Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

Autor: Łukasz Rzepnicki
Rok vydání: 2013
Předmět:
Zdroj: Integral Equations and Operator Theory. 76:565-588
ISSN: 1420-8989
0378-620X
DOI: 10.1007/s00020-013-2071-3
Popis: This paper is concerned with estimations of solutions of the Sturm–Liouville equation $$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$ where $${\mu\in\mathbb{C}}$$ is a spectral parameter. We assume that the strictly positive function $${\rho\in L_{\infty}[0,1]}$$ is of bounded variation, $${p\in W^1_1[0,1]}$$ is also strictly positive, while $${d\in L_1[0,1]}$$ and $${q\in L_1[0,1]}$$ are real functions. The main result states that for any r > 0 there exists a constant c r such that for any solution y of the Sturm–Liouville equation with μ satisfying $${|{\rm Im}\, \mu|\leq r}$$ , the inequality $${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$$ is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C 0-semigroup.
Databáze: OpenAIRE
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