# Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

- (i)
- The PDO $A(x,D)$ is elliptic at every point $x\in \overline{\Omega}$, i.e., ${A}^{\circ}(x,\xi )\ne 0$ whenever $0\ne \xi \in {\mathbb{R}}^{n}$.
- (ii)
- The PDO $A(x,D)$ is properly elliptic at every point $x\in \Gamma $; i.e., for an arbitrary tangent vector $\tau \ne 0$ to $\Gamma $ at x, the polynomial ${A}^{\circ}(x,\tau +\nu \left(x\right)\zeta )$ in $\zeta \in \mathbb{C}$ has q roots with positive imaginary part and q roots with negative imaginary part (of course, these roots are counted with regard for their multiplicity).
- (iii)
- The boundary conditions (2) cover $A(x,D)$ at every point $x\in \Gamma $. This means that, for each vector $\tau \ne 0$ from condition (ii), the boundary-value problem$$\begin{array}{c}{A}^{\circ}(x,\tau +\nu \left(x\right){D}_{t})\theta \left(t\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t>0,\\ {B}_{j}^{\circ}(x,\tau +\nu \left(x\right){D}_{t})\theta \left(t\right){|}_{t=0}+\sum _{k=1}^{\varkappa}{C}_{j,k}^{\circ}(x,\tau ){\lambda}_{k}=0,\phantom{\rule{1.em}{0ex}}j=1,\dots ,q+\varkappa ,\\ \theta \left(t\right)\to 0\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}t\to \infty \end{array}$$

## 3. Extended Sobolev Scale

## 4. The Main Results

**Theorem**

**1.**

**Lemma**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

## 5. The Interpolation between Hilbert Spaces

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

## 6. Proofs of the Main Results

**Proof**

**of**

**Theorem**

**1**

**Proof**

**Lemma**

**1**

**Proof**

**of**

**Theorem**

**2**

**Proof**

**of**

**Theorem**

**3**

**Proof**

**of**

**Theorem**

**4**

**Proof**

**of**

**Theorem**

**5**

## 7. Applications

**Proposition**

**5.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Remark**

**1.**

**Proof.**

**Theorem**

**8.**

**Proof**

**of**

**Theorem**

**8**

## 8. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Anop, A.; Chepurukhina, I.; Murach, A. Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces. *Axioms* **2021**, *10*, 292.
https://doi.org/10.3390/axioms10040292

**AMA Style**

Anop A, Chepurukhina I, Murach A. Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces. *Axioms*. 2021; 10(4):292.
https://doi.org/10.3390/axioms10040292

**Chicago/Turabian Style**

Anop, Anna, Iryna Chepurukhina, and Aleksandr Murach. 2021. "Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces" *Axioms* 10, no. 4: 292.
https://doi.org/10.3390/axioms10040292