# On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Remark**

**1.**

**Remark**

**2.**

## 2. Auxiliary Results

**Theorem**

**1.**

- m-strongly monotone, if there exists $m>0$ such that for all $u,v\in \mathcal{H}$ it holds$$\u2329A\left(u\right)-A\left(v\right),u-v\u232a\ge m{\u2225u-v\u2225}^{2};$$
- potential, if there exists a Gâteaux differentiable functional $\mathcal{A}:\mathcal{H}\u27f6\mathbb{R}$ called potential of A, such that ${\mathcal{A}}^{\prime}=A$;
- demicontinuous, if convergence ${u}_{n}\to {u}_{0}$ in $\mathcal{H}$ implies that $A\left({u}_{n}\right)\rightharpoonup A\left({u}_{0}\right)$ in ${\mathcal{H}}^{\ast}$.

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**2**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2**

**Theorem**

**3**

**Proof.**

**Remark**

**5.**

## 3. On the Solvability of the Continuous Problem

**Theorem**

**4.**

**Proof.**

## 4. On the Discretizations of Problem (1)

**Remark**

**6.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**1.**

**Proof.**

## 5. Convergence of Discretizations

**Lemma**

**4.**

**Proposition**

**2.**

**Proof.**

**Remark**

**7.**

**Theorem**

**5.**

**Theorem**

**6.**

**Remark**

**8.**

**Example**

**1.**

- ${f}_{1}(t,x)=\frac{t-{\pi}^{2}sin\left(x\right)}{2{x}^{2}+4}$;
- ${f}_{2}(t,x)={\pi}^{2}x{e}^{\pi -t}+arctan\left(x\right)+{e}^{t}$;
- ${f}_{3}(t,x)=({\pi}^{2}-1)x-{x}^{3}+{t}^{3}-sin\left(t\right)$.

#### Some Numerical Phenomena (Whole Subsection Has Been Changed)

**Example**

**2.**

**Example**

**3.**

## 6. Conclusions

- Usage of a Sobolev space setting for our problem allows us to consider a continuous and discrete problems as an elements in the same spaces. Therefore we can use a general and easy tools from functional analysis (Ritz and Direct Method of Variational Method together with some monotonicity relations).
- Algebraic discretizations are of use since, for sufficiently large n, the solvability of a continuous problem provides a solvability of a discrete one (at in least if we use the mentioned Direct Method).
- There may be significant differences with handling homogeneous (Example 2) and nonhomogeneous first eigenvalue problem and its discretizations.
- The above mentioned phenomena is not a general rule. It strongly depends on the precise form of the nonlinearity.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Plot of $ln\left(1+{10}^{6}\xb7{\u03f5}_{1,n,\alpha}\right)$ depending on $\alpha $ and n.

**Figure 4.**Plot of $ln\left(1+{10}^{6}\xb7{\u03f5}_{2,n,\alpha}\right)$ depending on $\alpha $ and n.

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**MDPI and ACS Style**

Bełdziński, M.; Gałaj, T.; Bednarski, R.; Pietrusiak, F.; Galewski, M.; Wojciechowski, A.
On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem. *Symmetry* **2021**, *13*, 231.
https://doi.org/10.3390/sym13020231

**AMA Style**

Bełdziński M, Gałaj T, Bednarski R, Pietrusiak F, Galewski M, Wojciechowski A.
On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem. *Symmetry*. 2021; 13(2):231.
https://doi.org/10.3390/sym13020231

**Chicago/Turabian Style**

Bełdziński, Michał, Tomasz Gałaj, Radosław Bednarski, Filip Pietrusiak, Marek Galewski, and Adam Wojciechowski.
2021. "On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem" *Symmetry* 13, no. 2: 231.
https://doi.org/10.3390/sym13020231