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• W2963429272 abstract "Abstract The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>Ω</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {Omega_{alpha}subsetmathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>ℍ</m:mi> <m:mi>p</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>Ω</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathbb{H}^{s}_{p}(Omega_{alpha})} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>s</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>p</m:mi> </m:mfrac> </m:mrow> </m:math> {s>frac{1}{p}} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>𝕎</m:mi> <m:mi>p</m:mi> <m:mrow> <m:mi>s</m:mi> <m:mo>-</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>𝕎</m:mi> <m:mi>p</m:mi> <m:mi>r</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathbb{W}^{r}_{p}(mathbb{R}^{+})} and Bessel potential <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>ℍ</m:mi> <m:mi>p</m:mi> <m:mi>r</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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." @default.
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• W2963429272 date "2019-07-12" @default.
• W2963429272 modified "2023-09-24" @default.
• W2963429272 title "Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain" @default.
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• W2963429272 doi "https://doi.org/10.1515/gmj-2019-2031" @default.
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