[Paper Review] Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces

Yevheniia Hnyp, Vladimir Mikhailets+1 more|arXiv (Cornell University)|Apr 12, 2017

Differential Equations and Boundary Problems7 references22 citations

TL;DR

This paper establishes a constructive necessary and sufficient criterion for the continuity of solutions to parameter-dependent one-dimensional boundary-value problems in Sobolev spaces, proving that solutions remain continuous in the Sobolev norm as the parameter approaches zero. It further provides a two-sided estimate for the convergence rate of solutions toward the nonperturbed problem, with applications to a new class of multipoint boundary-value problems.

ABSTRACT

We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.

Motivation & Objective

• To establish a necessary and sufficient condition for the continuity of solutions to parameter-dependent linear boundary-value problems in Sobolev spaces.
• To extend prior results—previously only sufficient—by proving the necessity of the conditions for continuity in the Sobolev norm.
• To derive a two-sided estimate for the rate of convergence of solutions as the parameter tends to zero.
• To apply the criterion to a new class of multipoint boundary-value problems with general parameter-dependent boundary conditions.
• To generalize the analysis to higher-order systems and complex-valued functions within the Sobolev space framework.

Proposed method

• Formulates a general class of linear boundary-value problems for higher-order ordinary differential systems where solutions and right-hand sides belong to complex Sobolev spaces $W_p^n$.
• Introduces boundary conditions via arbitrary continuous linear operators on the Sobolev space of solutions, ensuring generality and avoiding reliance on Green's formula or formal adjoints.
• Employs compactness and continuity arguments in Sobolev spaces, leveraging the continuous embedding of $W_p^{n+r}$ into Hölder spaces $C^{n+r-1,1/q}$ for point evaluation estimates.
• Uses the Lagrange mean value theorem and norm estimates to control differences in function values at perturbed points, ensuring convergence of boundary operators.
• Applies the Banach-Steinhaus theorem to prove uniform boundedness of the parameter-dependent boundary operators $B(\varepsilon)$ as $\varepsilon \to 0^+$.
• Establishes convergence $B(\varepsilon)y \to By$ in $\mathbb{C}^{rm}$ for all $y \in (C^\infty)^m$, using conditions on coefficient convergence and pointwise perturbation of evaluation points.

Experimental results

Research questions

• RQ1 Under what conditions is the solution operator of a parameter-dependent boundary-value problem continuous in the Sobolev space norm with respect to the parameter?
• RQ2 Can the previously known sufficient conditions for continuity be shown to be necessary as well?
• RQ3 What is the quantitative rate of convergence of solutions to the nonperturbed problem as the parameter $\varepsilon \to 0^+$?
• RQ4 How can the general framework be applied to multipoint boundary-value problems with parameter-dependent coefficients and evaluation points?
• RQ5 To what extent can the results be extended to other function spaces beyond Sobolev spaces?

Key findings

• A constructive necessary and sufficient criterion is established for the continuity of solutions in the Sobolev space $W_p^{n+r}$ with respect to the parameter $\varepsilon$.
• The solution operator $y(\cdot, \varepsilon)$ is continuous in $\varepsilon$ in the norm of $(W_p^{n+r})^m$ as $\varepsilon \to 0^+$, under the derived conditions.
• A two-sided estimate for the convergence rate is obtained: $\|y(\cdot, \varepsilon) - y(\cdot, 0)\|_{n+r,p} \leq C \varepsilon^{1/q}$ for some $C > 0$, with $q$ related to the Hölder exponent.
• The convergence of the boundary operator $B(\varepsilon)$ to $B(0)$ is proven in the operator norm on $W_p^{n+r}$, uniformly for smooth functions.
• The results are applied to a new class of multipoint boundary-value problems with $\varepsilon$-dependent evaluation points and coefficients, yielding explicit sufficient conditions.
• The analysis confirms that the conditions on coefficient convergence and pointwise perturbations are both necessary and sufficient for solution continuity in Sobolev norms.