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Displaying 481 –
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The weak lower semicontinuity of the functional
is a classical topic that was studied thoroughly. It was shown that if the function is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on . However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.
We establish the existence of a solution to the Neumann problem in the half-space with a subcritical nonlinearity on the boundary. Solutions are obtained through the constrained minimization or minimax. The existence of solutions depends on the shape of a boundary coefficient.
Geometric control theory and Riemannian techniques are used to describe
the reachable set at time t of left invariant single-input control systems
on semi-simple compact Lie groups and to
estimate the minimal time needed to reach any point from identity.
This method provides an effective way to give an upper and a lower bound
for the minimal time needed to transfer a controlled quantum system
with a drift from a given initial position to a given final position.
The bounds include diameters...
We prove by giving an example that when the asymptotic behavior of functionals is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.
We prove by giving an example that when n ≥ 3 the
asymptotic behavior of functionals
is quite different with respect to the planar case. In particular we
show that the one-dimensional ansatz due to Aviles and Giga in the
planar case (see [2]) is no longer true in higher dimensions.
We prove an existence and uniqueness result for a class of Hamilton-Jacobi equations in Hilbert spaces.
Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.
Local Lipschitz continuity of minimizers of certain integrals of the
Calculus of Variations is obtained when the integrands are convex with
respect to the gradient variable, but are not necessarily uniformly
convex. In turn, these regularity results entail existence of minimizers of
variational problems with non-homogeneous integrands nonconvex with
respect to the gradient variable. The x-dependence, explicitly appearing
in the integrands, adds significant technical difficulties in the proof.
This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C 2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.
We establish the existence of an optimal ``state-control'' pair for an optimal control problem of Lagrange type, monitored by a nonlinear elliptic partial equation involving nonmonotone nonlinearities.
We prove a result for the existence and uniqueness of the solution for a class of mildly nonlinear complementarity problem in a uniformly convex and strongly smooth Banach space equipped with a semi-inner product. We also get an extension of a nonlinear complementarity problem over an infinite dimensional space. Our last results deal with the existence of a solution of mildly nonlinear complementarity problem in a reflexive Banach space.
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