By George Boole

ISBN-10: 0828401284

ISBN-13: 9780828401289

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Example text

6. Let Ω be a bounded domain in RN , p ∈ (1, N ), and let f ∈ L+ (Ω), where q ∈ [Np/(Np − N + p), ∞]. Furthermore, let u, v be weak solutions of the following boundary value problems: 1,p u ∈ W0 (Ω), 1,p v ∈ W0 Ω |u| v − , p u = −∇ − pv |∇u|p−2 ∇u = f in Ω, = −∇ |∇v|p−2 ∇v = |f | in Ω . 26) Then 0 in Ω . 27) 1,p P ROOF. 25). ∞ (2) Assume that f ∈ L∞ + (Ω), and that ∂Ω ∈ C . g. [107,86]) that 1,α 1,α u ∈ C (Ω), and v ∈ C (Ω ), for some α ∈ (0, 1). 1 0,1 that u ∈ C0+ (Ω ). 57)), this reads as u ∈ C 0,1 ([0, |Ω|]).

Since 1,p (un )H → uH in Lp (ΩH ), and since the functions (un )H are equibounded in W0 (ΩH ), 1,p we find a function v ∈ W0 (ΩH ) and a subsequence (un )H which converges to v weakly in W 1,p (ΩH ). This means that for every ϕ ∈ C0∞ (ΩH ) and i ∈ {1, . . , N}, ΩH ϕvxi dx ←− ϕ ΩH −→ − ΩH ∂((un )H ) dx = − ∂xi ΩH ϕxi (un )H dx ϕxi uH dx, that is v = uH . The corollary is proved. 3. g. [106,68,96,45,39,44,20,72,69,112,37, 120] and the references cited therein). In this study we restrict ourselves to three types of symmetrizations on RN which are defined below.

13) follows from this in the general case, too. 13). The details are left to the reader. 2. Let Ω be a domain in RN and u ∈ W0+ (Ω) for some p ∈ [1, ∞). Then u ∈ W0 (Ω ) and u∗ ∈ W 1,p (Ω ∗ ). 1,p 0,1 (Ω) such that un → u in W 1,p (Ω). Since P ROOF. Choose a sequence {un } ⊂ C0+ supp un ⊂ Ω, it is easy to see that also supp u∗n = (supp un )∗ ⊂ Ω ∗ , so that u∗n ∈ W0 (Ω), 1,p n = 1, 2, . . Since u∗n u∗ in W 1,p (Ω), for a subsequence, we find that u∗ ∈ W0 (Ω ∗ ). The proof for the Schwarz symmetrization is analogous and will be omitted.

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A treatise on differential equations by George Boole


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