Two non-trivial solutions for semilinear elliptic resonant problems are obtained via the Lyapunov—Schmidt reduction and the three-critical-points theorem. The difficulty that the variational functional does not satisfy the Palais—Smale condition is overcome by taking advantage of the reduction and a careful analysis of the reduced functional.

We consider the periodic solutions of the equation

satisfying −1 ≤ x ≤ 1 ẋ(t−) = −ẋ(t+) if x(t) = ±1, where f is a continuous function, periodic in t. A sequence of periodic solutions is obtained via the variational method.

We study the existence and non-existence of classical solutions to a general Gierer—Meinhardt system with Dirichlet boundary condition. The main feature of this paper is that we are concerned with a model in which both the activator and the inhibitor have different sources given by general nonlinearities. Under some additional hypotheses and in the case of pure powers in nonlinearities, regularity and uniqueness of the solution in one dimension is also presented.

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by . This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.

The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in ℝn. Using variational techniques and in particular a method introduced by Alikakos and Fusco we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to bulk free energies in crystals, we establish existence for N ≥ 2 in various cases of more than two minima. Finally, we obtain a sufficient condition establishing existence of connections to potentials which are not necessarily symmetric for arbitrary N and three minima.

Taking the argument used by De Giorgi to obtain the rectifiability of the reduced boundary of a set of finite perimeter in ℝN, we prove that a set E of finite perimeter in an open set Ω of ℝN may be approached, in the sense of the BV(Ω) norm, by sets whose boundary is included in a finite union of 1 hypersurfaces; more precisely, arbitrarily large parts (for the N−1 measure in Ω) of the essential boundaries of E and of the approximating set coincide and are included in a single 1 hypersurface.

We investigate critical points of the functional

over a ball in ℝn. Here, W is radially symmetric but not convex. We embed the functional into a family of functionals

where E0,0(u) = E(u). A global bifurcation analysis yields a branch of non-trivial critical points depending on λ and positive ε, where we can set λ = 0. The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as ε ↘ 0 exists. Under natural conditions on W and G the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

We study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.