Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions
Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions
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混合边界条件椭圆算子的正性与反极大值原理
DOI：
10.4171/jems/104
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发表时间：
2008
影响因子：
2.6
通讯作者：
W. Reichel
中科院分区：
文献类型：
--
作者：
C. Bandle;J. Below;W. Reichel
考虑有界Lipschitz域D R N上线性椭圆方程1u + q（x）u = u + f，其混合边界条件为@u/@n =（x）u +g.这类边值问题的主要特点是在方程和边界条件中都出现了。一般来说，我们不对系数（x）的符号做任何假设。我们研究积极性原则和反最大原则。我们的一个主要结果是，如果q为负，且R D q（x）dx > 0，则存在两个特征值1，1，使得正性原理对2（1，1）成立，反极大值原理对2（1，1 +）或2（1，1）成立.一个类似的，但更复杂的结果成立，如果q 0。这是因为在这种情况下0 = 0成为特征值，并且当的平均值与值0 = 0相交时，作为的函数的1（）连接到1（）。|D|/|@D|.在维数N = 1，我们确定的最佳区间，使反最大值的原则保持一致的所有右手边f，g 0。最后，我们将所得结果应用于常系数问题1u +q（x）u = u +f inD，@u/@n =u +g on@D，2 R.
We consider linear elliptic equations 1u + q(x)u = u + f in bounded Lipschitz domainsD R N with mixed boundary conditions@u/@n =(x)u +g on@D. The main feature of this boundary value problem is the appearance of both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient (x) . We study positivity principles and anti-maximum principles. One of our main results states that if is somewhere negative, q 0 and R D q(x)dx > 0 then there exist two eigenvalues 1, 1 such the positivity principle holds for 2 ( 1, 1) and the anti-maximum principle holds if 2 ( 1, 1 + ) or 2 ( 1 , 1). A similar, but more complicated result holds if q 0. This is due to the fact that 0 = 0 becomes an eigenvalue in this case and that 1() as a function of connects to 1() when the mean value of crosses the value 0 = | D|/|@D|. In dimension N = 1 we determine the optimal -interval such that the anti-maximum principles holds uniformly for all right-hand sides f,g 0. Finally, we apply our result to the problem 1u +q(x)u = u +f inD,@u/@n =u +g on@D with constant coefficients , 2 R.