A Priori and A Posteriori Analysis of Mixed Finite Element Methods for Nonlinear Elliptic Equations

In this paper, we establish global Sobolev a priori estimates for [Formula: see text]-viscosity solutions of fully nonlinear elliptic equations as follows: [Formula: see text] by considering minimal integrability condition on the data, i.e. [Formula: see text] for [Formula: see text] and a regular domain [Formula: see text], and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting “fine” regularity estimates from a limiting operator, the Recession profile, associated to [Formula: see text] to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when [Formula: see text]. In such a scenery, we show that solutions admit [Formula: see text] type estimates for their second derivatives.

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Multiple boundary blow-up solutions for nonlinear elliptic equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500002377 ◽

2003 ◽

Vol 133 (2) ◽

pp. 225-235 ◽

Cited By ~ 15

Author(s):

Amandine Aftalion ◽

Manuel del Pino ◽

René Letelier

Keyword(s):

Elliptic Equations ◽

Blow Up ◽

A Priori ◽

Nonlinear Elliptic Equations ◽

Positive Parameter ◽

Number Of Solutions ◽

Bounded Smooth Domain ◽

Nonlinear Elliptic ◽

Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

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A priori and a posteriori analysis of the flow around a rectangular cylinder

Journal of Physics Conference Series ◽

10.1088/1742-6596/923/1/012028 ◽

2017 ◽

Vol 923 ◽

pp. 012028

Author(s):

A Cimarelli ◽

A Leonforte ◽

M Franciolini ◽

E De Angelis ◽

D Angeli ◽

...

Keyword(s):

A Priori ◽

A Posteriori ◽

Rectangular Cylinder ◽

Posteriori Analysis ◽

A Posteriori Analysis

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A Priori Error Estimates for Finite Element Methods forH(2,1)-Elliptic Equations

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2013.811600 ◽

2013 ◽

Vol 35 (2) ◽

pp. 153-176 ◽

Cited By ~ 2

Author(s):

Thomas Apel ◽

Thomas G. Flaig ◽

Serge Nicaise

Keyword(s):

Finite Element ◽

Error Estimates ◽

Finite Element Methods ◽

Elliptic Equations ◽

A Priori ◽

A Priori Error Estimates ◽

Priori Error Estimates

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On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

Advances in Calculus of Variations ◽

10.1515/acv-2016-0007 ◽

2019 ◽

Vol 12 (4) ◽

pp. 393-421

Author(s):

Tilak Bhattacharya ◽

Leonardo Marazzi

Keyword(s):

Viscosity Solutions ◽

Elliptic Equations ◽

Eigenvalue Problems ◽

A Priori ◽

Nonlinear Elliptic Equations ◽

First Eigenvalue ◽

Nonlinear Elliptic ◽

Degenerate Elliptic ◽

First Eigenfunction ◽

Nonlinear Degenerate

AbstractWe consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

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