Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Abstract

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.

On non-local reaction-diffusion system in a bounded domain

“…Shahmurov [13] studied the Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. The aim of that work was to measure smoothness of solutions for the boundary datum in Besov spaces.…”

Free boundary value problems for abstract elliptic equations and applications

“…α−β,∞ denotes the real interpolation. Consequently, invoking Theorem 2.3 in [24], we find that 20) where M (E 1 , E 2 ) denotes the class of all operators in s having the property of maximal regularity in the sense of Da Prato and Grisvard [6]. The existence of a unique maximal classical solution of (E)g o and the property of a smooth semiflow on W can now be obtained along the lines of the proofs of Proposition 3.5 and Theorem 3.2 in [24].…”

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

“…The specific problems studied in this note are the Cauchy problems of elliptic systems such as Laplace and Helmholtz [10,11] operators within annulus domains, where no information on the interior boundary is available.…”