On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

Křížek, Michal; Liu, Liping

doi:10.4064/am-24-1-97-107

Cited by 10 publications

(5 citation statements)

References 16 publications

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“…Finally, using (11) and (29), property (P2) of the quadratures and (19) imply thatd i ≤ 0 for all i, i.e., q ≤ 0. Hence, (21) yields Q(c)c +Q(c)c ≤ 0 and thus, Theorem 1 states that…”

Section: Theorem 3 Let Conditions (I)-(v) Of Theorem 2 Hold and Letmentioning

confidence: 92%

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On discrete maximum principles for nonlinear elliptic problems

Karátson

Korotov

Kříek

2007

Mathematics and Computers in Simulation

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In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short review of the most important results devoted to discrete counterparts of the maximum principle (called discrete maximum principles, DMPs), mainly in the framework of the finite element method, and also present our own recent results on DMPs for a class of second-order nonlinear elliptic problems with mixed boundary conditions.

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“…Finally, using (11) and (29), property (P2) of the quadratures and (19) imply thatd i ≤ 0 for all i, i.e., q ≤ 0. Hence, (21) yields Q(c)c +Q(c)c ≤ 0 and thus, Theorem 1 states that…”

Section: Theorem 3 Let Conditions (I)-(v) Of Theorem 2 Hold and Letmentioning

confidence: 92%

“…Other works devoted to various aspects of DMPs and related issues include [1], [8], [13], [21], [22], [31], [32], [33], and [34]. Several examples of real-life problems for which the validity of DMP is essential are given, e.g., in papers [3], [14], [25], and [37].…”

Section: Introductionmentioning

confidence: 99%

On discrete maximum principles for nonlinear elliptic problems

Karátson

Korotov

Kříek

2007

Mathematics and Computers in Simulation

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“…In this section, we prove a comparison theorem by the method recently proposed [32] for proving comparison theorems in the case of quasilinear elliptic equations. A modified method was first used for the problem of radiative-conductive heat transfer in a system of grey bodies in [24].…”

Section: Comparison Theoremmentioning

confidence: 99%

Stationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency

Амосов

2009

J Math Sci

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We consider a stationary nonlinear nonlocal boundary value problem governing radiative-conductive heat transfer in a system of opaque bodies with surfaces whose properties depend on the radiation frequency. We establish the existence and uniqueness of a solution and prove a comparison theorem. Results on the exponential summability and boundedness of the solution are also obtained. Bibliography: 32 titles.

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“…We shall see that a direct consequence of the comparison principle is the uniqueness of weak solutions. Our proof of the comparison principle is a generalization of that in [8], where an elliptic boundary value problem is studied. Similar results are obtained in another way in [1], where different boundary conditions are examined.…”

Section: Introductionmentioning

confidence: 96%

Comparison principle for a nonlinear parabolic problem of a nonmonotone type

Vejchodský

2002

Appl. Math. (Warsaw)

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Abstract.A nonlinear parabolic problem with the Newton boundary conditions and its weak formulation are examined. The problem describes nonstationary heat conduction in inhomogeneous and anisotropic media. We prove a comparison principle which guarantees that for greater data we obtain, in general, greater weak solutions. A new strategy of proving the comparison principle is presented. Introduction.Comparison principles are important features of second order partial differential equations. They hold for linear and many nonlinear problems of elliptic and parabolic types. It seems that the ellipticity of an appropriate operator is the main feature which assures the comparison principle (compare [5,10]).In this paper we deal with the following quasilinear parabolic problem:where [65]

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On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

Cited by 10 publications

References 16 publications

On discrete maximum principles for nonlinear elliptic problems

On discrete maximum principles for nonlinear elliptic problems

Stationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency

Comparison principle for a nonlinear parabolic problem of a nonmonotone type

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