C1,α-regularity for electrorheological fluids in two dimensions

Diening, Lars; Ettwein, Frank; Růžička, Michael

doi:10.1007/s00030-007-5026-z

Cited by 41 publications

(17 citation statements)

References 2 publications

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“…The constant exponent p(·) ≡ p is particularly known to have a threshold between two drastically different types of nonlinear diffusion, and moreover, the limit of p → ∞ exhibits a peculiar phenomena called fast/slow diffusion. Parabolic equations involving the p(·)-Laplacian have been proposed in the study of image restoration (see [17]) as well as in some model of electrorheological fluids (see [19,21,34]). A mathematical analysis was also done for such problems by Acerbi and Mingione [1,2] and by Acerbi et al [3].…”

Section: G Akagi and K Matsuuramentioning

confidence: 99%

Nonlinear diffusion equations driven by the p(·)-Laplacian

Akagi

Matsuura

2012

Nonlinear Differ. Equ. Appl.

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Abstract. This paper is concerned with nonlinear diffusion equations driven by the p(·)-Laplacian with variable exponents in space. The wellposedness is first checked for measurable exponents by setting up a subdifferential approach. The main purposes are to investigate the large-time behavior of solutions as well as to reveal the limiting behavior of solutions as p(·) diverges to the infinity in the whole or in a subset of the domain. To this end, the recent developments in the studies of variable exponent Lebesgue and Sobolev spaces are exploited, and moreover, the spatial inhomogeneity of variable exponents p(·) is appropriately controlled to obtain each result. Mathematics Subject Classification (2010). Primary 35K92; Secondary 35K55, 46E35.

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Section: G Akagi and K Matsuuramentioning

confidence: 99%

Nonlinear diffusion equations driven by the p(·)-Laplacian

Akagi

Matsuura

2012

Nonlinear Differ. Equ. Appl.

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“…The regularity for variational problems and differential equations with nonstandard growth has been studied extensively and many interesting results have been obtained, for example, we refer to [8,12,34] for general (p, q) growth case, refer to [1,[4][5][6][19][20][21][30][31][32][33] for − → p growth case and refer to [2,9,10,[13][14][15][16][17]25,41,44,45] for p(x) growth case. To our knowledge, the regularity for the − → p (x) growth case has not yet been studied specially and systematically.…”

Section: Fan Nodeamentioning

confidence: 99%

Local boundedness of quasi-minimizers of integral functionals with variable exponent anisotropic growth and applications

Fan

2010

Nonlinear Differ. Equ. Appl.

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Abstract. The local boundedness of local quasi-minimizers of integral functionals with variable exponent anisotropic − → p (x) growth under suitable assumptions is proved. Based on this result, the global boundedness and the Lipschitz continuity of weak solutions of Dirichlet or Neumann boundary value problems for the − → p (x)-Laplace type equations are obtained. Mathematics Subject Classification (2000). 49N60, 35J70.

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“…Equation (P) is a nonlinear diffusion equation which has been used to study image restoration and electrorheological fluids (see [4][5][6][7][8][9][10][11]). In particular, Bendahmane et al [12] proved the well-posedness of a solution, for 1 -data.…”

Section: Introductionmentioning

confidence: 99%