Affiliation:
1. Institute of Mathematics , The University of Georgia , 79A M. Kostava str.; and A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6 Tamarashvili str. , Tbilisi 0177 , Georgia
Abstract
Abstract
The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary
𝒞
{\mathcal{C}}
in the non-classical setting when solutions are sought in the Bessel potential spaces
ℍ
p
s
(
𝒞
)
{\mathbb{H}^{s}_{p}(\mathcal{C})}
,
1
p
<
s
<
1
+
1
p
{\frac{1}{p}<s<1+\frac{1}{p}}
,
1
<
p
<
∞
{1<p<\infty}
.
Fredholm criteria and unique solvability criteria are found.
By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain
Ω
α
⊂
ℝ
2
{\Omega_{\alpha}\subset\mathbb{R}^{2}}
of magnitude α.
The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting.
The problems are investigated by the potential method and reduction to locally equivalent
2
×
2
{2\times 2}
systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes
ℝ
+
{\mathbb{R}^{+}}
in the Bessel potential spaces.
Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].
Funder
Shota Rustaveli National Science Foundation
Reference29 articles.
1. T. Buchukuri, R. Duduchava, D. Kapanadze and M. Tsaava,
Localization of a Helmholtz boundary value problem in a domain with piecewise-smooth boundary,
Proc. A. Razmadze Math. Inst. 162 (2013), 37–44.
2. L. P. Castro, R. Duduchava and F.-O. Speck,
Localization and minimal normalization of some basic mixed boundary value problems,
Factorization, Singular Operators and Related Problems (Funchal 2002),
Kluwer Academic, Dordrecht (2003), 73–100.
3. L. P. Castro, R. Duduchava and F.-O. Speck,
Mixed impedance boundary value problems for the Laplace–Beltrami equation,
to appear in J. Integral Equations Appl., https://projecteuclid.org/euclid.jiea/1580958082.
4. L. P. Castro and D. Kapanadze,
Wave diffraction by wedges having arbitrary aperture angle,
J. Math. Anal. Appl. 421 (2015), no. 2, 1295–1314.
5. M. Costabel and E. Stephan,
Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation,
Mathematical Models and Methods in Mechanics,
Banach Center Publ. 15,
PWN, Warsaw (1985), 175–251.
Cited by
1 articles.
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