Embedding operators of Sobolev spaces with variable exponents and applications
Tóm tắt
We introduce the vector-valued Sobolev spaces W
m,p(x) (Ω;E
0,E) with variable exponent associated with two Banach spaces E
0 and E. The most regular space E
α
is found such that the differential operator D
α
is bounded and compact from W
m,p(x)(Ω;E
0,E) to L
q(x)(Ω;E
α
), where E
α
are interpolation spaces between E
0 and E is depending on α = (α
1, α
2,..., α
n
) and the positive integer m, where Ω ⊂ ℝ
n
is a region such that there exists a bounded linear extension operator from W
m,p(x) (Ω;E
0,E) to W
m,p(x) (ℝ
n
;E (A), E). The function p(x) is Lipschitz continuous on Ω and q(x) is a measurable function such that
$$1 < p(x) \leqslant q(x) \leqslant \tfrac{{np(x)}}
{{n - mp(x)}}$$
for a.e.
$$x \in \bar \Omega $$
. Ehrling–Nirenberg–Gagilardo type sharp estimates for mixed derivatives are obtained. Then, by using this embedding result, the separability properties of abstract differential equations are established.
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