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Tính khả giải L p của bài toán Dirichlet cho các phương trình elliptic trong mặt phẳng, Các kết quả chính xác
Tóm tắt
Giả sử rằng toán tử elliptic L=div (A(x)∇) là L
p
-khả giải, p>1, trên đĩa đơn vị
$\mathbb{D}\subset \mathbb {R}^{2}$
. Điều này có nghĩa là bài toán Dirichlet
$$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{trong }\mathbb{D},\\[3pt]u=g&\mbox{trên }\partial\mathbb{D}\end{array}\right.$$
có nghiệm duy nhất cho bất kỳ
$g\in L^{p}(\partial\mathbb{D})$
. Sau đó, tồn tại ε>0 sao cho L là L
r
- khả giải trong khoảng tối ưu p−ε
Từ khóa
Tài liệu tham khảo
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