Smoothness of the Solution of a Boundary Value Problem for Degenerate Elliptic Equations

Abstract

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of degeneracy. The analysis is grounded in several fundamental aspects of symmetry. Structural symmetry is reflected in the formulation of the differential operators; functional symmetry emerges in the properties of the associated weighted Sobolev spaces; and spectral symmetry plays a critical role in the behavior of the eigenvalues and eigenfunctions used to characterize solutions. By employing localization techniques, a priori estimates, and spectral theory, we establish new coefficient conditions ensuring smoothness in both semi-periodic and Dirichlet boundary settings. Moreover, we prove the boundedness and compactness of certain weighted operators, whose definitions and properties are tightly linked to underlying symmetries in the problem’s formulation. These results are not only of theoretical importance but also bear practical implications for numerical methods and models where symmetry principles influence solution regularity and operator behavior.

1. Introduction

The study of partial differential equations (PDEs) forms a cornerstone of contemporary mathematical analysis, with profound applications across physics, engineering, and finance. Within this broad class, elliptic equations are particularly notable due to their appearance in steady-state processes—such as electrostatics, heat conduction, and incompressible fluid flow—where symmetry often governs physical laws and boundary behaviors.

Symmetry in PDEs typically manifests in various ways: through the invariance of differential operators under spatial transformations, the uniformity of coefficients across domains, or the harmonic balance of solutions and boundary data. Classical elliptic equations exploit these symmetries to yield well-behaved solutions under smooth and regular conditions. However, real-world problems frequently deviate from such idealizations, resulting in degenerate elliptic equations where the symmetry is partially or fully broken. In such cases, standard assumptions—such as uniform ellipticity—no longer apply, and the analysis of solution regularity becomes significantly more delicate.

This paper focuses on boundary value problems for degenerate elliptic equations in which the coefficients may vanish or become singular in a non-uniform or non-monotonic manner. A prominent challenge in this setting lies in characterizing when and how smooth solutions exist, especially near regions where degeneracy intensifies. These issues are not merely technical; they reflect fundamental questions about how symmetry—or its controlled breaking—affects solution structure and operator behavior.

We approach this investigation by considering the interplay between structural and functional symmetry and the degeneracy of the elliptic operator. Structural symmetry is inherent in the form of the equation, particularly when the degeneracy is symmetric with respect to certain spatial variables or domains. Functional symmetry arises in the use of weighted Sobolev spaces, where the weights themselves reflect symmetries (or asymmetries) in the degeneration profile. Finally, spectral symmetry is evident in the decomposition of solutions into eigenfunctions, whose regularity and orthogonality properties are crucial for understanding solution behavior.

Historically, foundational work by Fichera, Keldysh, and Oleinik established criteria for regularity in degenerate elliptic equations under symmetry-related constraints. More recently, advanced tools from harmonic and spectral analysis have expanded our understanding of such problems, especially under weighted frameworks that preserve certain operator symmetries. Yet many open questions remain, particularly when dealing with complex boundary geometries, anisotropic degeneration, or non-monotonic coefficient behaviors.

In this study, we address some of these challenges by identifying new conditions under which smooth solutions can be obtained. Our analysis is based on

-

Deriving new a priori estimates under symmetry-aligned assumptions;

-

Establishing boundedness and compactness properties of key operators;

-

Providing weighted derivative estimates that respect the symmetry structure of the problem.

These contributions advance the theory of degenerate elliptic equations by showing that, even in the presence of degeneracy, solutions can exhibit smoothness and regularity when the underlying problem retains certain symmetric features. The findings are especially relevant for applications where physical, geometrical, or spectral symmetries play a role in governing the behavior of complex systems.

The main purpose of this work is to investigate the smoothness of solutions to boundary value problems for a class of degenerate elliptic equations. In contrast to uniformly elliptic problems, where classical Schauder or Calderón–Zygmund theory ensures regularity under appropriate conditions, the behavior of solutions in the degenerate setting is more subtle and may vary significantly depending on the nature and strength of the degeneracy. The fundamental question we address is as follows: under what conditions on the degeneracy and the boundary data can we guarantee smoothness (or certain regularity) of the solution up to the boundary? This question has attracted significant attention over the past decades. Early foundational results by Fichera, Keldysh, and Oleinik established frameworks for analyzing degeneracies in elliptic equations, particularly in divergence form. Later works, such as those by Fabes, Kenig, and Serapioni, extended regularity theory to degenerate and singular elliptic equations under weighted Sobolev space frameworks. More recent contributions have utilized tools from harmonic analysis, geometric measure theory, and potential theory to handle cases with highly non-uniform degeneracies or complex boundary geometries (e.g., [1,2,3,4]). Despite this progress, the regularity theory for degenerate elliptic equations remains incomplete and, in some aspects, controversial. For instance, while Muckenhoupt-weighted estimates provide strong results in certain settings, their applicability in highly degenerate or anisotropic contexts is debated. Diverging hypotheses exist regarding the optimal regularity of solutions under minimal assumptions on the coefficients or the degeneracy structure. Moreover, in many cases, boundary regularity is far less understood than interior regularity, which makes the study of boundary behavior especially important. Degenerate elliptic equations also occupy a central place in contemporary mechanics and theoretical physics, where PDEs are used not only to describe the evolution of systems but also to extract qualitative information about physical processes. In this broader context, three principal analytical problems typically arise: (1) the existence of solutions, (2) their uniqueness, and (3) the investigation of qualitative properties of solutions, including regularity, asymptotic behavior, and spectral characteristics. A comprehensive body of literature addresses the first two problems for both linear and nonlinear elliptic equations, as documented in [5,6,7,8,9,10,11,12,13,14,15]. In practical applications, one is often most concerned with understanding the qualitative behavior of solutions: what does a solution look like, how smooth is it, and what properties can we deduce about its structure? These questions become particularly pressing in degenerate cases, where standard methods break down. Among the various directions in qualitative analysis, particular attention is paid to the following: smoothness of solutions (including coercivity and separability); evaluation of solutions in different weighted norms; and spectral properties of the associated differential operators. This work focuses specifically on the first aspect—smoothness—involving the coercivity and separability of solutions in the presence of degeneracy. These properties are critical not only from a theoretical standpoint but also for practical computation, where regularity often underlies the convergence and accuracy of numerical methods. In cases where the domain is bounded and the coefficients are smooth, many of these issues have been addressed using classical tools of functional analysis and PDE theory. Notable contributions by T.Sh. Kalmenov, M. Otelbaev, M.B. Muratbekov, and E.A. Akzhigitov [6,7,8,9,10] have established standard methods for treating such equations, particularly under assumptions that guarantee uniform ellipticity or controlled degeneracy. However, in real-world applications, one often encounters equations with more complex degeneracies—those that are non-uniform across different variables or non-monotonic in nature. Such equations fall outside the scope of classical theory and demand new techniques. This study focuses on the first type of complexity: non-uniform degeneracy across variables. In these cases, the degeneracy may vary in direction or strength, complicating the analytical framework and the tools required to study regularity. The aim of this work is to develop a refined approach for analyzing the smoothness of solutions to boundary value problems for such degenerate elliptic equations. We aim to derive conditions on the degeneracy and boundary data under which coercivity and separability—key indicators of regularity—can still be ensured. Our results provide sufficient conditions for the smoothness of solutions in appropriate weighted or Sobolev-type spaces, even when classical assumptions are not met. In summary, this study contributes to the ongoing effort to extend the theory of elliptic PDEs into more realistic and complex domains. It highlights new challenges in dealing with variable degeneracy and proposes a framework for addressing it. The main conclusions demonstrate that with appropriate structural assumptions, it is possible to recover smoothness properties that mirror those in the uniformly elliptic setting. This has direct implications not only for theoretical analysis but also for the development of robust numerical methods and for the modeling of complex systems in applied sciences.

2. Preliminaries

The purpose of this study is to identify coefficient conditions that guarantee the eventual smoothness of solutions to boundary value problems for elliptic equations that degenerate either non-uniformly or uniformly, particularly when the degeneration exhibits a non-monotonic character. Additionally, this study seeks to establish weighted estimates for the solutions of the considered boundary value problems. The objectives of this study are as follows: determining novel coefficient conditions that ensure the limiting smoothness of solutions to the semi-periodic Dirichlet problem for a degenerate elliptic equation; deriving the limiting regularity of solutions to the first boundary value problem involving non-uniformly degenerate elliptic equations; and obtaining derivative estimates of the solutions using various weight functions.

3. Main Result

Smoothness of the Solution of a Boundary Value Problem for Degenerate Elliptic Equations

Let $\Omega$ be an open rectangle in ${\mathbb{R}}^2$:

$$\Omega =\left\{\right(x,y):0<x<1,0<y<1\}$$

In the region $\Omega$ we consider the degenerate elliptic equation

$$Lu+\lambda u=-{\displaystyle \frac{\partial }{\partial x}}\left(K\left(y\right)\alpha \left(x\right){\displaystyle \frac{\partial u}{\partial x}}\right)-{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+\lambda u=f(x,y),$$

(1)

where $K\left(y\right)\in {\mathcal{G}}_{{\delta }_0}$, $\alpha \left(x\right)$ is the continuous real function on $[0,1]$ that satisfies the condition

$$\alpha \left(0\right)=0,\alpha \left(x\right)>0,x>0,$$

(2)

From condition (2) it is evident that degeneration in the vicinity of point $(0,0)\in \partial \Omega$ occurs non-uniformly. The rate of approach to zero of the coefficient $\alpha \left(x\right)$ is of particular importance for the behavior of Equation (1). As a result, the types of boundary value problems formulated typically depend on the nature of the coefficient $\alpha \left(x\right)$:

• If

  $${\int }_0^1{\displaystyle \frac{dx}{\alpha \left(x\right)}}<\infty ,$$

  (3)

  then it is required to find a function $u(x,y)$ that satisfies Equation (1) and the constraints

  $$\left\{\begin{array}{cc}\hfill u(x,0)& =0\hfill \\ \hfill u(0,y)& =0\hfill \end{array}\right.\mathrm{and}\left\{\begin{array}{cc}\hfill u(x,1)& =0\hfill \\ \hfill u(1,y)& =0\hfill \end{array}\right.$$

  (4)
• If

  $${\int }_0^1{\displaystyle \frac{dx}{\alpha \left(x\right)}}=\infty ,{\int }_0^1{\displaystyle \frac{xdx}{\alpha \left(x\right)}}<\infty ,$$

  (5)

  then it is required to find a function $u(x,y)$ that satisfies Equation (1) and the conditions

  $$\left\{\begin{array}{cc}\hfill u(x,0)& =u(x,1)=0\hfill \\ \hfill u(1,y)& =0\hfill \end{array}\right.$$

  (6)

Note that in case (2) on the line $x=0$, the boundary condition is not specified.

Let $L_{2,K}^{0T2}(\Omega )$ be the space obtained by completing it according to the norm

$${∥u∥}_{L_{2,K}^{T,2}(\Omega )}={\left({\int }_{\Omega }{\left|K\left(y\right){\displaystyle \frac{\partial }{\partial x}}\left(\alpha \left(x\right){\displaystyle \frac{\partial u}{\partial x}}\right)\right|}^2+{\left|{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right|}^2+{\left|u\right|}^{2}dxdy\right)}^{1/2},$$

subsets of the space $C^{\infty }\left(\overline{\Omega }\right)$ with elements $u(x,y)$ for which ${\left.u\right|}_{\partial \Omega }=0$, and ${}^{0}L_{2,K}^{T,2}$ is the completion, according to the same norm, of the set of functions from $C^{\infty }\left(\overline{\Omega }\right)$ satisfying condition (6).

If $u(x,y)\in L_2(\Omega )$, then the following representation holds:

$$u(x,y)=\sum _{n=1}^{\infty }{v}_n\left(y\right)z_n\left(x\right),$$

where $z_n\left(x\right)$ are the eigenfunctions of the operator T and $v_n\left(y\right)$ are the Fourier coefficients.

Definition 1.

The fractional derivative [1] with respect to x of order $\alpha \ge 0$ of the function $u(x,y)$ is called the function

$$T_x^{\alpha }u=\sum _{n=1}^{\infty }{\lambda }_n^{\alpha }{v}_n\left(y\right)z_n\left(x\right),$$

${\lambda }_n$ and ${\overline{z}}_n\left(x\right)$ are the eigenvalues and the corresponding eigenfunctions of the operator T.

If $\alpha =1$, then

$$T_{x}u={\displaystyle \frac{\partial }{\partial x}}\left(a\left(x\right){\displaystyle \frac{\partial u}{\partial x}}\right)=\sum _{n=1}^{\infty }{\lambda }_n{\nu }_n{\overline{z}}_n\left(x\right).$$

Definition 2.

If for a function $u\in L_2(\Omega )$ there exists a sequence $\left\{u_n\right\}$ of functions from $C^{\infty }\left(\overline{\Omega }\right)$ satisfying conditions (4) and (6) such that $\left\{u_n\right\}$ and $\left\{(L+\lambda E)u_n\right\}$ converge in the norm of $L_2(\Omega )$ to u and f, respectively, then u is called a strong solution of problems (1), (4) ((1), (6)).

The main result is as follows.

Theorem 1.

Let $K\left(y\right)\in G_{{\delta }_0}$ and $\rho \left(y\right)\in G_{{\delta }_0}^{\ast }$, and let the function $a\left(x\right)$ satisfy conditions (3) and (5). Then, when $\lambda >0$, the following statements are true: for any $f\in L_2(\Omega )$ there exists in $L_2(\Omega )$ a unique strong solution of problems (1), (4) ((1), (6)).

The operator $K\left(y\right)T_x{(L+\lambda E)}^{-1}$ is bounded in $L_2(\Omega )$;

• the operator $\rho \left(y\right)T_x^1{(L+\lambda E)}^{-1}$ is completely continuous in $L_2(\Omega )$ if $\rho \left(y\right),K\left(y\right)$ and $0\le \alpha <1$.

To prove the theorem, we present two auxiliary lemmas with proofs. Consider the operator defined by the equality

$$(L+\lambda E)u=-u^{″}\left(y\right)+(a_{x}K\left(y\right)+\lambda )u,$$

and boundary conditions

$$u\left(0\right)=u\left(1\right)=0.$$

Lemma 1.

Let $K\left(y\right)\in G_x$. Then there exists a continuous inverse operator ${(L+\lambda E)}^{-1}$ defined in $L_2(0,1)$.

Proof. 

Integrating by parts for all $u\in C_0^2(0,1)$ we have

$$⟨(L+\lambda E)u,u⟩={\int }_0^1\left[{\left(u^{\prime }\right)}^2+(a_{x}K\left(y\right)+\lambda )u^2\right]dy,$$

From here

$${\parallel (L+\lambda E)u\parallel }_{L_2}\ge C{\parallel u\parallel }_{L_2},C>0.$$

Now, if we show that the set $(L+\lambda E)D\left(L\right)$ is dense in $L_2$, then it follows that the operator $L+\lambda E$ has a continuous or the inverse operator ${(L+\lambda E)}^{-1}$.

• We will prove this by contradiction. Suppose that the set $(L+\lambda E)D$ is not dense in $L_2(0,1)$.
• Then there exists an element $v\in L_2(0,1)$ such that

  $$⟨(L+\lambda E)v,u⟩=⟨(L+\lambda E)u,v⟩=0,\mathrm{for}\mathrm{all}u\in D\left(L\right).$$

  Since the set $D\left(L\right)$ is dense in $L_1(0,1)$, according to the Dubois-Reymond lemma [2]

  $${(L+\lambda E)}^{*}v=-v^{″}+(a_{x}K\left(y\right)+\lambda )v=0,v\in L_2(0,1).$$

Since the function $K\left(y\right)$ is a bounded function in the segment $[0,1]$, then $K\left(y\right)v\in L_2(0,1)$ and therefore $v^{\prime }\in L_2(0,1)$. Hence, $v\equiv 0$. □

Lemma 1 is proven.

Lemma 2.

Let $K\left(y\right)\in G_x$ and let conditions (3), (5) be satisfied. Then the following inequality holds:

$${\mu }_{\left|\rho \right|K\left(y\right)}\ge {\displaystyle \frac{C_0}{4^2}}{\lambda }_n^{2-2\alpha },$$

where

$${\mu }_{\left|\rho \right|K\left(y\right)}=\underset{\begin{array}{c}z\in C_0^2(0,1)\\ {\int }_0^1{\left|K\left(x\right)z\right|}^{2}dx\ne 0\end{array}}{inf}{\displaystyle \frac{{\int }_0^1{\left|(L+\lambda E)z\right|}^{2}dx}{{\int }_0^1{|\rho ·K\left(x\right)z|}^{2}dx}}=\underset{\begin{array}{c}{\psi }_n\in C_0^2(0,1)\\ \parallel \rho ·K\left(y\right){\psi }_n{\parallel }_{L_2(0,1)}\ne 0\end{array}}{inf}{\displaystyle \frac{\parallel (L+\lambda E){\psi }_n{\parallel }_{L_2(0,1)}^2}{\parallel \rho ·K\left(y\right){\psi }_n{\parallel }_{L_2(0,1)}^2}},n=1,2,\dots$$

(7)

Proof. 

Let us make the substitution

$$y={\displaystyle \frac{x}{{\lambda }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}}}.$$

After this substitution, (7) will take the following form:

$${\mu }_{\left|\rho \right|K\left(y\right)}={\displaystyle \frac{{\lambda }^2}{{\lambda }_n^{2\alpha }}}·\underset{{\psi }_n\in C_0^2[0,1]}{inf}{\displaystyle \frac{{\int }_0^{\frac{{\lambda }_n^{1/2}}{{\lambda }^{1/2}}}{\left[z^{″}+{\psi }_n\left(x\right)\right]}^{2}dx}{{\int }_0^{\frac{{\lambda }_n^{1/2}}{{\lambda }^{1/2}}}{\left|K\left(x\right){\psi }_n\right|}^{2}dx}}$$

(8)

where

$$K\left(x\right)=K\left({\displaystyle \frac{x}{{\lambda }_n^{1/2}{\lambda }^{1/2}}}\right),{\psi }_n\left(x\right)={\lambda }_n^{-1}(a_{x}K\left(x\right)+\lambda ).$$

Now we estimate $\parallel z^{″}+{\psi }_n{\parallel }_{L_2(0,1)}$ from below. To achieve this, by composing a quadratic form and using condition (2), we obtain

$$\begin{array}{cc}\hfill \parallel z^{″}+{\psi }_n\left(x\right){\parallel }_{L_2(0,1)}^2& \ge C\parallel z{\parallel }_{L_2(0,1)}^2,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \parallel z^{″}+{\psi }_n\left(x\right){\parallel }_{L_2(0,1)}^2& \ge C\parallel {\psi }_n{\parallel }_{L_2(0,1)}^2.\hfill \end{array}$$

(10)

Let us take two functions ${\phi }_1\left(x\right),{\phi }_2\left(x\right)\in C^2(-\infty ,\infty )$ satisfying the conditions

$${\phi }_1^2+{\phi }_2^2=1,sup\sum \left(|{\phi }_1^{″}|+|{\phi }_2^{″}|+|{\phi }_1^{\prime }|+|{\phi }_2^{\prime }|\right)\le C.$$

$$\underset{x}{sup}\sum _{i=1}^2\left(|{\phi }_i^{\prime }|+|{\phi }_i^{″}|+|{\phi }_i|\right)\le C$$

$${\phi }_k\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x\in \left[{\displaystyle \frac{3}{2}}k-{\displaystyle \frac{3}{4}},{\displaystyle \frac{3}{2}}k+{\displaystyle \frac{1}{4}}\right]\hfill \\ \in [0,1],\hfill & x\in \left[{\displaystyle \frac{3}{2}}k+{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{2}}k+{\displaystyle \frac{3}{2}}\right]\hfill \\ 0,\hfill & \mathrm{elsewhere}\hfill \end{array}\right.,(k=0,\pm 1,\pm 2,\dots )$$

If $z\in C^2[0,{\lambda }_n^2]$, $z\left(0\right)=z\left({\lambda }_n^2\right)=0$, then using (9) and (10) we have

$$\begin{array}{cc}\hfill \parallel z{\left({\phi }_i\right)}^{″}+{\psi }_n\left(x\right)z\left({\phi }_i\right){\parallel }_{L^2}& =\parallel z^{″}{\phi }_i-2z^{\prime }{\phi }_i^{\prime }-z{\phi }_i^{″}+{\psi }_n\left(x\right)z{\phi }_i{\parallel }_{L^2}\hfill \\ \hfill & =\parallel \left({\phi }_i{z}^{″}\right)+{\psi }_n\left(x\right)z{\phi }_i-2z^{\prime }{\phi }_i^{\prime }-z{\phi }_i^{″}{\parallel }_{L^2}\hfill \\ \hfill & \le \parallel {\phi }_i{\parallel }_{L^{\infty }}\parallel z^{″}{\parallel }_{L^2}+\parallel {\psi }_n{\left(x\right)\parallel }_{L^{\infty }}\parallel z{\phi }_i{\parallel }_{L^2}+2\parallel z^{\prime }{\parallel }_{L^2}\parallel {\phi }_i^{\prime }{\parallel }_{L^{\infty }}+{\parallel z\parallel }_{L^2}{\parallel {\phi }_i^{″}\parallel }_{L^{\infty }}\hfill \\ \hfill & \le C\left(\parallel z^{″}{\parallel }_{L^2}+\parallel z^{\prime }{\parallel }_{L^2}+{\parallel z\parallel }_{L^2}\right),(i=1,2)\hfill \end{array}$$

Let us substitute these inequalities into (8):

$${\mu }_n^{{\left|K\left(x\right)\right|}^2}={\displaystyle \frac{{\lambda }_n^2}{C{\lambda }_n^2}}\underset{z\in C_0^2[0,{\lambda }_n^2]}{inf}{\displaystyle \frac{{\int }_0^{{\lambda }_n^2}{|z^{″}+{\psi }_n\left(x\right)z|}^{2}dx}{{\int }_0^{{\lambda }_n^2}{\left|K\left(x\right)z\left(x\right)\right|}^{2}dx}}\ge {\displaystyle \frac{1}{C{\lambda }_n^2}}\sum _{i=1}^2\underset{z\in C_0^2[0,{\lambda }_n^2]}{inf}{\displaystyle \frac{{\int }_0^{{\lambda }_n^2}{|z^{″}+{\psi }_n\left(x\right)z|}^{2}dx}{{\int }_0^{{\lambda }_n^2}{\left|K\left(x\right)z{\phi }_i\right|}^{2}dx}}$$

$$\ge {\displaystyle \frac{{\lambda }_n^2}{C{\lambda }_n^2}}\underset{\xi (·)\in C_0^2[0,1]}{inf}\sum _{k=-N_n}^{N_n}{\int }_{{\Delta }_k^n}{\left|z^{″}+{\psi }_n\left(x\right)z\right|}^{2}dx\ge {\displaystyle \frac{{\lambda }_n^2}{C{\lambda }_n^2}}\underset{\xi (·)\in C_0^2[0,1]}{inf}\sum _{k=-N_n}^{N_n}{\int }_{{\Delta }_k^n}{\left|z^{″}+{\psi }_n\left(x\right)z\right|}^{2}dx$$

$$\ge {\displaystyle \frac{{\lambda }_n^2}{C{\lambda }_n^2}}\underset{\xi (·)\in C_0^2[0,1]}{inf}\sum _{k=-N_n}^{N_n}{\int }_{{\Delta }_k^n}{\left|K\left(x\right)z{\phi }_i\right|}^{2}dx$$

$$\ge {\displaystyle \frac{{\lambda }_n^2}{C_n{\lambda }_n^2}}\underset{\xi (·)\in C_0^2[0,1]}{inf}\underset{{\phi }_i\in C_0^2,\mathrm{supp}\left({\phi }_i\right)={\Delta }_k^n}{inf}{\displaystyle \frac{{\int }_{{\Delta }_k^n}{\left|{\left({\phi }_{i}z\right)}^{″}+{\psi }_n\left(x\right)\left({\phi }_{i}z\right)\right|}^{2}dx}{{\int }_{{\Delta }_k^n}{\left|K\left(x\right)z{\phi }_i\right|}^{2}dx}}$$

(11)

Let $\Delta =({\Delta }^{-},{\Delta }^{+})\subset (0,\frac{1}{2}{\lambda }_n^2)$, where ${\Delta }^{+}-{\Delta }^{-}\le 1$ and ${\Delta }^{-},{\Delta }^{+}$ are the left and right ends of $\Delta$, respectively. Using condition (2) for the function $z\in C^2(\Delta ),z\left({\Delta }^{-}\right)=z\left({\Delta }^{+}\right)=0$, repeating the calculations used to obtain inequality (9), we obtain

$${\int }_{\Delta }|z^{″}+{\psi }_n\left(x\right)z{|}^{2}dx\ge {\int }_{\Delta }{\left|K\left(x\right)z\right|}^{2}dx$$

(12)

Inequality (11) together with the known inequality

$${\displaystyle \frac{a_1+a_2+\dots +a_n}{b_1+b_2+\dots +b_n}}\ge \underset{1\le i\le n}{inf}{\displaystyle \frac{a_i}{b_i}},a_i\ge 0,b_i\ge 0$$

gives

$${\mu }_{{\left|K\left(x\right)\right|}^2}\ge {\displaystyle \frac{{\lambda }_n^2}{2{\lambda }_n}}\underset{\begin{array}{c}\Delta \subset C^2[0,{\lambda }_n^{\frac{1}{2}}]\\ {\Delta }^{-}<\Delta <{\Delta }^{+}\\ {\Delta }^{+}-{\Delta }^{-}\le 1\end{array}}{inf}\underset{z\in C_0^2(\Delta )}{inf}{\displaystyle \frac{{\int }_{\Delta }{\left|z^{″}+{\psi }_n\left(x\right)z\right|}^{2}dx}{{\int }_{\Delta }{\left|K\left(x\right)z\right|}^{2}dx}}$$

Therefore, by virtue of (12) we have

$${\mu }_{{\left|K\left(x\right)\right|}^2}\ge {\displaystyle \frac{{\lambda }_n^2}{2{\lambda }_n}}\underset{\begin{array}{c}\Delta \subset C^2[0,{\lambda }_n^{\frac{1}{2}}]\\ {\Delta }^{+}-{\Delta }^{-}\le 1\end{array}}{inf}{\displaystyle \frac{{\int }_{\Delta }{\left|K\left(x\right)\right|}^{2}dx}{{\int }_{\Delta }{\left|K\left(x\right)\right|}^{2}dx}}$$

From here, taking into account (2), we find the following lower bound for the numbers ${\mu }_{{\left|K\left(x\right)\right|}^2}$:

$${\mu }_{{\left|K\left(x\right)\right|}^2}\ge {\displaystyle \frac{{\lambda }_n^2}{2{\lambda }_n}}$$

□

This inequality proves Lemma 2.

Proof of Theorem 1.

Consider the quadratic form

$$(Lu+\lambda u,u),\mathrm{for}u\in C_0^{\infty }(\Omega )$$

By virtue of (4) we have

$$(Lu+\lambda u,u)={\int }_{\Omega }K\left(\nu \right)\alpha \left(x\right)|u_x{|}^{2}dxd\gamma +{\int }_{\Omega }|u_y{|}^{2}dxd\gamma +{\int }_{\Omega }\lambda {\left|u\right|}^{2}dxd\gamma$$

Using the Cauchy–Bunyakovsky inequality and condition (4), we obtain

$${\parallel Lu+\lambda u\parallel }_{L_2(\Omega )}\ge \lambda {\parallel u\parallel }_{L_2(\Omega )}$$

(13)

If

$$f_k(x,y)=\sum _{\nu =-k}^k{f}_n\left(\nu \right){\psi }_n\left(x\right)$$

(14)

where $z_n\left(x\right)$ are eigenfunctions of the operator T, then it is easy to see that the function

$$u_k(x,y)={(L+\lambda E)}^{-1}{f}_k(x,y)=\sum _{\nu =-k}^k\left[{(h_n+\lambda E)}^{-1}{f}_n\left(y\right)\right]f_n\left(x\right)$$

is a solution to problems (1), (4) and (1), (6).

Since a set of the form (14) is dense in $L_2(\Omega )$, then by virtue of inequality (13) the sequence $\left\{u_k(x,y)\right\}$ is fundamental in $L_2(\Omega )$, and by virtue of the completeness of $L_2(\Omega )$, it converges to $u(x,y)\in L_2(\Omega )$. From the obtained relations and Definition 2 we obtain that $u(x,y)$ is a solution to problems (1), (4) and (1), (6). □

Proof of Points (b) and (c). Note that if

$$f(x,y)=\sum _{n=-\infty }^{\infty }{f}_n\left(y\right)e^{inx}$$

then

$$u(x,y)={(L+\lambda E)}^{-1}f=\sum _{n=-\infty }^{\infty }\left[{(h_n+\lambda E)}^{-1}{f}_n\left(y\right)\right]e^{inx}$$

From this and from the orthonormality of the system ${\left\{e^{inx}\right\}}_{n=-\infty }^{\infty }$ in $L_2(0,1)$, the relation follows

$${∥\rho \left(y\right)T_x^n{(L+\lambda E)}^{-1}∥}_{L_2\to L_2}=\underset{\left\{f\right\}}{sup}{∥{\lambda }_n^n\rho \left(y\right){(h_n+\lambda E)}^{-1}∥}_{L_2\to L_2}$$

(15)

where $\rho \left(y\right)=K\left(y\right)$.

Since for each n the operator ${(h_n+\lambda E)}^{-1}$ is completely continuous, it follows from (15) and the known theorems for completely continuous operators that the operator

$$\rho \left(y\right)T_x^n{(L+\lambda E)}^{-1}$$

is completely continuous if and only if

$$\underset{\left|n\right|\to \infty }{lim}{∥\left(\rho \left(y\right){\lambda }_n^n{(h_n+\lambda E)}^{-1}\right)∥}_{L_2(0,1)\to L_2(0,1)}=0,$$

or what is the same thing:

$$\underset{\left|n\right|\to \infty }{lim}\left({∥\left(\rho \left(y\right){\lambda }_n^n{(h_n+\lambda E)}^{-1}\right)∥}_{L_2(0,1)\to L_2(0,1)}^{-1}\right)=\infty .$$

But the magnitude

$${\mu }_{\left|n\right|,\rho \left(y\right)}={∥\rho \left(y\right){\lambda }_n^{\alpha }{\left(L_n+\lambda E\right)}^{-1}∥}_{L_2(0,1)\to L_2(0,1)}^{-1},$$

(16)

according to Lemma 2 allows for the estimates

$$C^{-1}{\lambda }_n^{2-2\alpha }\le {\mu }_{\left|n\right|,\rho \left(y\right)}.$$

from which the statement of the theorem on the complete continuity of the operator

$$\rho \left(y\right)T_x^{\alpha }{(L+\lambda E)}^{-1}$$

follows. The statement on the boundedness of

$$K\left(y\right)T_x{(L+\lambda E)}^{-1}$$

follows from (15) and (16). The theorem is completely proven.

If we recall the history of the formulation of this problem, then T.Sh. Kalmenov and M. Otelbaev considered the case when $K\left(y\right),a\left(y\right)$ are monotonically non-decreasing and satisfy the conditions

$$\underset{y\to 0}{lim}\left\{{\displaystyle \frac{K\left(2y\right)}{K\left(y\right)}},{\displaystyle \frac{a\left(2y\right)}{a\left(y\right)}}\right\}<\infty$$

and for such classes of problems they obtained necessary and sufficient conditions of coercivity and solvability. M.B. Muratbekov considered a non-classical-type operator and proved the coercivity of the solvability of the Dirichlet problem, and under restrictions (3), (5) on the coefficients $K\left(y\right),a\left(y\right)$, the coercivity of the Dirichlet problem was proved.

The case of non-uniformly degenerate elliptic equations and their coercive solvability was considered by M.B. Muratbekov.

Let us recall that an elliptic operator is called a non-uniformly degenerating elliptic operator in a domain $\Omega$ or on some part of the boundary if the rate of degeneration of its coefficients occurs non-uniformly with respect to the coordinate variables. □

4. Conclusions

In this article, we investigate the limiting smoothness of solutions to the first boundary value problem for non-uniformly degenerating elliptic equations within a broader class of coefficients. Specifically, we consider the problem under assumptions (3), (5) and on the coefficients $K\left(y\right)$ and $a\left(y\right)$, which allow for a non-monotonic and possibly irregular nature of degeneration. We establish the existence and uniqueness of strong solutions to the problem and further demonstrate the complete continuity of the operator

$$\rho \left(y\right)T_x^{\alpha }{(L+\lambda E)}^{-1}$$

as well as the boundedness of the operator

$$K\left(y\right)T_x{(L+\lambda E)}^{-1}$$

These results are obtained through techniques involving a priori estimates, spectral theory, and localization methods. In particular, we analyze the behavior of solutions in weighted Sobolev spaces adapted to the degeneracy structure of the equation. The findings contribute to the broader understanding of degenerate elliptic operators and may have applications in the spectral analysis of corresponding differential operators, particularly in settings where standard ellipticity conditions do not hold.