scholarly journals Convolution Type Operators with Symmetry in Bessel Potential Spaces

2017 ◽  
pp. 21-49
Author(s):  
Luís Pinheiro de Castro ◽  
Frank-Olme Speck
Keyword(s):  
Convolution Type ◽  
Bessel Potential ◽  
Bessel Potential Spaces

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Beltrami Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
Classical Setting ◽  
Type Boundary ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe 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 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{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 \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\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \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].

scholarly journals Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

2020 ◽  
Vol 27 (2) ◽  
pp. 211-231
Author(s):  
Roland Duduchava ◽  
Medea Tsaava
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Bessel Potential ◽  
Angular Domain ◽  
Mixed Boundary Value Problems ◽  
Classical Setting ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

Boundary Problems of Thermopiezoelectricity in Domains with Cuspidal Edges

2000 ◽  
Vol 7 (3) ◽  
pp. 441-460 ◽  
Author(s):  
T. Buchukuri ◽  
O. Chkadua
Keyword(s):  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Bessel Potential ◽  
Manifolds With Boundary ◽  
Asymptotics Of Solutions ◽  
Boundary Problems ◽  
Neumann Type ◽  
Type Boundary ◽  
Homogeneous Anisotropic Medium ◽  
Bessel Potential Spaces

Abstract Dirichlet- and Neumann-type boundary value problems of statics are considered in three-dimensional domains with cuspidal edges filled with a homogeneous anisotropic medium. Using the method of the theory of a potential and the theory of pseudodifferential equations on manifolds with boundary, we prove the existence and uniqueness theorems in Besov and Bessel-potential spaces, and study the smoothness and a complete asymptotics of solutions near the cuspidal edges.

scholarly journals Estimates for capacities and traces of potentials

1984 ◽  
Vol 7 (1) ◽  
pp. 41-63 ◽  
Author(s):  
V. G. Maz'ja ◽  
S. P. Preobrazenskii
Keyword(s):  
Sufficient Conditions ◽  
Isoperimetric Inequalities ◽  
Necessary And Sufficient Conditions ◽  
Sobolev Type ◽  
Bessel Potential ◽  
Necessary And Sufficient ◽  
Bessel Potential Spaces

It is shown that “isoperimetric” inequalities, relating measures and capacities, hold for all sets inℝnif they are valid for all balls. As a corollary, the necessary and sufficient conditions for the continuity of some imbeddings of M. Riesz and Bessel potential spaces are obtained. The introduction contains a survey of known results on inequalities of Sobolev type.

scholarly journals Norms of embeddings of logarithmic Bessel potential spaces

1998 ◽  
Vol 126 (8) ◽  
pp. 2417-2425 ◽  
Author(s):  
David E. Edmunds ◽  
Petr Gurka ◽  
Bohumír Opic
Keyword(s):  
Bessel Potential ◽  
Bessel Potential Spaces
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