| 牛培平.一类具多重特征的无解算子[J].数学研究及应用,1982,2(3):15~18 |
| 一类具多重特征的无解算子 |
| A Class of Unsolvable Operators wilh Multiple Characteristics |
| 投稿时间：1981-06-09 |
| DOI：10.3770/j.issn:1000-341X.1982.03.004 |
| 中文关键词: |
| 英文关键词: |
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| 摘要点击次数: 2188 |
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| 中文摘要: |
| 本文给出一类型如P(x,D)=D14 x14D24-(i1/2 (-i)1/2)D12D2 4x1D1D22-i(i1/2-(-i)1/2)x12D23 (1 2i)D22 C 或更一般地p(x,D)=LtL(x,D) C(L为无解算子)的多重特征算子。指出包括零阶项在内的低阶项对局部可解性能具有决定性影响。具体地说,在原点邻域上面所给算子p(x,D)的主部D14 x14D24为可解算子,当C=0时P(x,D)为不可解算子。但当C>0时又变为局部可解算子。类似地讨论了算子附加零阶项的一些情况。文章最后证明了当自由项f具形|x1|ψ＇(x2)(ψ为实函数)时,在原点邻域有古典解的充要条件为ψ(x2)解析。 |
| 英文摘要: |
| In This paper we consider a class of operators with multiple characteristics:P(x,D)=D14 x14D24-(i1/2 (-i)1/2)D12D2 4x1D1D22-i(i1/2-(-i)1/2)x12D23 (1 2i)D22 C. Generally P(x,D) = LtL(x,D) C where L is a operator which has no solutions. We conclude that the lower order terms, including the zero order term, influence essentially the local solvability, the principal part p4(x,D)=D14 x14D24 of p(x,D) is a solvable operator in a neighbourhood of origin, p(x,D) isn't solvable when C = 0, and p(x,D) becomes again solvable when C>0. We also discuss Грущин operator with zero term. Finally, we prove that if the nonhomogeneous term f(x1,x2) is |x1|ψ＇(x2)(ψ is a real function), there are solutions if and only if ψ(x2) is analytic. |
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