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Some application of the implicit function theorem to the stationary Navier-Stokes equations

Konstanty Holly (1991)

Annales Polonici Mathematici

We prove that - in the case of typical external forces - the set of stationary solutions of the Navier-Stokes equations is the limit of the (full) sequence of sets of solutions of the appropriate Galerkin equations, in the sense of the Hausdorff metric (for every inner approximation of the space of velocities). Then the uniqueness of the N-S equations is equivalent to the uniqueness of almost every of these Galerkin equations.

Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

Leszek Gęba, Tadeusz Pruszko (1991)

Annales Polonici Mathematici

This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in Ω ⊂ ℝ n , u = D u = . . . = D m - 1 u on ∂Ω in the Sobolev space W 0 m , 2 ( Ω ) , where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.

Some aspects of the variational nature of mean curvature flow

Giovanni Bellettini, Luca Mugnai (2008)

Journal of the European Mathematical Society

We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional ℱ on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with ℱ . We show some connections between minimizers of ℱ and mean curvature flow.

Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Peter Poláčik (2002)

Mathematica Bohemica

We consider three types of semilinear second order PDEs on a cylindrical domain Ω × ( 0 , ∞ ) , where Ω is a bounded domain in ℝ N , N ≥ 2 . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of Ω × ( 0 , ∞ ) is reserved for time t , the third type is an elliptic equation with a singled out unbounded variable t . We discuss the asymptotic behavior, as t → ∞ , of solutions which are defined and bounded on Ω × ( 0 , ∞ ) .

Some Computational Aspects of the Consistent Mass Finite Element Method for a (semi-)periodic Eigenvalue Problem

De Schepper, H. (1999)

Serdica Mathematical Journal

We consider a model eigenvalue problem (EVP) in 1D, with periodic or semi–periodic boundary conditions (BCs). The discretization of this type of EVP by consistent mass finite element methods (FEMs) leads to the generalized matrix EVP Kc = λ M c, where K and M are real, symmetric matrices, with a certain (skew–)circulant structure. In this paper we fix our attention to the use of a quadratic FE–mesh. Explicit expressions for the eigenvalues of the resulting algebraic EVP are established. This leads...

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