On Unique Minimal Sobolev Norm Element of Banach Spaces of Functions which Takes a Given Value in a Fixed Point
Download PDF
Authors: T. Ł. ŻYNDA
DOI: 10.46793/KgJMat2610.1685Z
Abstract:
First, it will be shown that some Banach spaces V of functions, which are subspaces subspaces of Sobolev spaces satisfy the c-minimal norm property, i.e., in any set
if non-empty, there is exactly one element with t minimal Sobolev norm. Later, it will be proved that this element depends continuously on the deformation of the norm and on an increasing sequence of domains in a precisely defined sense. We conclude with applications to the theory of linear partial differential equations.
Keywords:
Sobolev spaces, minimal norm element, continuous dependence, linear partial differential equation
References:
[1] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, Rhode Island, 1998.
[2] R. Figueroa and R. L. Pouso, Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments, Bound. Value Probl. 2012 (2012), Paper ID 7. https://doi.org/10.1186/1687-2770-2012-7
[3] Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, International Journal of Mathematics and Mathematical Sciences 15(1) (1992), 1–14. https://doi.org/10.1155/S0161171292000012
[4] I. P. Ramadanov, Sur une propriété de la fonction de Bergman, C. R. Acad. Bulgare Sci. 20 (1967).
[5] G. M. Troianiello, Maximal and minimal solutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91(1) (1984), 95–101. https://doi.org/10.2307/2045277
[6] T. Ł. Żynda, J. J. Sadowski, P. M. Wójcicki and S. G. Krantz, Reproducing kernels and minimal solutions of elliptic equations, Georgian Math. J. 30(2) (2023), 303–320.
Login:
