We set a coupled boundary value problem between two domains of different dimension. The first one is the unit cube of Rn, n C [2,3], with a crack and the second one is the crack. this problem comes from Ciarlet et al. (1989), that obtained an analogous coupled problem. We show that the solution has singularities due to the crack. As in Grisvard (1989), we adapt the Hilbert uniqueness method of J.-L. Lions (1968,1988) in order to obtain the exact controllability of the associated wave equation with...

This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form $\frac{\partial y}{\partial n}+f\left(y\right)=0$. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one...

The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincare-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling...

Continuing the previous Part I, the degenerate first order variational integrals depending on two functions of one independent variable are investigated.

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is 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 main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type $$\begin{array}{c}f(x,\eta ,\xi )\ge a\left(x\right)\frac{{\left|\xi \right|}^p}{{(1+|\eta \left|\right)}^{\alpha }}-b_1\left(x\right){\left|\eta \right|}^{{\beta }_1}-g_1\left(x\right),\\ f(x,\eta ,0)\le b_2\left(x\right){\left|\eta \right|}^{{\beta }_2}+g_2\left(x\right),\end{array}$$ where$0\le \alpha <p$, $1\le {\beta }_1<p$, $0\le {\beta }_2<p$, $\alpha +{\beta }_i\le p$, $a\left(x\right)$, $b_i\left(x\right)$, $g_i\left(x\right)$ ($i=1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}\left(a\right)$ , which assume a boundary datum $u_0\in W^{1,p}\left(a\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ $\Delta u^{\text{'}}\left(t_j\right)=u^{\text{'}}(t{⁺}_j-u^{\text{'}}\left(t{¯}_j\right)=I_j\left(u\left(t_j\right)\right)$, j = 1,...,p, are established, where $t₀=0<t₁<\cdots <t_p<t_{p+1}=T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_j:ℝ\to ℝ$, j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results....

We show that local minimizers of functionals of the form ${\int }_{\Omega }\left[f\left(Du\right(x\left)\right)+g(x,u(x\left)\right)\right]\mathrm{d}x$,  $u\in u_0+W_0^{1,p}\left(\Omega \right)$, are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.