Critical Phenomena in Linear Elliptic Problems

Non-coercive Linear Elliptic Problems

We study here some linear elliptic partial differential equations (with Dirichlet, Fourier or mixed boundary conditions), to which convection terms (first order perturbations) are added that entail the loss of the classical coercivity property. We prove the existence, uniqueness and regularity results for the solutions to these problems. Mathematics Subject Classification (2000): 35J25.

Concentration Phenomena for Fourth-order Elliptic Equations with Critical Exponent

We consider the nonlinear equation ∆u = u n+4 n−4 − εu with u > 0 in Ω and u = ∆u = 0 on ∂Ω. Where Ω is a smooth bounded domain in Rn, n ≥ 9, and ε is a small positive parameter. We study the existence of solutions which concentrate around one or two points of Ω. We show that this problem has no solutions that concentrate around a point of Ω as ε approaches 0. In contrast to this, we construct ...

Quasi-linear elliptic boundary value problems

Equivalent operator preconditioning for linear elliptic problems

Finite element or finite difference approximations to linear partial differential equations of elliptic type lead to algebraic systems, normally of very large size. However, since the matrices are sparse, even extremely large-sized systems can be handled. To save computer memory and elapsed time, such equations are normally solved by iteration, most commonly using a preconditioned conjugate gra...

Normal Solvability of General Linear Elliptic Problems

The paper is devoted to general elliptic problems in the Douglis-Nirenberg sense. We obtain a necessary and sufficient condition of normal solvability in the case of unbounded domains. Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more condition formulated in terms of limiting prob...