Spectral properties of the operator pencils generated by second order elliptic boundary-Value problems depending on parameter in an angle

Abstract. In this paper we consider spectral properties of the operator pencils generated by boundary-value problems for second order elliptic equation depending parameter in a plane angle. We receive explicit formula for the eigenvalues and the eigenvectors of the operator pencil, the multiplicity of the eigenvalues, and the differentiability of the eigenvalues and the eigenvectors with respect to the parameter.

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JOURNAL OF SCIENCE OF HNUE Natural Sci., 2008, Vol. 53, N ◦ . 5, pp. 41-48 SPECTRAL PROPERTIES OF THE OPERATOR PENCILS GENERATED BY SECOND ORDER ELLIPTIC BOUNDARY-VALUE PROBLEMS DEPENDING ON PARAMETER IN AN ANGLE Nguyen Manh Hung and Nguyen Thanh Anh Hanoi National University of education Abstract. In this paper we consider spectral properties of the operator pen- cils generated by boundary-value problems for second order elliptic equation depending parameter in a plane angle. We receive explicit formula for the eigenvalues and the eigenvectors of the operator pencil, the multiplicity of the eigenvalues, and the differentiability of the eigenvalues and the eigen- vectors with respect to the parameter. Keywords: Elliptic equation, boundary-value problem, nonsmooth domains, pencil of operators. 1. Introduction Boundary value problems and initial boundary value problems have been con- sidered for various types of partial differential operators in domains with various types of nonsmooth boundaries. For the boundary-value problems for elliptic equa- tion in domains with conical points, the solvability of the problem, the smoothness and the asymptotic of the solutions are determined by the spectral properties the op- erator pencils generated by the problems such as the location and the multiplicities of the eigenvalues (see [1,2,3]). Elliptic boundary-value problems depending on a pa- rameter arise when ones consider initial-boundary-value problems for nonstationary equations with coefficients depending on time (see, for example, [4,5]). In general, it is not easy to locate the eigenvalues and calculate thei multiplicities. However, for the second order equation in a plane angle, these can be done by establishing explicit formulas for the eigenvalues and the eigenvectors. This is the main purpose of the present paper. Our paper is organized as follows. In Section 2, we introduce some needed notations and definitions. The main results are presented in Section 3 and 4. 2. Preliminaries Let K = {x = (x1, x2) ∈ R2 : r > 0, 0 < ω < ϕ} be a plane angle with vertex at the origin. Here r, ω are the polar coordinates of the point x = (x1, x2) and 0 < ϕ < 2pi, ϕ 6= pi. Let T be a positive real number or T = +∞. Set Γ0 = {x ∈ 41 Nguyen Manh Hung and Nguyen Thanh Anh R2 : r > 0, ω = 0}, Γ1 = {x ∈ R2 : r > 0, ω = ϕ}, ΓjT = Γ j × [0, T ], j = 0, 1. For each multi-index α = (α1, α2), set |α| = α1 + α2 and ∂ α x = ∂ |α|/∂α1x1 ∂ α2 x2 . Let L = L(t, ∂x) be a homogeneous elliptic operator of second order in K with coefficients depending on a parameter t ∈ [0, T ] L = L(t, ∂x) = 2∑ j,k=1 ajk(t)∂xj∂xk . We consider the elliptic boundary-value problem depending on parameter: L(t, ∂x)u = f in KT , (2.1) Bj(t, ∂x)u = gj on ,Γ j T j = 0, 1 (2.2) where Bj(t, ∂x) are homogeneous differential operators on the boundaries Γ j T Bj(t, ∂x) = ∑ |α|=µj bj,α(t)∂ α x , µj = 0 or µj = 1. We can write L(t, ∂x), Bj(t, ∂x) in the form L(t, ∂x) = r −2 L(ω, t, ∂ω, r∂r), Bj(t, ∂x) = r −µjBj(ω, t, ∂ω, r∂r). The operator U(λ, t) = (L(ω, t, ∂ω, λ),B0(ω, t, ∂ω, λ),B1(ω, t, ∂ω, λ)), λ ∈ C, t ∈ [0, T ] is called the pencil of operators generated by the problem (2.1)-(2.2). For every fixed λ ∈ C, t ∈ [0, T ] this operator continuously maps X := H l((0, ϕ)) into Y := H l−2((0, ϕ))× C2 (l > 2), where H l((0, ϕ)) is the usual Sobolev space of functions defined in (0, ϕ). We mention now some well-known definitions in [2]. Let t0 ∈ [0, T ] be fixed. Let λ0 be an eigenvalue of U(λ, t0) with geometric multiplicity I and ϕ0 ∈ X be an eigenvector corresponding to λ0, that means, ϕ0 6= 0,U(λ0, t0)ϕ0 = 0 and I = dimkerU(λ0, t0). If the elements ϕ1, . . . , ϕs of X satisfy the equations σ∑ q=0 1 q! ∂q dλq U(λ0, t0)ϕσ−q = 0 for σ = 1, . . . , s, (2.3) then the ordered collection ϕ0, ϕ1, . . . , ϕs is said to be a Jordan chain corresponding to the eigenvalue λ0 of the length s+1. The elements ϕ1, . . . , ϕs are called generalized 42 Spectral properties of the operator pencils generated by second order elliptic... eigenvectors. The rank of the eigenvector ϕ0 (rankϕ0) is the maximal length of the Jordan chains corresponding to the eigenvector ϕ0. The sum of ranks of all eigenvectors linearly independent of U(λ0, t0) is called the algebraic multiplicity of the eigenvalue λ0. An eigenvalue is said to be simple if its algebraic multiplicity is equal to one. It is well-known that for each t ∈ [0, T ] the spectrum of the operator pencil U(λ, t) is an enumerable set of eigenvalues with the finite algebraic multiplicities (see [2, Theo. 5.2.1]). To investigate boundary-value problems for elliptic equations in domains with conical points we are concerned with understanding properties of eigenvalues of the pencil U(λ, t) such as the location, the multiplicity and the smoothness with respect to the parameter t. 3. The Dirichlet boundary problem We can take advantage that the operator L is written in the form L(t, ∂x) = (∂x2 − a1(t)∂x1)(∂x2 − a2(t)∂x1), where aj(t) (j = 1, 2) are complex-value functions defined on [0, T ] and Im a1(t) > 0, Im a2(t) < 0 for all t ∈ [0, T ]. In this section, we consider the Dirichlet boundary problem depending on a param- eter: L(t, ∂x)u = f in KT , (3.1) u =gj on Γ j T ,j = 0, 1. (3.2) In the polar coordinates (r, ω) the problem (3.1), (3.2) has the form r−2L(t, ω, ∂ω, r∂r)u = f in KT , u|ω=0 = g0, u|ω=α = g1, where L(t, ω, ∂ω, λ) = 2∏ j=1 (( sinω − aj(t) cosω ) λ+ ( cosω + aj(t) sinω ) ∂ω ) . (3.3) Thus, a complex number λ is an eigenvalue of the operator pencil U(λ, t) generated by the problem (3.1)-(3.2) if the following problem L(t, ω, ∂ω, λ)u = 0 in (0, ϕ), (3.4) u|ω=0 = 0, (3.5) u|ω=α = 0 (3.6) 43 Nguyen Manh Hung and Nguyen Thanh Anh has a nontrivial solution. If λ = 0, w1(ω, t) = 1, w2(ω, t) = ω are two linearly independent solutions of the equation (3.4). We can verify directly that for each t ∈ [0, T ] there are no constants c1(t), c2(t) such that the function u = c1(t)u1 + c2(t)u2 satisfies the boundary conditions (3.5), (3.6). Hence, λ = 0 is not an eigenvalue of U(λ, t). For λ 6= 0, by calculating directly we can see that the functions vj(ω, t) = e λFj(ω,t), j = 1, 2, (3.7) are solutions of the equation (3.4), where Fj(ω, t) = ∫ ω 0 fj(θ, t)dθ, fj(ω, t) = − sinω + aj(t) cosω cosω + aj(t) sinω . The Wronskian of v1(ω, t) and v2(ω, t) at ω = 0 is∣∣∣∣ 1 1λa1(t) λa2(t) ∣∣∣∣ = λ(a2(t)− a1(t)) 6= 0 for all t ∈ [0, T ]. Thus, for λ 6= 0, v1(ω, t), v2(ω, t) are two linearly independent solutions of the equa- tion (3.4), and therefore, the its general solution is u = c1(t)v1 + c2(t)v2. Inserting this function into the boundary conditions (3.5), (3.6), we get the system{ c1(t) + c2(t) = 0 c1(t)e λF1(α,t) + c2(t)e λF2(α,t) = 0. (3.8) The coefficients determinant of this system is equal to D(λ, t) = eλF2(α,t) − eλF1(α,t). A complex number λ is an eigenvalue of U(λ, t) if the system has nontrivial solutions c1(t), c2(t) or, in other words, λ is a solution of the following equation D(λ, t) = eλF2(α,t) − eλF1(α,t) = 0. (3.9) This equation is equivalent to eλ(F2(α,t)−F1(α,t)) = 1. (3.10) Since Im a1(t) > 0, Im a2(t) < 0, t ∈ [0, T ], by calculating directly, we have (see [1, Theo. 14.6]) Im ( F2(α, t)− F1(α, t) ) < 0 for all t ∈ [0, T ]. Therefore, the equation (3.10) has solutions λk(t) = ik2pi F2(α, t)− F1(α, t) , k = ±1,±2, . . . (3.11) 44 Spectral properties of the operator pencils generated by second order elliptic... Thus, the eigenvalues of U(λ, t) are λk(t), k = ±1,±2, . . . For each eigenvalue λk(t), choosing c1(t) = 1, c2(t) = −1 in (3.8), we have the corresponding eigenvector is ψk(ω, t) = e λk(t)F1(ω,t) − eλk(t)F2(ω,t), k = ±1,±2, . . . (3.12) For each t0 ∈ [0, T ] fixed, by lemmas 3.1.1 and 3.1.2 of [3], the algebraic multiplicity of an eigenvalue λ0 of U(λ, t0) coincides with the multiplicity of the zero λ = λ0 of the function D(λ, t0). Moreover, we have d dλ D(λ, t)|λ=λk(t) = e λk(t)F2(α,t)f2(α, t)− e λF1(α,t)f1(α, t) = eλk(t)F1(α,t) ( f2(α, t)− f1(α, t) ) = eλk(t)F1(α,t) a2(t)− a1(t) (cosω + a1(t) sinω)(cosω + a2(t) sinω) 6= 0 (3.13) for all t ∈ [0, T ]. This means the zeros λk(t) of D(λ, t) have multiplicity one, and therefore, the eigenvalues λk(t)(k = ±1,±2, . . .) ofU(λ, t) are simple for all t ∈ [0, T ]. From the facts above, we get Theorem 3.1. Suppose that the functions a1(t), a2(t) belong to the class C h([0, T ]), where h is a natural number. Then all eigenvalues of the pencil U(λ, t) are defined by the formula (3.11); these eigenvalues are simple and belong to class the Ch([0, T ]). The corresponding eigenvectors are given by the formula (3.12); they are infinitely differentiable with respect to the variable ω and h-times continuously differentiable with respect to the variable t. 4. The oblique derivative problem In this section, we consider the oblique derivative problem depending on a parameter: L(t, ∂x)u = f in KT , (4.1) ∂νju+ bj(t)∂τju = gj on Γ j T , j = 0, 1, (4.2) where τj are the directions of the rays Γ j , and νj are the exterior normals to the sides Γj of the angle K, bj(t) are real-value functions defined in [0, T ]. If bj ≡ 0, j = 0, 1, then (4.1)-(4.2) is the Newmann boundary problem. In the polar coordinates (r, ω) the problem (4.1), (4.2) has the form r−2L(t, ω, ∂ω, r∂r)u = f in KT , r−1(−∂ω + b0(t)∂r)u = g0 on Γ 0 T , r−1(∂ω + b1(t)∂r)u = g1 on Γ 1 T , 45 Nguyen Manh Hung and Nguyen Thanh Anh where L(t, ω, ∂ω, λ) is defined as in (3.3). Thus a complex number λ is an eigenvalue of the operator pencil U(λ, t) generated by the problem (4.1)-(4.2) if the following problem L(t, ω, ∂ω, λ)u = 0 in (0, ϕ), (4.3) (−∂ωu+ b0(t)λu)|ω=0 = 0, (4.4) (∂ωu+ b1(t)λu)|ω=α = 0 (4.5) has a nontrivial solution. If λ 6= 0, as in Section 3, the general solution of the equation (4.3) has the form u = c1(t)v1 + c2(t)v2, (4.6) where v1, v2 are defined as in (3.7). Inserting the function (4.6) into the boundary conditions (4.4), (4.5), we get the system{ λ(a1(t) + b0(t))c1(t) + λ(a2(t) + b0(t))c2(t) = 0, λeλF1(α,t)(f1(α, t) + b1(t))c1(t) + λe λF2(α,t)(f2(α, t) + b1(t))c2(t) = 0. (4.7) The coefficients determinant D(λ, t) of this system is equal to λ2 ( (a1(t) + b0(t))(f1(α, t) + b1(t))e λF2(α,t) − (a2(t) + b0(t))(f2(α, t) + b1(t))e λF1(α,t) ) . Since Im a1(t) > 0, Im a2(t) < 0 for all t ∈ [0, T ] and b0(t) is a real function, then aj(t) + b0(t) 6= 0 for all t ∈ [0, T ] (j = 1, 2). (4.8) Moreover, we have fj(ω, t) = (− sinω + aj(t) cosω)(cosω + aj(t) sinω) | cosω + aj(t) sinω|2 = (|aj(t)| 2 − 1) sinω cosω +Re aj(t)(cos 2 ω − sin2 ω) + i Im aj(t) | cosω + aj(t) sinω|2 (j=1,2). It follows from this equality that fj(ω, t) (j=1,2) are not real for all ω ∈ [0, α] and all t ∈ [0, T ]. Therefore, fj(α, t) + b1(t) 6= 0 for all t ∈ [0, T ] (j = 1, 2). (4.9) Thus, D(λ, t) = 0 if and only if eλ(F2(α,t)−F1(α,t)) = (a2(t) + b0(t))(f2(α, t) + b1(t)) (a1(t) + b0(t))(f1(α, t) + b1(t)) . (4.10) 46 Spectral properties of the operator pencils generated by second order elliptic... Denote by A(t) the right-hand side of (4.10) and set Z(t) = ln |A(t)|+ iArgA(t), where Arg z is the argument of the complex number z belonging to [0, 2pi). Then the equation (4.10) is equivalent to λ(F2(α, t)− F1(α, t)) = Z(t) + ik2pi, (4.11) and therefore, the zeros of D(λ, t) except λ = 0 are λk(t) = Z(t) + ik2pi F2(α, t)− F1(α, t) , k = ±1,±2, . . . (4.12) By calculating directly and using (3.13), (4.8) and (4.9), we get d dλ D(λ, t)|λ=λk(t) = λ2k(t)e λk(t)F1(α,t) ( a2(t) + b0(t) )( f2(α, t) + b1(t) )( f2(α, t)− f1(α, t) ) 6= 0 for all t ∈ [0, T ]. Hence, as in Section 3, the zeros λk(t) (k = ±1,±2, . . .) of D(λ, t) have the multiplicity one, and therefore, they are simple eigenvalues of the pencil U(λ, t) for all t ∈ [0, T ]. For each eigenvalue λk, choosing c1(t) = a2(t) + b0(t), c2(t) = −(a1(t) + b0(t)) in (4.7) we have the corresponding eigenvector is ψk(ω, t) = (a2(t) + b0(t))e λk(t)F1(ω,t) − (a1(t) + b0(t))e λk(t)F2(ω,t). (4.13) If λ = 0, as in Section 3, the equation (4.3) has the general solution in the form u(ω, t) = c1(t) + c2(t)ω. Inserting this function into the boundary conditions (4.4), (4.5), we get c2(t) ≡ 0 and c1(t) is arbitrary. Thus, λ(t) = 0 is an eigenvalue of the pencil U(λ, t) with the geometric multiplicity one, and u0 = 1 is an eigenvector corresponding to this eigenvalue. To find a necessary and sufficient condition for the simplicity of the eigenvalue λ(t) = 0, we consider the equation U(0, t)u1 = −U ′(0, t)u0 (4.14) where U′(0, t) = (L′(t, ω, ∂ω, 0), b0(t), b1(t)), L ′(t, ω, ∂ω, 0) = d dλ L(t, ω, ∂ω, λ)|λ=0 = 2∑ j,h=1 j 6=h (sinω − aj(t) cosω)(cosω + ah(t) sinω)∂ω. 47 Nguyen Manh Hung and Nguyen Thanh Anh Rewrite the equation (4.14) in the equivalent form ∂2ωu1 = 0 in (0, α), (4.15) −∂ωu1|ω=0 = −b0(t) on Γ 0 T , (4.16) ∂ωu1|ω=α = −b1(t) on Γ 1 T . (4.17) The equation (4.15) has the general solution u1(ω, t) = c1(t) + c2(t)ω. Inserting this function into the boundary conditions (4.16), (4.17), we have −c2(t) = −b0(t), c2(t) = −b1(t). Thus, if b2(t)+b1(t) 6= 0, then the problem (4.15)-(4.17) has no solutions. Otherwise, u1 = b0(t)ω is an its solution. Summarizing the results above, we obtain Theorem 4.1. Suppose that the functions a1(t), a2(t), b0(t), b1(t) belong to the class Ch([0, T ]), where h is a natural number. Then 1) All nonzero eigenvalues of the pencil U(λ, t) are given by the formula (4.12); these eigenvalues are simple and belong to class the Ch([0, T ]). The corresponding eigenvectors are given by the formula (4.13), which are infinitely differentiable with respect to the variable ω and h-times continuously differentiable with respect to the variable t. 2) λ(t) ≡ 0 is an eigenvalue of the pencil U(λ, t) with a corresponding eigen- vector u0 = 1. This eigenvalue is simple on [0, T ] if and only if b0(t) + b1(t) 6= 0 for all t ∈ [0, T ]. REFERENCES [1] M. Dauge, 1988. Elliptic boundary value problems on corner domains, Lec- ture Notes in Mathematics, Springer Verlag, Berlin. [2] V.A. Kozlov, V.G. Maz'ya and J. Rossmann, 1997. Elliptic boundary prob- lems in domains with point singularities, Mathematical Surveys and Monographs 52, Amer. Math. Soc., Providence, Rhode Island. [3] Kozlov V.A., Maz'ya V.G., Rossmann J., 2005. 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