Reproducing kernels and minimal solutions of elliptic equations

被引:3
|
作者
Zynda, Tomasz Lukasz [1 ]
Sadowski, Jacek Jozef [2 ]
Wojcicki, Pawel Marian [3 ]
Krantz, Steven George [4 ]
机构
[1] Mil Univ Technol, Fac Cybernet, Radiowa 22, PL-01485 Warsaw, Poland
[2] Univ Warsaw, Fac Math Informat & Mech, Stefana Banacha 2, PL-02097 Warsaw, Poland
[3] Warsaw Univ Technol, Fac Math & Informat Sci, Koszykowa 75, PL-00662 Warsaw, Poland
[4] Washington Univ, Dept Math, Campus Box 1146,One Brookings Dr, St Louis, MO 63130 USA
关键词
Reproducing kernel Hilbert space; elliptic operator; solutions of elliptic equations; Ramadanov theorem; continuous dependence of reproducing kernel;
D O I
10.1515/gmj-2022-2202
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L-2-norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczynski, while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere.
引用
收藏
页码:303 / 320
页数:18
相关论文
共 50 条