The notion of $\tilde{\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde{\ell }$-stable functions coincides with the class of C${}^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral ${\int }_{\Omega }F\left(\nabla u\left(x\right)\right)\mathrm{d}x$ where $\Omega \subset {ℝ}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition$F\left(\xi \right)\le c(1+|\xi {|}^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space ${ℒ}^{p,\mu }(\Omega ,{ℝ}^{N\times n})$, µ < n, Campanato space ${ℒ}^{p,n}(\Omega ,{ℝ}^{N\times n})$ and the space of bounded mean oscillation $\mathrm{BMO}\Omega ,{ℝ}^{N\times n})$. The admissible maps $u:\Omega \to {ℝ}^N$ are of Sobolev class W1,p, satisfying a Dirichlet boundary...

We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $-\Delta u+u^3+\beta u{v}^2=\lambda u,-\Delta v+v^3+\beta u^{2}v=\mu v,u,v\in H_0^1\left(\Omega \right),u,v>0$, as the interspecies scattering length $\beta$ goes to $+\infty$. For this system we consider the associated energy functionals $J_{\beta },\beta \in (0,+\infty )$, with $L^2$-mass constraints, which limit $J_{\infty }$ (as $\beta \to +\infty$) is strongly irregular. For such functionals, we construct multiple critical points via a common...

In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of ℓ-stability for vector functions.

Vsevolod I. Ivanov stated (Nonlinear Analysis 125 (2015), 270-289) the general second-order optimality condition for the constrained vector problem in terms of Hadamard derivatives. We will consider its special case for a scalar problem and show some corollaries for example for $\ell$-stable at feasible point functions. Then we show the advantages of obtained results with respect to the previously obtained results.

We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as an almost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of terms obtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the various versions of the concentration-compactness principle.

In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer...

We will deal with a new geometrical interpretation of the classical Legendre and Jacobi conditions: they are represented by the rate and the magnitude of rotation of certain linear subspaces of the tangent space around the tangents to the extremals. (The linear subspaces can be replaced by conical subsets of the tangent space.) This interpretation can be carried over to nondegenerate Lagrange problems but applies also to the degenerate variational integrals mentioned in the preceding Part II.