We express our sincere gratitude to the referee for a careful reading of the manuscript.International audienceWe investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$

We consider a special class of radial solutions of semilinear equations − u = g(u) in the unit ball of R n . It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, L q , and W k,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution u ∈ H 1 0 is bounded if n 9 (for every g), and belongs to H 3 = W 3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.

We study a three-dimensional model for alloys that undergo a cubic-to-tetragonal phase transition in the martensitic phase. Any pair of the three martensitic variants can form a stress-free laminate. However, this laminate is only compatible on average with the remaining variant. The resulting local stresses favor a microstructure if all three variants are present (for instance, because of an externally imposed average strain).Next to the linearized elastic energy, the variational model features an interfacial energy between the three variants. This introduces a material length scale, which together with the sample size (which we mimic by periodic boundary conditions) gives rise to a nondimensional parameter Á.We rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in Á. More precisely, we show e Á 2=3 . The upper bound construction is achieved by a few patches of branched laminates; the lower bound relies on suitable interpolation inequalities. This is in the spirit of a celebrated work by Kohn and Müller and relies on techniques developed for domain branching in micromagnetics.We also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than Á 2=3 , the configuration is approximately a simple unbranched laminate. In particular, one of the three volume fractions has to be small. This is related to similar rigidity results by Dolzmann and Müller and by Kirchheim.

It is shown that the three-body trigonometric G 2 integrable system is exactly solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly. * 3885 Int. J. Mod. Phys. A 1998.13:3885-3903. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/04/15. For personal use only. 3 k

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