2 Some estimates for the first eigenvalue of a Sturm-Liouville problem with a weighted integral condition

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1 On some estimates for the first eigenvalue of a Sturm-Liouville problem with Dirichlet boundary conditions and a weighted integral condition S Ezhak, M Telnova Plekhanov Russian University of Economics, svetlanaezhak@gmailcom, mytelnova@yandexru Introduction We consider a problem which origin was the Lagrange problem of finding the form of the firmest column of given volume The Lagrange problem was the source for different extremal eigenvalue problems Some of them are eigenvalue problems for second-order differential equations with integral conditions on the potential The Dirichlet problem for the equation y + λqx)y = with a non-negative summable on [, ] function Q satisfying the condition Q x)dx =, as R,, was considered in [] The Dirichlet problem for the equation y Qx)y + λy = with a real integrable on, ) by Lebesgue function Q was considered in [] for In this paper we consider a problem of that kind in the case when the integral condition contains a weight function Consider the Sturm Liouville problem y + Qx)y + λy =, x, ), ) y) = y) =, ) where Q belongs to the set T α,β, of all real valued locally integrable functions on, ) with non negative values such that the following integral condition holds x α x) β Q x)dx =, α, β, R, 3) A function y is a solution to problem ), ) if it is absolutely continuous on the segment [, ], satisfies ), its derivative y is absolutely continuous on any segment [ρ, ρ], where < ρ <, and equality ) holds almost everywhere in the interval, ) We give estimates for λ Q), M α,β, = sup λ Q) Q T α,β, Q T α,β, For any function Q T α,β, by H Q we denote the closure of the set C, ) in the norm y HQ = y dx + ) Qx)y dx For any function Q T α,β, we can prove see, for example, [7], [9]) that ) y Qx)y dx λ Q) = R[Q, y], where R[Q, y] = y H Q \{} y dx Some estimates for the first eigenvalue of a Sturm-Liouville problem with a weighted integral condition For problem ) 3) for α = β = the following theorem was proved see, for example, [4] [6]) Theorem If > then m,, π, M,, = π and there exist functions Q T,, and u H, ) such that m,, = R[Q, u] π

2 If = then M,, = π, m,, = λ, where λ, π ) is the solution to the equation λ ) λ = tg Here m,, is attained at Qx) = δ x ) 3 If / < then m,, =, M,, = π 4 If /3 < / then m,, =, M,, π 5 If < < /3 then m,, =, M,, < π 6 If < then m,, =, M,, < π, and there exist functions Q T,, and u H, ) such that M,, = R[Q, u] Remark The result M,, < π for < < / was obtained in [3] The following theorem gives some results for arbitrary real numbers α, β Theorem For any α, β,,, we have M α,β, π If < or < < then 3 If = and α, β then m α,β, 3 4 π 3 If =, β < α or α < β then m α,β, 33 if =, < α, β then π m α,β, π ; 34 if > and < α, β then m α,β, 3 ) ) π ; 35 if > and β < α or α < β then m α,β, 36 if > and α, β then m α,β, Proof Let α, β, be arbitrary real numbers, Let us prove that M α,β, π Consider the function, < x < ; v x) =, x ;, < x <, ) ) π ; where < < By average processing we obtain the function y x) = v ρ x) sin πx of C, ), ρ < Since y x) sin πx H,) as and then for any we have V [y ] = V [y] = y H,) y H,) y dx y = π y dx y dx π and V [y ] π as Then for any function Q T α,β, we have R[Q, y ] < V [y ] = we obtain and therefore, R[Q, y] lim R[Q, y ] lim V [y ] = π, y H Q \{} M α,β, = sup Q T α,β, y dx Consequently, for any function Q T y dx α,β, R[Q, y] y H Q \{} π

3 Let us prove that if < or < < then ) Suppose that <, α, β > Put α β Consider the function α ) α ) x α x) β, < x < ; Q,α,β, x) = α ) α ) x α x) β, x ; α ) α ) x α x) β, < x <, where < < Then y dx x α x) α α ) α α = ) x) α β y dx = α ) α β x x) y dx = α ) α α = ) Q,α,β, x)y dx For any y H Q,α,β, by the the Hölder inequality we have y dx = y dx + y dx + y dx α y dx + ) Q,α,β, x)y dx + y dx Then Q,α,β, x)y dx Q,α,β, x)y dx y dx y dx + ) y dx α ) For any the function y = sin πx belongs to the space H Q,α,β, and Thus y dx =, y dx = π, y dx = π ) π sin π, R[Q,α,β,, y] R[Q,α,β,, y ] = π 3 π π ) y H Q,α,β, \{} sin π) Q T α,β, R[Q, y] lim y H Q \{} α ) R[Q,α,β,, y] lim R[Q,α,β,, y ] = y H Q,α,β, \{} If β α then on the segment [, ] we define the function Q,α,β, as follows β ) β Q,α,β, = ) x α x) β = β ) β x x) β β ) β x β α, similarly we obtain ) Suppose that <, β < α Consider the function α + ) α+ ) x) β, < x ; Q,α,β, x) = α + ) α+ ) x) β, < x <, 3

4 where < < For any function y H Q,α,β, by the the Hölder inequality we obtain y dx = Then y dx + y dx ) y dx + α+ ) α + Q,α, x)y dx α + ) α+ ) y dx For any the function y = x) β sin πx belongs to the space HQ,α,β, and Q,α,β, x)y dx ) y dx R[Q,α,, y ] y dx + α + ) α+ ) y dx y dx) y dx Q T α,β, R[Q, y] lim y H Q \{} R[Q,α,β,, y] lim R[Q,α,β,, y ] = y H Q,α,β, \{} Note that the case <, α < β is symmetrical with the case <, β < α 3) Suppose that <, α, β Consider the function ) x α x) β, < x ; Q,α,β, x) = ) x α x) β, < x <, where < < For any the function y = x α x) β sin πx belongs to the space HQ,α,β, and similarly to the cases ) and ) we obtain 4) Suppose that < < and α, β are arbitrary real numbers For < < consider the function, x < ; Q,α,β, x) = x α x) β, x + ;, + < x For any the function y = sin πx belongs to the space H Q,α,β, constant C = min [, + ] x α x) β such that Q,α,β, x)y dx + x α x) β sin πxdx C similarly to the cases ) 3) we obtain + and for any, α, β, there exists a ) cos πx dx = C sin π + π 3 Suppose that = and α, β It was proved see, for example, []) that for any function y H, ) the inequality sup y 4 y dx holds For any function Q T α,β, and for any function y H Q [,] we obtain Qx)y y [,] x α Q T α,β, Qx)x α y [,] x α y H Q \{} R[Q, y] 3 4 Q T α,β, Qx)x α x) β y sup y [,] xα [,] 4 y H Q \{} y dx y dx 3 4 y H,)\{} y dx y dx y dx = 3 4 π 4

5 3 Suppose that =, β < α For any function Q T α,β, we have Qx)y dx = Qx)y x α x α y [,] x α For any x, ) by the the Hölder inequality we obtain Qx)x α x) β y [,] x x y x) = y t)dt) x x y t)dt x y t)dt y and sup [,] x y dx Then Q T α,β, y H Q \{} y Qx)y dx y dx Note that the case α < β = is symmetrical with the case β < α = Similarly we can prove the results of 33) 36) References [] Yu V Egorov, V A Kondratiev, On Spectral theory of elliptic operators in Operator theory: Advances and Applications Birkhouser 996), 35 [] V A Vinokurov, V A Sadovnichii, On the Range of Variation of an Eigenvalue When the Potential Is Varied Doklady Mathematics Translated from Doklady Akademii Nauk, Vol 68, No, 3), Russian) [3] A A Vladimirov, On some a priori majorant of eigenvalues of Sturm Liouville problems arxiv preprint arxiv:658, 6 - arxivorg [4] S S Ezhak, On estimates for the first eigenvalue of the Sturm - Liouville problem with Dirichlet boundary conditions In: Astashova I V ed) Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition, M: UNITY-DANA, ), Russian) [5] S S Ezhak, On estimates for the first eigenvalue of the Sturm - Liouville problem with Dirichlet boundary conditions and integral condition Differential and Difference Equations with Applications, Springer Proceedings in Mathematics and Statistics, Springer New York LLC, V 47, 3), [6] S S Ezhak, On a minimization problem for a functional generated by the Sturm - Liouville problem with integral condition on the potential Vestnik of Samara State University, No 6 8), 5), 57 6 Russian) [7] M Yu Telnova, Some estimates for the first eigenvalue of the Sturm Liouville problem with a weight integral condition Mathematica Bohemica, V 37 ),), 9 38 [8] I V Astashova, Qualitative properties of solutions to quasilinear ordinary differential equations In: Astashova I V ed) Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition, M: UNITY-DANA, ), 9 Russian) [9] M Yu Telnova, Estimates for the first eigenvalue of a Sturm Liouville problem with Dirichlet conditions and a weight integral condition In: Astashova I V ed) Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition, M: UNITY-DANA, ), Russian) 5