Spaces admissible for the Sturm-Liouville equation

N. A. Chernyavskaya; L. A. Shuster

doi:10.3934/cpaa.2018050

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May 2018, 17(3): 1023-1052. doi: 10.3934/cpaa.2018050

Spaces admissible for the Sturm-Liouville equation

N. A. Chernyavskaya ^1, and L. A. Shuster ^2,

1.	Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel
2.	Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

Received August 2016 Revised November 2017 Published January 2018

We consider the equation

$-{y}''(x)+q(x)y(x)=f(x),\ \ \ \ x\in \mathbb{R}\text{ }\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where

$f∈ L_p^{\text{loc}}(\mathbb R),$

$p∈[1,∞)$

and

$0≤ q∈ L_1^{\text{loc}}(\mathbb R).$

By a solution of (1) we mean any function

$y,$

absolutely continuous together with its derivative and satisfying (1) almost everywhere in

$\mathbb R.$

Let positive and continuous functions

$μ(x)$

and

$θ(x)$

for

$x∈\mathbb R$

be given. Let us introduce the spaces

$\begin{align} & {{L}_{p}}(\mathbb{R},\mu )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\mu )}^{p}=\int_{-\infty }^{\infty }{|}\mu (x)f(x){{|}^{p}}dx < \infty \right\}, \\ & {{L}_{p}}(\mathbb{R},\theta )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\theta )}^{p}=\int_{-\infty }^{\infty }{|}\theta (x)f(x){{|}^{p}}dx <\infty \right\}. \\ \end{align}$

In the present paper, we obtain requirements to the functions

$μ,θ$

and

$q$

under which

1) for every function

$f∈ L_p(\mathbb R,θ)$

there exists a unique solution (1)

$y∈ L_p(\mathbb R,μ)$

of (1);

2) there is an absolute constant

$c(p)∈(0,∞)$

such that regardless of the choice of a function

$f∈ L_p(\mathbb R,θ)$

the solution of (1) satisfies the inequality

$\|y\|_{L_p(\mathbb R,μ)}≤ c(p)\|f\|_{L_p(\mathbb R,θ)}.$

Keywords: Admissible spaces, Sturm-Liouville equation.

Mathematics Subject Classification: 34B05, 34B24, 34K06.

Citation: N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050

References:

[1]	N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.
[2]	N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426.
[3]	N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space L_p(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054.
[4]	N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.
[5]	N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021.
[6]	N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120.
[7]	R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.
[8]	E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.
[9]	E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S $75, New York, Dover Publications, 1959.
[10]	L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.
[11]	A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.
[12]	J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966.
[13]	K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.
[14]	M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.
[15]	C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.

show all references

References:

[1]	N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.
[2]	N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426.
[3]	N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space L_p(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054.
[4]	N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.
[5]	N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021.
[6]	N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120.
[7]	R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.
[8]	E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.
[9]	E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S $75, New York, Dover Publications, 1959.
[10]	L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.
[11]	A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.
[12]	J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966.
[13]	K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.
[14]	M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.
[15]	C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.

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