Degenerate anisotropic variational inequalities with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math>-data

Kovalevsky, Alexander A.; Gorban, Yuliya

doi:10.1016/j.crma.2007.09.004

Cited by 12 publications

(17 citation statements)

References 9 publications

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“…In the last decades, increasing attention has been given to the investigation of existence and properties of solutions to various problems for integral functionals with anisotropic integrands and for elliptic equations and variational inequalities with anisotropic coefficients (see for instance [1][2][3][4][5][6][7][8][9][10][11]13,15]). On the whole, the study of these problems requires more delicate procedures as compared with those used in the isotropic case.…”

Section: Introduction and Exposition Of Known Resultsmentioning

confidence: 99%

Integrability and boundedness of solutions to some anisotropic problems

Kovalevsky

2015

Journal of Mathematical Analysis and Applications

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In this article, we give a series of results on the improved integrability and boundedness of solutions to several anisotropic problems. In particular, we consider the nonhomogeneous Dirichlet problem for anisotropic elliptic second-order equations. Moreover, we study anisotropic variational inequalities with unilateral and bilateral obstacles as well as gradient constraints. Finally, variational problems for integral functionals with anisotropic integrands are considered.

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Section: Introduction and Exposition Of Known Resultsmentioning

confidence: 99%

Integrability and boundedness of solutions to some anisotropic problems

Kovalevsky

2015

Journal of Mathematical Analysis and Applications

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“…In view of the observation given above, we adopt the classification of the solutions to the boundary valued problem (9)-(10) following Pastukhova & Zhikov [34] (for more details and other types of solutions we refer to [3,16,33]).…”

Section: Classification Of Optimal Solutionsmentioning

confidence: 99%

Optimal $L^1$-Control in Coefficients for Dirichlet Elliptic Problems: $H$-Optimal Solutions

Kogut¹,

Leugering²

2011

Z. Anal. Anwend.

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In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in L 1 (Ω). In particular, when the coefficient matrix is taken to satisfy the decomposition B(x) = ρ(x)A(x) with a scalar function ρ, we allow the ρ to degenerate. Such problems are related to various applications in mechanics, conductivity and to an approach in topology optimization, the SIMP-method. Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that the optimal control problem in the coefficients can be stated in different forms depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the so-called class of H-admissible solutions. Keywords. Degenerate elliptic equations, control in coefficients, weighted Sobolev spaces, Lavrentieff phenomenon, direct method in the Calculus of variations.

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“…/ under the fixed value of the second variable. In our case we have no opportunity to use the results [21][22][23] directly. We follow a general approach for proving the above-mentioned theorem.…”

Section: Introductionmentioning

confidence: 99%

“…This approach has been proposed in [1] to the investigation on the existence and properties of solutions for nonlinear elliptic second-order equations with isotropic nondegenerate (with respect to the independent variables) coefficients and L 1 -right-hand sides. In [21,23] this approach has been taken to the anisotropic degenerate case. Also we use some ideas of [24].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the existence results of some other types of solutions to the given problem were also obtained. Observe that the proofs of these theorems are based on use of some results of [21][22][23] on the existence and properties of solutions of second-order variational inequalities with L 1 -right-hand sides and sufficiently general constraints. Note that in [20][21][22][23] right-hand sides to the investigated variational inequalities and equations depend on independent variables only, and belong to the class L 1 .…”

Section: Introductionmentioning

confidence: 99%

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Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

Gorban

2017

Open Mathematics

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Abstract:In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L 1 -right-hand sides in a bounded domain of R n .n > 2/. This class is described by the presence of a set of exponents q 1 ; : : : ; q n and a set of weighted functions 1 ; : : : ; n in growth and coercitivity conditions on coefficients of the equations. The exponents q i characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions i characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.

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Degenerate anisotropic variational inequalities with L1-data

Cited by 12 publications

References 9 publications

Integrability and boundedness of solutions to some anisotropic problems

Integrability and boundedness of solutions to some anisotropic problems

Optimal $L^1$-Control in Coefficients for Dirichlet Elliptic Problems: $H$-Optimal Solutions

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

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