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On a Class of Elliptic Equations for the N-Laplacian in R^n with One-Sided Exponential Growth

Candela, Anna Maria, Squassina, Marco (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35J40, 49J52, 49J40, 46E30By means of a suitable nonsmooth critical point theory for lower semicontinuous functionals we prove the existence of infinitely many solutions for a class of quasilinear Dirichlet problems with symmetric non-linearities having a one-sided growth condition of exponential type.The research of the authors was partially supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (COFIN 2001)....

On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Jean Dolbeault, Régis Monneau (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in ℝ 2 . We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

On exact results in the finite element method

Ivan Hlaváček, Michal Křížek (2001)

Applications of Mathematics

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u . We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.

On Korn's second inequality

J. A. Nitsche (1981)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

On monotone-like mappings in Orlicz-Sobolev spaces

Vesa Mustonen, Matti Tienari (1999)

Mathematica Bohemica

We study the mappings of monotone type in Orlicz-Sobolev spaces. We introduce a new class ( S m ) as a generalization of ( S + ) and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.

On singular perturbation problems with Robin boundary condition

Henri Berestycki, Juncheng Wei (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the following singularly perturbed elliptic problem ϵ 2 Δ u - u + f ( u ) = 0 , u > 0 in Ω , ϵ ∂ u ∂ ν + λ u = 0 on ∂ Ω , where f satisfies some growth conditions, 0 ≤ λ ≤ + ∞ , and Ω ⊂ ℝ N ( N > 1 ) is a smooth and bounded domain. The cases λ = 0 (Neumann problem) and λ = + ∞ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant λ * > 1 such that, as ϵ → 0 , the least energy solution has a spike near the boundary if λ ≤ λ * , and has an interior spike near the innermost part of the domain if λ > λ * . Central to our study is the corresponding problem...

On solution to an optimal shape design problem in 3-dimensional linear magnetostatics

Dalibor Lukáš (2004)

Applications of Mathematics

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization...

On the combined effect of boundary approximation and numerical integration on mixed finite element solution of 4th order elliptic problems with variable coefficients

Pulin K. Bhattacharyya, Neela Nataraj (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Error estimates for the mixed finite element solution of 4th order elliptic problems with variable coefficients, which, in the particular case of aniso-/ortho-/isotropic plate bending problems, gives a direct, simultaneous approximation to bending moment tensor field Ψ = ( ψ i j ) 1 ≤ i , j ≤ 2 and displacement field 'u', have been developed considering the combined effect of boundary approximation and numerical integration.

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