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We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...

Deuxième microlocalisation simultanée et applications

Jean-Marc Delort (1990)

Journées équations aux dérivées partielles

Differential inequalities of parabolic type in the Sobolev space

J. Chabrowski (1973)

Annales Polonici Mathematici

Differential-functional inequalities of parabolic type in unbounded regions

P. Besala, G. Paszek (1980)

Annales Polonici Mathematici

Div-curl lemma revisited: Applications in electromagnetism

Marián Slodička, Ján Jr. Buša (2010)

Kybernetika

Two new time-dependent versions of div-curl results in a bounded domain $Ω \subset ℝ^{3}$ are presented. We study a limit of the product $v_{k} w_{k}$ , where the sequences $v_{k}$ and $w_{k}$ belong to $Ł_{2} (Ω)$ . In Theorem 2.1 we assume that $\nabla \times v_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. In Theorem 2.2 we suppose that $\nabla \times w_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. The time derivative of $w_{k}$ is bounded in both cases in the norm of $- 1 (Ω)$ . The convergence (in the sense of distributions) of $v_{k} w_{k}$ to the product $v w$ of weak limits...

Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle.

Peter Polácik (1989)

Journal für die reine und angewandte Mathematik

Droplet spreading under weak slippage : the waiting time phenomenon

Günther Grün (2004)

Annales de l'I.H.P. Analyse non linéaire

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