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On the equivalence of variational problems. II

Jan Chrastina (1993)

Archivum Mathematicum

Elements of general theory of infinitely prolonged underdetermined systems of ordinary differential equations are outlined and applied to the equivalence of one-dimensional constrained variational integrals. The relevant infinite-dimensional variant of Cartan’s moving frame method expressed in quite elementary terms proves to be surprisingly efficient in solution of particular equivalence problems, however, most of the principal questions of the general theory remains unanswered. New concepts of...

On the existence of a weak solution of the boundary value problem for the equilibrium of a shallow shell reinforced with stiffening ribs

Igor Bock, Ján Lovíšek (1978)

Aplikace matematiky

The existence and the unicity of a weak solution of the boundary value problem for a shallow shell reinforced with stiffening ribs is proved by the direct variational method. The boundary value problem is solved in the space W ( Ω ) ⊂ H 0 1 ( Ω ) × H 0 1 ( Ω ) × H 0 2 ( Ω ) , on which the corresponding bilinear form is coercive. A finite element method for numerical solution is introduced. The approximate solutions converge to a weak solution in the space Q ( Ω ) .

On the lower semicontinuity of supremal functionals

Michele Gori, Francesco Maggi (2003)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the lower semicontinuity problem for a supremal functional of the form F ( u , Ω ) = ess sup x ∈ Ω f ( x , u ( x ) , D u ( x ) ) with respect to the strong convergence in L ∞ ( Ω ) , furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur’s lemma for gradients of uniformly converging sequences is proved.

On the Lower Semicontinuity of Supremal Functionals

Michele Gori, Francesco Maggi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the lower semicontinuity problem for a supremal functional of the form F ( u , Ω ) = ess sup x ∈ Ω f ( x , u ( x ) , D u ( x ) ) with respect to the strong convergence in L∞(Ω), furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

On the numerical approximation of first-order Hamilton-Jacobi equations

Rémi Abgrall, Vincent Perrier (2007)

International Journal of Applied Mathematics and Computer Science

Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.

On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

Nalini Anantharaman (2004)

Journal of the European Mathematical Society

We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on ( ℝ d ) ℤ / ℤ d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...

On valuation of derivative securities: A Lie group analytical approach

Phillip S. C. Yam, Hailiang Yang (2006)

Applications of Mathematics

This paper proposes a Lie group analytical approach to tackle the problem of pricing derivative securities. By exploiting the infinitesimal symmetries of the Boundary Value Problem (BVP) satisfied by the price of a derivative security, our method provides an effective algorithm for obtaining its explicit solution.

On variational approach to the Hamilton-Jacobi PDE

Jan H. Chabrowski, Ke Wei Zhang (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper we construct a minimizing sequence for the problem (1). In particular, we show that for any subsolution of the Hamilton-Jacobi equation ( * ) there exists a minimizing sequence weakly convergent to this subsolution. The variational problem (1) arises from the theory of computer vision equations.

Optimal control of a stochastic heat equation with boundary-noise and boundary-control

Arnaud Debussche, Marco Fuhrman, Gianmario Tessitore (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C1 regularity...

Optimal control problems with upper semicontinuous Hamiltonians

Arkadiusz Misztela (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we give examples of value functions in Bolza problem that are not bilateral or viscosity solutions and an example of a smooth value function that is even not a classic solution (in particular, it can be neither the viscosity nor the bilateral solution) of Hamilton-Jacobi-Bellman equation with upper semicontinuous Hamiltonian. Good properties of value functions motivate us to introduce approximate solutions of equations with such type Hamiltonians. We show that the value function is...

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