By G. V. Caffarelli et al.

Haim Brezis has made major contributions within the fields of partial differential equations and sensible research, and this quantity collects contributions by means of his former scholars and collaborators in honor of his sixtieth anniversary at a convention in Gaeta. It offers new advancements within the idea of partial differential equations with emphasis on elliptic and parabolic difficulties.

**Read Online or Download Elliptic and Parabolic Problems: A Special Tribute to the Work of Haim Brezis (Progress in Nonlinear Differential Equations and Their Applications) PDF**

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**Additional resources for Elliptic and Parabolic Problems: A Special Tribute to the Work of Haim Brezis (Progress in Nonlinear Differential Equations and Their Applications)**

**Sample text**

For some references see, for instance, [8] and [3]. 3) are proved in [8], where new ideas and techniques are developed. 17. 2) 24 H. 1) mostly concern the nonslip boundary condition u|Γ = 0 . 4) β uτ + τ (u)|Γ = b(x), appears to be quite important in many ﬁelds. Here n is the unit outward normal to the domain’s boundary Γ, β ≥ 0 is a given constant and b(x) is a given tangential vector ﬁeld. We denote by t = T · n the normal component of the tensor T , by uτ = u− (u· n) n the tangential component of u and by τ the tangential component of t τ (u) = t − (t · n)n.

2 in the “half-space version” given in reference [3] consists on a sequence of steps denoted below by (a), (b), (c), (b1) and (c1). 1. 2. In each step we prove the results shown in the following box. D∗2 u (a) ∈ L2 (Ω). |Du|p−2 ∇∗ Du ⎫ ⎧ 2 ⎬ ⎨ D u p−2 (b) ∈ Lp (Ω). |Du| 2 ∇∗ Du ⎭ ⎩ ∗ ∇ π ∇ π ∈ Lp (Ω). (c) (b1) in step (b) replace p (c1) in step (c) replace p by by l. m. Note that there is a loss of regularity in going from tangential to normal derivatives and in going from u to π. 2 (bounded set Ω) we do not take into account “additional regularity” in the tangential directions.

We remark that silicon solidiﬁes around the walls of the inner enclosure at the beginning of the process, and then the solidiﬁcation front is progressively getting ﬂatter. With Figure 7 c), showing the solidiﬁcation front at time t = 128000 s, we emphasize the fact that the solidiﬁcation front grows upwards as time increases; thus the top of the silicon ingot is the last part to solidify. 42 A. Berm´ udez, R. C. Mu˜ niz and F. Pena a) b) c) Figure 7. Solidiﬁcation front at a) t = 1000 s, b) t = 36000 s, c) t = 128000 s.