Sturm–Liouville theory

Summary

In mathematics and its applications, classical Sturm–Liouville theory (developed by Joseph Liouville and Jacques Charles François Sturm) is the theory of real second-order linear ordinary differential equations of the form: for given coefficient functions p(x), q(x), and w(x), an unknown function y=y(x) of the free variable x, and an unknown constant \lambda. All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution y is typically required to satisfy some boundary conditions at extreme values of x. Each such equation () together with its boundary conditions constitutes a Sturm–Liouville problem. In the simplest case where all coefficients are continuous on the finite closed interval and p has continuous derivative, a function y = y(x) is called a solution if it is cont

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Liouville's theorem and Laurent Series Expansions for Solutions of the Heat Equation

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We give a Liouville theorem for entire solutions and Laurent series expansions for solutions with isolated singularities of the heat equation.

Springer Basel Ag2014

Recovering a Potential from Cauchy Data via Complex Geometrical Optics Solutions

Hoài-Minh Nguyên

This paper is devoted to the problem of recovering a potential q in a domain in ℝd for d≥3 from the Dirichlet to Neumann map. This problem is related to the inverse Calder'on conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of W2,d/2 conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvester and Uhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of the second result, in the work of Ferreira et al. [10], is based on the construction of CGO solutions via Carleman estimates. The main goal of the paper is to understand whether or not an approach which is based on the construction of CGO solutions in the spirit of Sylvester and Uhlmann and involves only standard Sobolev spaces can be used to obtain these results. In fact, we are able to obtain a new proof of uniqueness for the Calder'on problem for 1) a slightly different class as the one in [11], and for 2) the class of W2,d/2 conductivities. The proof of statement 1) is based on a new estimate for CGO solutions and some averaging estimates in the same spirit as in [11]. The proof of statement 2) is on the one hand based on a generalized Sobolev inequality due to Kenig et al. [14] and on another hand, only involves standard estimates for CGO solutions [22]. We are also able to prove the uniqueness of a potential for 3) the class of Ws,3/s (⫌W2,3/2) conductivities with 3/2

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Solutions of large norm for non-linear Sturm-Liouville problems

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