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Hamilton-Jacobi equations for control problems of parabolic equations

Sophie Gombao, Jean-Pierre Raymond (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ( - A ) β – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already...

Higher-order phase transitions with line-tension effect

Bernardo Galvão-Sousa (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and...

Higher-order phase transitions with line-tension effect

Bernardo Galvão-Sousa (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order...

High-order angles in almost-Riemannian geometry

Ugo Boscain, Mario Sigalotti (2006/2007)

Séminaire de théorie spectrale et géométrie

Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M , then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way...

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