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This paper derives upper and lower bounds for the $ℓ^{p}$ -condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin–Least Squares technique or by means of a non-standard Galerkin technique in L1(Ω). Numerical...

Everywhere regularity for vectorial functionals with general growth

Elvira Mascolo, Anna Paola Migliorini (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is $F (u) = \int_{Ω} a (x) {[h (| D u |)]}^{p (x)} d x$ with $h$ a convex function with general growth (also exponential behaviour is allowed).

Everywhere regularity for vectorial functionals with general growth

Elvira Mascolo, Anna Paola Migliorini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is $F (u) = \int_{Ω} a (x) {[h (| D u |)]}^{p (x)} d x$ with h a convex function with general growth (also exponential behaviour is allowed).

Évolution d'une interface par capillarité et diffusion de volume. Estimations a priori et résultats d'existence

Jean Duchon, Raoul Robert (1984)

Journées équations aux dérivées partielles

Exact multiplicity of positive solutions for a class of semilinear equations on a ball.

Korman, Philip (1999)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Exact multiplicity of positive solutions in semipositone problems with concave-convex type nonlinearities.

Gadam, Sudhasree, Iaia, Joseph A. (2001)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Exact multiplicity of positive solutions to a superlinear problem.

Shi, Junping (2003)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Exact solutions to some external mixed problems in potential theory

Valery I. Fabrikant (1986)

Aplikace matematiky

A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved; i) An arbitrary potential is specified at the boundary outside a circle, and its normal derivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and be potential is zero inside. Several...

Example of an $\infty$ -harmonic function which is not $C^{2}$ on a dense subset.

Mikayelyan, Hayk (2005)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Excitation of shear waves in a piezoceramic medium with partially electrodized tunnel openings strengthened by a rigid stringer.

Bardzokas, D.I., Sfyris, G.I. (2005)

Mathematical Problems in Engineering

Existence and Approximation of Solutions of Non-Linear Elliptic Equations.

Charles A. Stuart (1976)

Mathematische Zeitschrift

Existence and asymptotic behavior of boundary blow-up solutions for weighted $p (x)$ -Laplacian equations with exponential nonlinearities.

Yin, Li, Guo, Yunrui, Yang, Jing, Lu, Bibo, Zhang, Qihu (2010)

Abstract and Applied Analysis

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

Our main purpose is to establish the existence of a positive solution of the system ⎧ $- ∆_{p (x)} u = F (x, u, v)$ , x ∈ Ω, ⎨ $- ∆_{q (x)} v = H (x, u, v)$ , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded domain with C² boundary, $F (x, u, v) = λ^{p (x)} [g (x) a (u) + f (v)]$ , $H (x, u, v) = λ^{q (x}) [g (x) b (v) + h (u)]$ , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $- ∆_{p (x)} u = - {d i v (| \nabla u |}^{p (x) - 2} \nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Existence and asymptotic behavior of positive solutions to $p (x)$ -Laplacian equations with singular nonlinearities.

Zhang, Qihu (2007)

Journal of Inequalities and Applications [electronic only]

Existence and asymptotic behavior of solutions for weighted $p (t)$ -Laplacian system multipoint boundary value problems in half line.

Qiu, Zhimei, Zhang, Qihu, Wang, Yan (2009)

Journal of Inequalities and Applications [electronic only]

Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in $R^{3}$

Khalid Benmlih, Otared Kavian (2008)

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

Existence and asymptotics for selfdual periodic vortices of topological-type

Macrí, Marta, Nolasco, Margherita (2007)

Proceedings of Equadiff 11

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