We consider a model elliptic pseudo-differential equation in a 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of a special boundary value problem for this equation can have a limit with respect to endpoint values of the parameters in appropriate Sobolev - Slobodetskii space if the boundary function is a solution of a special functional singular integral equatio

Chernova, O.

Lukinova, O.

Vasilyev, V.

DSpace at Belgorod State University

ViewOnlineExportCitationCrossMarkRESEARCH ARTICLE |  JUNE 08 2023On solvability of certain non-standard elliptic problems Vladimir Vasilyev ; Olga Chernova; Oksana LukinovaAIP Conference Proceedings 2781, 020004 (2023)https://doi.org/10.1063/5.0144767Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdfOn Solvability of Certain Non-Standard Elliptic ProblemsVladimir Vasilyev,a) Olga Chernova,b) and Oksana Lukinovac)Applied Mathematics and Computer Modeling, Belgorod State National Research University, Belgorod, Russian Federationa)Corresponding author: vladimir.b.vasilyev@gmail.comb)chernova_olga@bsu.edu.ruc)lukinova@bsu.edu.ruAbstract. We consider a model elliptic pseudo-differential equation in a 4-wedge conical canonical 3D singular domain with twoparameters. It is shown that the solution of a special boundary value problem for this equation can have a limit with respect toendpoint values of the parameters in appropriate Sobolev – Slobodetskii space if the boundary function is a solution of a specialfunctional singular integral equation.INTRODUCTIONA lot of papers (see, for example, [1, 2, 3]) are devoted to constructing and developing the theory of elliptic pseudo-differential operators and equations on non-smooth manifolds or manifolds with non-smooth boundaries. The term"theory" means an existence in these stidies Fredholmness and index theorems.Using the local principle and studying invertibility properties for model operators in so called canonical domainswe at the same time investigate Fredholm properties for general elliptic pseudo-differential operators [4, 5, 6, 7]. Letus note that distinct model operators and canonical domains generate distinct Fredholm theories.By canonical domain as usual we mean a certain cone in m-dimensional space. It may be a whole space Rm, ahalf-space Rm+ = {x ∈ Rm : x = (x′,xm),xm > 0} or a certain typical cone in Rm.Let C be a convex cone in Rm non including a straight line. Let us consider a pseudo-differential operator of thefollowing type(Au)(x) =∫C∫RmA(ξ )ei(x−y)·ξ u(y)dξ dy, x ∈C,and a model equation(Au)(x) = v(x), x ∈C, (1)assuming that the symbol A(ξ ) of the operator A satisfies the conditionc1(1+ |ξ |)α ≤ |A(ξ )| ≤ c2(1+ |ξ |)α ,α ∈ R. (2)We would like to remind that new approach to studying pseudo-differential equations on manifolds with a non-smooth boundary was developed in [8]. It is based on studying invertibility conditions for a model pseudo-differentialoperator A or conditions of unique solvability for the model equation (1) in appropriate functional spaces. To describethese conditions the concept of wave factorization for an elliptic symbol was used [8].ELLIPTIC EQUATION IN A MULTIDIMENSIONAL CONEWe will start from three-dimensional case. Let Cab+ be a conical canonical domain of the following typeCab+ = {x ∈R3 : x = (x1,x2,x3),x3 > a|x1|+ b|x2|,a,b > 0}.As usual Ã(ξ ) denotes symbol of the pseudo-differential operator A, it does not depend on a spatial variable x andsatisfies the condition (2). Here we remind some definitions from [9].A radial tube domain over the cone Cab+ is called a domain in 3-dimensional complex space C3 of the followingtypeT (Cab+ )≡ {z ∈ C3 : z = x+ iy,x ∈ R3,y ∈Cab+ }.Modern Problems of Applied Mathematics and Information Technology (MPAMIT 2021)AIP Conf. Proc. 2781, 020004-1–020004-5; https://doi.org/10.1063/5.0144767Published by AIP Publishing. 978-0-7354-4535-2/$30.00020004-1Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdfA conjugate cone∗Cab+ is called such a cone in which for all points the conditionx · y > 0, ∀y ∈Cab+ ,holds; x · y means inner product for x and y.According to [10] we introduce the wave factorization for a symbol.Definition 1. Wave factorization of elliptic symbol Ã(ξ ) with respect to the cone Cab+ is called its representation inthe formÃ(ξ ) = Ã 6=(ξ )Ã=(ξ ),where the factors Ã 6=(ξ ), Ã=(ξ ) satisfy the following conditions:1) Ã 6=(ξ ), Ã=(ξ ) are defined everywhere excluding may be the points {ξ ∈ R3 : ξ3 =12a|ξ1|+12b|ξ2|};2) Ã 6=(ξ ), Ã=(ξ ) admit analytic continuation into radial tube domains T (∗Cab+ ),T (−∗Cab+ ) respectively with estimates|A±16= (ξ + iτ)| ≤ c1(1+ |ξ |+ |τ|)±,|A±1= (ξ − iτ)| ≤ c2(1+ |ξ |+ |τ|)±(α−), ∀τ ∈∗Cab+ .The real number is called an index of the wave factorization.Remark 1. Everywhere below we assume that such a factorization exists.We consider multidimensional equation (1) in analogous Sobolev–Slobodetskii space Hs(Cab+ ) [10]. For simplicitywe study the homogeneous equation(Au)(x) = 0,x ∈Cab+ . (3)To describe a general solution of the equation (1) for the case − s = 1+δ , |δ |< 1/2 we use some results from [11].Further, we introduce the following singular integral operators [12, 13](S1u)(ξ1,ξ2,ξ3) = v.pi2π+∞∫−∞u(τ,ξ2,ξ3)dτξ1 − τ,(S2u)(ξ1,ξ2,ξ3) = v.pi2π+∞∫−∞u(ξ1,η ,ξ3)dηξ2 −ηand writeA 6=(ξ )ũ(ξ ) = C̃1(ξ1 − aξ3,ξ2 − bξ3)+ C̃2(ξ1 − aξ3,ξ2 + bξ3)+C̃3(ξ1 + aξ3,ξ2 − bξ3)+ C̃4(ξ1 + aξ3,ξ2 + bξ3), (4)whereC̃1(ξ1 − aξ3,ξ2 − bξ3) =14c̃0(ξ1 − aξ3,ξ2 − bξ3)−12(S1c̃0)(ξ1 − aξ3,ξ2 − bξ3)−12(S2c̃0)(ξ1 − aξ3,ξ2 − bξ3)+ (S1S2c̃0)(ξ1 − aξ3,ξ2 − bξ3);020004-2Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdfC̃2(ξ1 − aξ3,ξ2 + bξ3) =14c̃0(ξ1 − aξ3,ξ2 + bξ3)−12(S1c̃0)(ξ1 − aξ3,ξ2 + bξ3)+12(S2c̃0)(ξ1 − aξ3,ξ2 + bξ3)− (S1S2c̃0)(ξ1 − aξ3,ξ2 + bξ3);C̃3(ξ1 + aξ3,ξ2 − bξ3) =14c̃0(ξ1 + aξ3,ξ2 − bξ3)+12(S1c̃0)(ξ1 + aξ3,ξ2 − bξ3)−12(S2c̃0)(ξ1 + aξ3,ξ2 − bξ3)− (S1S2c̃0)(ξ1 + aξ3,ξ2 − bξ3);C̃4(ξ1 + aξ3,ξ2 + bξ3) =14c̃0(ξ1 + aξ3,ξ2 + bξ3)+12(S1c̃0)(ξ1 + aξ3,ξ2 + bξ3)+12(S2c̃0)(ξ1 + aξ3,ξ2 + bξ3)+ (S1S2c̃0)(ξ1 + aξ3,ξ2 + bξ3).To determine uniquely the arbitrary function c0(ξ1,ξ2) we require certain additional condition. for example, weassume that the restriction ũ(ξ1,ξ2,0) is given, i.e. the following integral+∞∫−∞u(x1,x2,x3)dx3 ≡ g(x1,x2), (5)it gives the equalityũ(ξ1,ξ2,0) = g̃(ξ1,ξ2). (6)Substituting (6) into (4) we obtain all summands. Indeed,Ã 6=(ξ )ũ(ξ ) =4∑k=1C̃k(ξ1,ξ2)=14c̃0(ξ1,ξ2)−12(S1c̃0)(ξ1,ξ2)−12(S2c̃0)(ξ1,ξ2)+ (S1S2c̃0)(ξ1,ξ2)+14c̃0(ξ1,ξ2)−12(S1c̃0)(ξ1,ξ2)+12(S2c̃0)(ξ1,ξ2)− (S1S2c̃0)(ξ1,ξ2)+14c̃0(ξ1,ξ2)+12(S1c̃0)(ξ1,ξ2)−12(S2c̃0)(ξ1,ξ2)− (S1S2c̃0)(ξ1,ξ2)+14c̃0(ξ1,ξ2)+12(S1c̃0)(ξ1,ξ2)+12(S2c̃0)(ξ1,ξ2)+ (S1S2c̃0)(ξ1,ξ2) = c̃0(ξ1,ξ2).Taking into account the condition (6) we findc̃0(ξ′) = Ã 6=(ξ′,0)g̃(ξ ′). (7)Theorem 1. Let − s = 1+ δ , |δ |< 1/2,g ∈ Hs+1/2(R2). Then the unique solution of the problem (3),(5) is given bythe formula (4), and c0(x1,x2) is determined by the formula (7).020004-3Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdfDEGENERATED CASEIn the above section we have two parameters of the cone a and b. Degenerated case corresponds to that case whenone of the parameters (or both) tends to 0 or +∞. Let us note that case a,b → 0 was studied in [4], the casesa → 0,b = const and a = const,b → 0 were studied in [10].Here we consider limit case a →+∞,b = const, the casea = const,b →+∞ looks almost the same..Starting point for this consideration will be the equality (4). We use the change of variables ξ1−aξ3 = t1,ξ1+aξ3 =t3 from which we have ξ1 =t3+t12,ξ3 =t3−t12a. So, we have new variables t1,ξ2, t3. If we use the condition (5) we canfind the unknown function c̃0 by the formula (7). Now let us write the formula (4) for the new variables t1,ξ2, t3. Thenwe obtainA 6=(t2 + t12,ξ2,t3 − t12a)ũ(t2 + t12,ξ2,t3 − t12a)= C̃1(t1,ξ2 − bt3− t12a)+ C̃2(t1,ξ2 + bt3 − t12a)+C̃3(t3,ξ2 − bt3 − t12a)+ C̃4(t3,ξ2 + bt3 − t12a). (8)Tending a to +∞ we obtain the following relationA 6=(t2 + t12,ξ2,0)ũ(t2 + t12,ξ2,0)= C̃1 (t1,ξ2)+ C̃2 (t1,ξ2)+ C̃3 (t3,ξ2)+ C̃4 (t3,ξ2) .After accurate calculations we findC̃1 (t1,ξ2)+ C̃2 (t1,ξ2)+ C̃3 (t3,ξ2)+ C̃4 (t3,ξ2) =c̃0(t1,ξ2)+ c̃0(t3,ξ2)2− (S1c̃0)(t1,ξ2)+ (S1c̃0)(t3,ξ2).Taking into account the condition (6), the formula (7) and new notationÃ 6=(ξ1,ξ2,0)g̃(ξ1,ξ2)≡ h(ξ1,ξ2),we obtain the following equation with parameter ξ2h(t2 + t12,ξ2)=h(t1,ξ2)+ h(t3,ξ2)2− (S1h)(t1,ξ2)+ (S1h)(t3,ξ2). (9)Thus, we have the following property.Theorem 2. If the symbol A(ξ ) admits the wave factorization with respect to Cab+ with the index such that − s =1+ δ , |δ |< 1/2 for enough large a then the unique solution of the boundary value problem (3),(5) has a limit undera →+∞ if and only if the boundary function g ∈ Hs+1/2(R2) satisfies the equation (9).CONCLUSIONWe have considered here a homogeneous equation only, it was done for simplicity. More general and more interestingsituation is considering non-homogeneous equations. Some steps in this direction were suggested, but this approachshould be verified.REFERENCES1. J. V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications (Birkhauser, Basel, 1997).2. V. E. Nazaikinski, A. Y. Savin, B.-W. Schulze, and B. Y. Sternin, Elliptic Theory on Singular Manifolds (Chapman and Hall/CRC, BocaRaton, 2006).3. B.-W. Schulze, B. Y. Sternin, and V. Shatalov, Differential Equations on Singular Manifolds; Semiclassical Theory and Operator Algebras(Wiley, Berlin, 1998).020004-4Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdf4. G. Eskin, Boundar Value Problems for Elliptic Pseudodifferential Equations (Providence (RI): AMS, 1981).5. V. B. Vasilyev, Math. Bohem. 139, 333–340 (2014).6. V. B. Vasilyev, Adv. Dyn. Syst. Appl. 9, 227–237 (2014).7. V. B. Vasilyev, Math. Meth. Appl. Sci. 41, 9252–9263 (2018).8. V. B. Vasil’ev, Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems inNon-Smooth Domains (Kluwer Academic Publishers, Dordrecht–Boston–London, 2000).9. V. S. Vladimirov, Methods of the Theory of Generalized Functions (Taylor & Francis, 2002).10. V. B. Vasilyev, Lobachevskii Journal of Mathematics 41, 917–925 (2020).11. V. B. Vasilyev, Opusc. Math. 39, 109–124 (2019).12. F. D. Gakhov, Boundary Value Problems (Dover Publications, Mineola, NY, 1981).13. N. I. Muskhelishvili, Singular Integral Equations (North Holland, Amsterdam, 1976).020004-5Downloaded from http://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0144767/17985537/020004_1_5.0144767.pdf

On Solvability of Certain Non-Standard Elliptic Problems

http://dspace.bsu.edu.ru/bitstream/123456789/61798/1/Vasilev_Solvability_23.pdf

On Solvability of Certain Non-Standard Elliptic Problems

Abstract

Similar works

Full text

Available Versions

DSpace at Belgorod State University