Free Boundary Problems by Isabel Narra Figueiredo, Isabel N. Figueiredo;José F.
By Isabel Narra Figueiredo, Isabel N. Figueiredo;José F. Rodrigues;Lisa Santos
This ebook gathers a set of refereed articles containing unique effects reporting the hot contributions of the lectures and communications provided on the loose Boundary difficulties convention that happened on the collage of Coimbra, Portugal, from June 7 to twelve, 2005 (FBP2005). They take care of the math of a extensive category of versions and difficulties concerning nonlinear partial differential equations bobbing up in physics, engineering, biology and finance. one of the major issues, the talks thought of loose boundary difficulties in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in snapshot processing, in monetary arithmetic or in computations for inter-scale problems.
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Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. , 156 (2001), pp. 121–140. [2] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. , 164 (2002), pp. 213–259. [3] E. Acerbi, G. A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 21 (2004), pp. 25–60. [4] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods (to appear).
E. on (0, T ). By the Gagliardo-Nirenberg interpolation inequality (cf. [11]) and the result of (i), the following inequalities hold: |∇θµ,λ |4L4 (Ω) ≤ C1 |θµ,λ |2H 2 (Ω) |θµ,λ |2L∞ (Ω) ≤ C2 |θµ,λ |2L2 (Ω) + |∇θµ,λ |2L2 (Ω) + |∆0 θµ,λ |2L2 (Ω) |θµ,λ |2L∞ (Ω) ≤ M4 |∆0 θµ,λ |2L2 (Ω) + |∇θµ,λ |2L2 (Ω) + 1 . e. on (0, T ). 1) together with the above inequalities we find a required constant M2 . 2. There exists a positive constant M1 (λ) depending only on λ ∈ (0, 1] (and on the data p, f, h, θ0 and w0 as well, but not on µ ∈ (0, 1]) such that βˆλ (wµ,λ )dx |θµ,λ |2H 2 (Ω) + |Aµ wµ,λ |2L2 (Ω) + ϕµ (wµ,λ ) + sup t∈[0,T ] Ω T +|θµ,λ |2L2 (0,T ;L2 (Ω)) + |∇θµ,λ |2L2 (0,T ;L2 (Ω)) + ϕµ (wµ,λ )dt ≤ M1 (λ), 0 where βˆ is a primitive of β.
I By W(Q) we define the Banach space W(Q) = u(x, t)|u ∈ Lσ(x,t) (Q), Di u ∈ Lpi (x,t) (Q), u = 0 on Γ , u W(Q) = Di u Lpi (x,t) (Q) + u Lσ(x,t) (Q) i with the dual W (Q) = w| w ∈ Lσ (x,t) (Q) ∩ Lpi (x,t) (0, T ; W −1,pi (x,t) (Ω)) . Here and throughout the text we use the notation 1/s + 1/s = 1. 3. 1) is understood in the following sense. 1. 1) if for every test-function ζ ∈ Lσ(x,t) (Q) ∩ L∞ 0, T ; L2(Ω) such that ζ = 0 on ∂Ω × (0, T ), Di ζ ∈ Lpi (x,t) (Q), ζt ∈ L2 (Q), and every t1 , t2 ∈ [0, T ] the following identity holds: t2 ai |Di u|pi −2 Di u Di ζ − c|u|σ−2 u ζ + f ζ dz = uζt − t1 Ω i uζdx t2 t1 Ω .