞) be the star-shaped combination of the starshaped anisotropies φ1 , . . , φm : (R N )∗ → [0, +∞). Set for notational convenience α := 2 , α j := φ 2j , j = 1, .

1. We formulate the theorem in an independent way. 4) u = u0, on ∂ , for u : → Rm , where is a smooth open domain and W is a C 2 potential. We have the following hypotheses on W and . (H1) The potential W is symmetric, with W (u 1 , u 2 , . . , u m ) = W (−u 1 , u 2 , . . , u m ). (H2) There exists a nondegenerate minimum a+ of the potential W (cf. Hypothesis 1 in Section 3) such that a1 > 0 and 0 = W (a1 ) ≤ W (u), for u ∈ Rm , and such that W (a+ ) < W (u), for |u − a+ | ≤ q0 , for some q0 > 0, with u = a+ .

2) with initial condition u = u and m (Dirichlet) boundary conditions wr = d r for r = 1, . . 4) (with boundary condition given by m r u = r=1 d ) and subsequently solving the first m − 1 linear elliptic equations at each time t to recover the unknowns w1 , . . , wm−1 , and m−1 r w . 5) λr2 div Tφ (∇wr ) = div T (∇u) , with (Dirichlet) boundary condition wr = d r . In the special case λr2 d r = λ2s d s , or equivalently m λr2 s=1 r, s = 1, . . 7) s=1 we can recover the unknowns wr as 1 w := 2 λr m r s=1 1 λ2s −1 u, r = 1, .

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