The paper deals with an initial problem of a parabolic variational inequality whichcontains a nonlinear elliptic form $a(v,w)$ having a potential $J\left(v\right)$, which is twice $G$-differentiable at arbitrary $v\in H^1\left(\Omega \right)$. This property of $a(v,w)$ makes it possible to prove convergence of an approximate solution defined by a linearized scheme which is fully discretized - in space by the finite elements method and in time by a one-step finite-difference method. Strong convergence of the approximate solution is proved without any regularity...

We prove that minimizers $u\in W^{1,n}$ of the functional $E_{}\left(u\right)=1/n{\int }_{|}{\nabla u|}^{n}dx+1/\left(4^n\right){\int }_{(}1-{\left|u\right|}^2{)}^{2}dx$, ⊂ ${ℝ}^n$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{}=g$ on for g: → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C_{loc}^{\alpha }$ for any α<1 - upon passing to a subsequence ${}_k\to 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.

Si studiano problemi di autovalori per disequazioni variazionali semilineari ellittiche con un ostacolo puntuale sulla derivata prima della funzione incognita. Si mette in particolare in evidenza il ruolo della «ipotesi di non tangenza» tra il convesso, che viene definito dalla condizione di ostacolo, e la sfera dello spazio funzionale, su cui è naturale studiare un problema di autovalori. Tale condizione viene analizzata in alcuni casi concreti e si indicano alcune ipotesi che, garantendone la...

Variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\in [0,T)$$ are studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some ${\lambda }_0$ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some ${\lambda }_I\ne {\lambda }_0$. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at ${\lambda }_0$ constructed...

Bifurcation and eigenvalue theorems are proved for a certain type of quasivariational inequalities using the method of a jump in the Leray-Schauder degree.

A new hybrid of block-pulse functions and Boubaker polynomials is constructed to solve the inequality constrained fractional optimal control problems (FOCPs) with quadratic performance index and fractional variational problems (FVPs). First, the general formulation of the Riemann-Liouville integral operator for Boubaker hybrid function is presented for the first time. Then it is applied to reduce the problems to optimization problems, which can be solved by the existing method. In this way we find...

We study integrals of the form ${\int }_{\Omega }f\left(d\omega \right)$, where $1\le k\le n$, $f:{\Lambda }^k\to ℝ$ is continuous and $\omega$ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where