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On the Dirichlet and Neumann problems in multi-dimensional cone

Vladimir Vasilyev (2014)

Mathematica Bohemica

We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems,...

On the existence and the stability of solutions for higher-order semilinear Dirichlet problems

Marek Galewski, M. Płócienniczak (2007)

Czechoslovak Mathematical Journal

We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.

On the structure of layers for singularly perturbed equations in the case of unbounded energy

E. Sanchez-Palencia (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We consider singular perturbation variational problems depending on a small parameter ε . The right hand side is such that the energy does not remain bounded as ε → 0 . The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with ε > 0 are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating...

On the structure of layers for singularly perturbed equations in the case of unbounded energy

E. Sanchez–Palencia (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider singular perturbation variational problems depending on a small parameter ε. The right hand side is such that the energy does not remain bounded as ε → 0. The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with ε > 0 are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after...

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