Singular Sturm–Liouville Theory

• Singular Sturm–Liouville theory is a framework for analyzing differential equations with singular or degenerate coefficients using weighted Sobolev spaces.
• The approach employs weak formulations, energy estimates, and variational methods to establish existence, uniqueness, and optimal regularity of solutions.
• It has practical applications in physics and engineering, modeling phenomena such as beams and rods with varying material properties.

A singular Sturm–Liouville equation is a differential equation of the type

$- \bigl( x^{2\alpha} u'(x) \bigr)' + u(x) = f(x)$

defined on the interval $(0,1]$ with a singular or degenerate coefficient at $x = 0$. Here, $\alpha < 1$ is a fixed real parameter, $f$ is a given function in $L^p(0,1)$, and boundary conditions are imposed at $x = 1$ (classically $u(1) = 0$) and, depending on $\alpha$, a weighted Dirichlet or Neumann-type condition at $x = 0$. Such problems arise naturally in mathematical physics, including in the modeling of rods or beams with varying material properties, and require careful functional analytic treatment due to degeneracy or blow-up of the weight. The following sections summarize the principal analytical features and results for this class of equations (Castro et al., 10 Dec 2024).

1. Problem Formulation and Functional Setting

The equation is given by

$$\begin{cases} - (x^{2\alpha} u'(x))' + u(x) = f(x) & \text{in } (0,1], \ u(1) = 0, \end{cases}$$

where $f \in L^p(0,1)$ for $1 \le p < \infty$, $\alpha < 1$, and $\alpha \ne 0$. The coefficient $x^{2\alpha}$ is singular at $x=0$, with the nature of the singularity depending on the sign of $\alpha$. For $\alpha > 0$, the weight vanishes at the origin, inducing degeneracy; for $\alpha < 0$, the weight diverges, causing a strong singularity.

To treat such problems rigorously, solutions are sought in weighted Sobolev spaces $X^\alpha$ or $X_0^\alpha$, reflecting the structure of the operator and boundary conditions. A weighted Dirichlet boundary condition at $x=0$ is frequently applied, typically specified as $\lim_{x\to 0^+} u(x) = 0$ or, in a weighted sense,

$$\lim_{x \to 0^+} x^{2\alpha-1} u(x) \quad \text{is bounded}.$$

For certain parameter regimes, a weighted Neumann-type condition $\lim_{x \to 0^+} x^{2\alpha} u'(x) = 0$ is also considered. This distinction is crucial for the well-posedness and regularity theory.

2. Role and Effect of the Singularity Parameter $\alpha$

The parameter $\alpha$ determines the strength and type of singularity at $x = 0$:

• If $\alpha > 0$: $x^{2\alpha} \to 0$ as $x \to 0^+$, leading to degeneracy of the elliptic part. This admits "absorbing" boundary phenomena and necessitates consideration of vanishing rates.
• If $\alpha < 0$: $x^{2\alpha} \to \infty$ as $x\to 0^+$, resulting in an inverse type singularity, which can dominate solution behavior near the origin.

The assumption $\alpha < 1$ is made to ensure that the singularity is not excessively strong, i.e., the operator remains coercive in appropriate weighted spaces and standard theory can be adapted.

These behaviors influence both the admissible function spaces and the formulation of boundary conditions, as pointwise conditions on $u(0)$ may be ill-defined for certain parameter ranges.

3. Weighted Sobolev Spaces and $L^p$ Theory

The existence and uniqueness theory is established for $f \in L^p(0,1)$. The use of $L^p$ data broadens the classical $L^2$ theory to accommodate more general source terms, including those with localized or singular behavior.

Weighted Sobolev spaces, denoted $X_0^\alpha$ (enforcing zero boundary condition at $x=0$ in a weighted sense) and $X^\alpha$ (with weaker or no condition), are defined so that the norm captures both the $L^2(0,1)$ (or $L^p$) norm of $u$ and the $L^2$ (or $L^p$) norm of the weighted derivative $x^\alpha u'(x)$. This structure is essential for a priori estimates, regularity analysis, and formulation of variational solutions.

Embeddings from these weighted spaces into spaces of continuous functions near the origin are controlled by the parameter $\alpha$ and the integrability exponent $p$, dictating the optimal boundary behavior and regularity obtainable.

4. Existence, Uniqueness, and Regularity of Solutions

The main analytical results are as follows:

• Weighted Dirichlet Problem: With the boundary condition $\lim_{x\to 0^+} u(x) = 0$, for every $\alpha < 1$ and each $f \in L^p(0,1)$ (for suitable $p$), there exists a unique weak solution $u$ in $X^\alpha$.

The solution $u$ satisfies: - $x^{2\alpha} u' \in W^{1,p}(0,1)$, - $x^{2\alpha-1} u$ possesses further differentiability, - $u(x)$ vanishes at the origin at the optimal rate $u(x) \sim x^{1-2\alpha}$ as $x\to 0^+$.

• Weighted Neumann Problem: When a weighted Neumann condition $\lim_{x\to 0^+} x^{2\alpha} u'(x) = 0$ is imposed (relevant for certain physical settings or when strong singularity precludes Dirichlet data), existence and uniqueness in a corresponding function space are also obtained.

Uniqueness is established using energy identities adapted to the singular weight, and existence follows from variational arguments, with the coercivity allowed by the restriction $\alpha < 1$. Regularity properties are optimal in the sense that the solution exhibits the maximal rate of vanishing or blow-up compatible with the singularity.

5. Techniques and Analytical Methods

The proofs utilize several key functional analytic and PDE tools:

• Weak Formulation and Variational Methods: The problem is posed in terms of weighted Sobolev spaces, leading to a weak (variational) formulation. Coercivity and boundedness of the associated bilinear form follow from weighted Hardy-type inequalities and the assumption on $\alpha$.
• Energy and A Priori Estimates: Carefully chosen test functions yield energy identities, from which regularity and uniqueness are deduced. Moser iteration (or similar) is used to improve regularity and obtain optimal pointwise vanishing (or growth) properties near $x = 0$.
• Explicit Representation (Appendix): While the main approach is abstract, the paper also demonstrates that explicit solution formulas are possible in terms of Bessel functions. However, the generality of the approach is seen in avoiding reliance on such formulas, thus permitting extension to more general weights or higher dimensions.

6. Applications and Extensions

The singular Sturm–Liouville equation studied here relates directly to models in mechanics, such as rods or beams with spatially varying stiffness, for which $x^{2\alpha}$ represents a cross-sectional area or moment of inertia with degeneracy at the endpoint.

The $L^p$ theory developed extends previous $L^2$-based results, handling more general sources and enabling applications to problems with rough data. The use of weighted Sobolev spaces and functional analytic machinery also provides a model for analyzing analogous equations in higher dimensions with singular weights of the form $|x|^{2\alpha}$ or for matrix-valued problems.

7. Summary Table of Key Analytical Outcomes

Parameter	Boundary Condition at $x=0$	Regularity of Solution
$\alpha\le 0$	$u(0)=0$ (classical Dirichlet)	$u \sim x^{1-2\alpha}$
$\alpha>0$	weighted Dirichlet (see text)	$u \sim x^{1-2\alpha}$
Any $\alpha$	weighted Neumann possible	$x^{2\alpha} u'(x) \to 0$

The conditions guarantee existence, uniqueness, and optimal boundary regularity for all $\alpha<1$ and $f\in L^p(0,1)$, with extensions contingent on the boundary regime and singularity strength.

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These results provide a rigorous analytic foundation for the theory of singular Sturm–Liouville equations in $L^p$ spaces, covering both Dirichlet and Neumann-type boundary behaviors and exhibiting precise control over the effects of the singular weight (Castro et al., 10 Dec 2024).