# Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

## Abstract

**:**

## 1. Introduction

## 2. Surjectivity for the Operators λJ_{ϕ} − S. Applications to Partial Differential Equations

#### 2.1. Theoretical Results

**Statement for the Problem**

**Proposition 1.**

**R**

^{N}, N ≥ 2, p ∈ (1, +∞), h from ${W}^{-1,p\prime}$(Ω) and f: Ω ×

**R**→

**R**a Carathéodory function with the properties

^{0}f (x, − s) = − f (x, s) ∀s from

**R**, ∀x from Ω,

^{0}| f (x, s)| ≤ ${c}_{1}{\left|s\right|}^{q-1}$+ β(x) ∀s from

**R**, ∀x from Ω \ A, μ(A) = 0,

- where${c}_{1}$ ≥ 0, q ∈ (1, p), β ∈ ${L}^{q\prime}$(Ω),$\frac{1}{q}+\frac{1}{q\prime}=1$.

**Proposition 2.**

**R**

^{N}, N ≥ 2, p ∈ (1, +∞), h from ${W}^{-1,p\prime}$(Ω) and f : Ω ×

**R**→

**R**Carathéodory function having the properties

^{0}f (x, −s) = − f (x, s) ∀x from Ω, ∀s from

**R**,

^{0}| f (x, s)| ≤ ${c}_{1}{\left|s\right|}^{p-1}$+ β(x) ∀s from

**R**, ∀x from Ω \ A, μ(A) = 0,

- where c
_{1}≥ 0, β ∈ ${L}^{p\prime}$(Ω),$\frac{1}{p}+\frac{1}{p\prime}$= 1.

**Statement for the Problem**

**Proposition 3.**

**R**

^{N}, N ≥ 2, p ∈ [2, +∞), h from ${W}^{-1,p\prime}$(Ω) and f : Ω ×

**R**→

**R**Carathéodory function with the properties

^{0}f (x, − s) = − f (x, s) ∀x from Ω, ∀s from

**R**,

^{0}| f (x, s)| ≤ ${c}_{1}{\left|s\right|}^{q-1}$+ β(x) ∀s from

**R**, ∀x from Ω \ A, μ(A) = 0,

- where${c}_{1}$ ≥ 0, q ∈ (1, p), β ∈ ${L}^{q\prime}$(Ω), $\frac{1}{q}+\frac{1}{q\prime}=1$.

**Proposition 4.**

**R**

^{N}, N ≥ 2, p ∈ [2, +∞), h from ${W}^{-1,p\prime}$(Ω) and f : Ω ×

**R**→

**R**Carathéodory function having the properties

^{0}f (x, −s) = − f (x, s) ∀x from Ω, ∀s from

**R**,

^{0}$|$ f (x, s)$|$ ≤ ${c}_{1}|$s${|}^{p-1}$+ β(x) ∀s from

**R**, ∀x from Ω \ A, μ(A) = 0,

- where${c}_{1}$ ≥ 0, β ∈ ${L}^{p\prime}$(Ω),$\frac{1}{p}+\frac{1}{p\prime}$= 1.

#### 2.2. Applications in Solving some Real Models

#### 2.2.1. Application in Glaciology

#### 2.2.2. Nonlinear Elastic Membrane

**I.**To describe a nonlinear elastic membrane under the load f, we can use the mathematical model:

^{0}. Therefore, for any λ, e.g., |λ| ≥ ${\lambda}_{1}^{-1}$, the last problem has a solution in ${W}_{0}^{1,p}$(Ω) in the sense of ${W}^{-1,p\prime}$(Ω).

**II.**In the paper [31], a model was proposed for the vibration of a nonhomogeneous membrane which is fixed along the boundary. Several materials (with different densities) were investigated there, following the location of these materials inside Ω by studying the first mode in the vibration of the membrane. Ω is a bounded smooth domain in**R**^{N}and g is a Lebesgue measurable function verifying the condition 0 ≤ g(x) ≤ H, ∀x ∈ Ω, where H is a positive constant, g$\overline{)\equiv}$ 0 and H. g can be replaced by any Lebesgue measurable function, equal to it almost everywhere. Consider the eigenvalue Dirichlet problem:

**|**∂Ω = 0.

^{0}is fulfilled with h ≡ 0, ${c}_{1}$= H, and, for any λ with $\left|\frac{1}{\lambda}\right|$≥ H${\mathsf{\lambda}}_{1}^{-1}$, equivalent |λ| ≤$\frac{{\lambda}_{1}}{H}$, ${\mathsf{\lambda}}_{1}$: = inf $\left\{\frac{\left|\right|u|{|}_{1,p}^{p}}{\left|\right|i(u)|{|}_{0,p}^{p}}:u\in {W}_{0}^{1,p}(\mathsf{\Omega})\backslash \{0\}\right\}$, the above problem has a solution in ${W}_{{0}_{}}^{1,p}$(Ω) in the sense of ${W}^{-1,p\prime}$(Ω). in the sense of ${W}^{-1,p\prime}(\mathsf{\Omega})$.

**III.**Using Proposition 2, we can propose a proof for the existence of the solution of the nonlinear problem of elastic membrane under the load f + h, in the general case when f is a Carathéodory function which fulfills the conditions 1^{0}and 2^{0}, h from ${W}^{-1,p\prime}$(Ω) and λ such that$$\left|\mathsf{\lambda}\right|\text{}{c}_{1}{\mathsf{\lambda}}_{1}^{-1},{\mathsf{\lambda}}_{1}\text{:}=\mathrm{inf}\left\{\frac{\left|\right|u|{|}_{1,p}^{p}}{\left|\right|i(u)|{|}_{0,p}^{p}}:u\in {W}_{0}^{1,p}(\mathsf{\Omega})\backslash \{0\}\right\}$$

#### 2.2.3. The Pseudo Torsion Problem

#### 2.2.4. Nonlinear Elastic Membrane with p-Pseudo-Laplacian

**I.**In [40], the expression $\underset{\mathsf{\Omega}}{\int}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left|\frac{\partial u}{\partial {x}_{\mathrm{i}}}\right|}^{\mathrm{p}}}dx$ was proposed for the deformation energy of the membrane and was woven out of elastic strings in a rectangular form. The phenomenon can be modelled with a Dirichlet problem for the p-pseudo-Laplacian:$$-\mathsf{\lambda}{\mathsf{\Delta}}_{p}^{s}u=|u{|}^{p-2}u,x\in \mathsf{\Omega},\mathsf{\lambda}\in \mathsf{R}$$u|∂Ω = 0.

^{0}, to obtain that the last exposed problem has a solution in ${W}_{0}^{1,p}$(Ω) in the sense of ${W}^{-1,p\prime}$(Ω), for any λ, with

**II.**Take in**I**the load f + h, in the general case when f is a Carathéodory function, which fulfills the conditions 1^{0}and 2^{0}from Proposition 4, h from ${W}^{-1,p\prime}$(Ω) and λ e.g., |$\mathsf{\lambda}$| > ${c}_{1}{\mathsf{\lambda}}_{1}^{-1}$,$${\mathsf{\lambda}}_{1}\text{:}=\mathrm{inf}\left\{\frac{{\u2759u\u2759}_{1,p}^{p}}{\left|\right|i(u)|{|}_{0,p}^{p}}:u\in {W}_{0}^{1,p}(\mathsf{\Omega})\backslash \{0\}\right\}$$

## 3. Applications for Results of the Fredholm Alternative Type for operators λJ_{ϕ} − S

#### 3.1. Results

**Proposition 5.**

**Proposition 6.**

**Proposition 7.**

#### 3.2. Applications for Real Phenomena

#### 3.2.1. Nonlinear Elastic Membrane

**Remark.**

#### 3.2.2. Nonlinear Elastic Membrane with p-Pseudo-Laplacian

## 4. Problems Solved Using Surjectivity to Different Homogeneity Degrees

#### 4.1. Theoretical Results

**Proposition 8.**

**Proposition 9.**

#### 4.2. Applications to Real Phenomena Models

#### 4.2.1. Nonlinear Elastic Membrane

**I.**Use Proposition 8 to prove the existence of the solution for the problem of a nonlinear elastic membrane under the load f + h, where we can use the mathematical model:

**II.**Similarly, for the problem:

**R**→

**R**an odd Carathéodory function and (q − 1) - homogeneous in the second variable, which verifies the growth condition:

#### 4.2.2. Nonlinear Elastic Membrane with p-Pseudo-Laplacian

**I.**Demonstrate the existence of the solution from ${W}_{{0}_{}}^{1,p}$(Ω) in the sense of ${W}^{-1,p\prime}$(Ω) for the problem:

**II.**For a similar problem:

**R**→

**R**an odd Carathéodory function and (q − 1) - homogeneous in the second variable, which verifies the growth condition:

## 5. Problems Having Weak Solutions Starting from Ekeland Variational Principle

#### 5.1. Critical Points and Weak Solutions for Elliptic Type Equations − Applications

#### 5.1.1. Theoretical Support

**R**

^{N}and let f : Ω ×

**R**→

**R**be a Carathéodory function with the growth condition:

**Proposition 10.**

**R**

^{N}, N ≥ 3, f : Ω ×

**R**→

**R**a Carathéodory function and ${u}_{1}$, ${u}_{2}$ from ${W}_{{0}_{}}^{1,p}(\mathsf{\Omega})$ bounded weak subsolution and weak supersolution of (∗), respectively, with ${u}_{1}$(x) ≤ ${u}_{2}$(x) a.e. on Ω. Suppose that f verifies (1) and there is ρ > 0 such that the function g: g(x, s) = f (x, s) + ρs is strictly increasing in s on [inf ${u}_{1}$(Ω), sup ${u}_{2}$(Ω)]. Then there is a weak solution $\overline{u}$ of (∗) in ${W}_{0}^{1,p}(\mathsf{\Omega})$ with the property

**Proposition 11.**

**R**

^{N}, N ≥ 3 and f : Ω ×

**R**→

**R**a Carathéodory function and ${u}_{1}$, ${u}_{2}$ from ${W}_{{0}_{}}^{1,p}(\mathsf{\Omega})$ bounded weak subsolution and weak supersolution of (∗∗), respectively, with ${u}_{1}$(x) ≤ ${u}_{2}$(x) a.e. on Ω. Suppose that f verifies (1) and there is ρ > 0 such that the function g: g(x, s) = f (x, s) + ρs is strictly increasing in s on [inf ${u}_{1}$(Ω), sup ${u}_{2}$(Ω)]. Additionally, there is a weak solution $\overline{u}$ of (∗∗) in ${W}_{0}^{1,p}(\mathsf{\Omega})$ with the property:

#### 5.1.2. Applications to Real Phenomena

- I.
- Application in Glaciology

- II.
- Nonlinear Elastic Membrane

- III.
- The Pseudo Torsion Problem

- IV.
- Nonlinear Elastic Membrane with p-Pseudo-Laplacian

#### 5.2. Applications of Critical Points for Nondifferentiable Functionals

#### 5.2.1. Theoretical Results

**R**

^{N}with the smooth boundary ∂Ω (topological boun-dary). Consider the nonlinear boundary value problems (∗) and (∗∗) from Section 5.1.2 above, where f : Ω ×

**R → R**is a measurable function with subcritical growth, i.e.,

**R**,

**Proposition 12.**

#### 5.2.2. Applications − Real Phenomena Modeling

**I.**We propose here an immediate application to characterize the solution of the modeling given in [47] for thermal transfer. Among cryogenic fluids used in industrial or laboratory applications, helium II offers remarkable properties. However, its behavior, both mechanical and thermal, appears particularly complex. Regarding the heat transfer at the heart of this fluid and through the interface with a solid, the two phenomenological laws have been used in [47] in order to obtain a mathematical model which should make it possible to optimally ensure the desired temperature in the problems associated with cooling by helium II; this goal is particularly crucial for multifilamentary superconductors.

^{−3}) such that:

**II.**One can give characterizations of the solutions using this kind of definition for the Dirichlet problems issued from the already-presented problems of the movement of the gla-cier, nonlinear elastic membrane, pseudo torsion problem, or nonlinear elastic membrane with p-pseudo-Laplacian.

#### 5.3. Other Solutions Starting from Ekeland Principle

#### 5.3.1. Theoretical Results

**Definition 1.**

**Proposition 13.**

**R**→

**R**bounded measurable T-periodic,

#### 5.3.2. Related Applications

**I.**We can apply Proposition 13 with g ≡ 1 and h ≡ 0 to prove that there exists a solution for the velocity problem under the assumption of solid friction, see [26],

**II.**Involve Proposition 13 to give another solution for the problem studied in [48]. Accor-ding to [42], the physical problem is as follows: the polymer is injected, over a period of time, ${t}_{0}$ < t < ${t}_{1}$ at some point ${x}_{1}$ ∈ Ω. It is not necessary to have the domain Ω simply connected. Some notations: ${\mathsf{\Omega}}_{t}$ = the part of Ω which is filled by fluid at time t; φ = pressure; v = fluid velocity (averaged over − h ≤ z ≤ h); ${\mathsf{\Gamma}}_{0}$ = ∂${\mathsf{\Omega}}_{t}$ ∩ Ω, and ${\mathsf{\Gamma}}_{1}$ = ∂${\mathsf{\Omega}}_{t}$ ∩ ∂Ω. ${\mathsf{\Gamma}}_{0}$ is the flow front. It is assumed that ∂Ω is solid (except for air vents). The equation for φ is:

_{t}filled by fluid is controlled by φ, which is determined from an elliptic partial differential equation. Note that, in this physically oriented description, all considered curves, functions, and vector fields are assumed “smooth”, such that each of the crucial expressions has a well-defined pointwise meaning. Following [48], the mathematical problem means to obtain the solution φ of the instantaneous flow problem which can be obtained as the solution φ* of a convex extremum problem by an appropriate and rather obvious choice of that problem. Taking g ≡ 1 and h ≡ 0, ${\mathsf{\Omega}}_{t}$ instead of Ω in (N), we are placed under the conditions of Proposition 13 and give, via this result, another proof for the existence of the solution for the mentioned problem.

**III.**We can add here the example of the solution of the modeling given in [47] for thermal transfer described at Section 5.2.2 with the same equation under Neumann boundary conditions (flux imposed): $|$∇u${|}^{p-2}$$\frac{\partial u}{\partial n}$= ψ on ∂Ω. Proposition 13 establishes the existence of the solution.

**IV.**Consider the pseudo torsion problem:

## 6. Weak Solutions Using a Perturbed Variational Principle

#### 6.1. Theoretical Results

**R**

^{N}, N ≥ 3. Consider the problems (∗) and (∗∗) from Section 5.1.1, where f : Ω ×

**R**→

**R**is a Carathéodory function with the growth condition

**R**,

- Problem (∗). Let ${\mathsf{\lambda}}_{1}$ be the first eigenvalue of –${\u2206}_{p}$ in ${W}_{0}^{1,p}$(Ω) with a homogeneous boun-dary condition. We have (see, for instance, in the Section 2)$${\mathsf{\lambda}}_{1}=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\frac{\left|\right|u|{|}_{1,p}^{p}}{\left|\right|i(u)|{|}_{0,p}^{p}}:u\in {W}_{0}^{1,p}(\mathsf{\Omega})\backslash \{0\}\right\}(\mathrm{the}\mathrm{Rayleigh}\text{-}\mathrm{Ritz}\mathrm{quotient}).$$

**Proposition 14.**

**R**,

- Problem (∗∗). Let ${\mathsf{\lambda}}_{1}$ be the first eigenvalue of –${\Delta}_{\mathrm{p}}^{\mathrm{s}}$ in ${W}_{0}^{1,p}$(Ω) with a homogeneous boundary condition. We have (see in Section 2)$${\mathsf{\lambda}}_{1}=\mathrm{inf}\left\{\frac{\u2759u{\u2759}_{1,p}^{p}}{\left|\right|i(u)|{|}_{0,p}^{p}}:u\in {W}_{0}^{1,p}(\mathsf{\Omega})\backslash \{0\}\right\}(\mathrm{the}\mathrm{Rayleigh}\text{-}\mathrm{Ritz}\mathrm{quotient}).$$

**Proposition 15.**

**R**,

#### 6.2. Applications

**I.**As the first application, we propose characterizing the solution of the Dirichlet problem which models the compression molding of polymers. This means to study a generalized Helle-Show flow of a power-law fluid which leads to the p-Poisson equation for the instantaneous pressure in the fluid. Therefore, this pressure u is the solution of:

**II.**We can also prove the existence and the uniqueness of the solution of the pseudo torsion problem:

## 7. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Su, Y.; Feng, Z. Lions-type theorem of the p-Laplacian with applications. Adv. Nonlinear Anal.
**2021**, 10, 1178–1200. [Google Scholar] [CrossRef] - Kaushik, B.; Garain, P. Weighted anisotropic Sobolev inequality with extremal and associated singular problems. Differ. Integral Equ.
**2023**, 36, 59–92. [Google Scholar] [CrossRef] - Ávarez-Caudevilla, P. Asymptotic behavior of cooperative systems involving p-Laplacian operators. Electron. J. Differ. Equ.
**2022**, 2022, 1–23. [Google Scholar] - Palencia, J.L.D.; Otero, A. Oscillatory solutions and smoothing of a high-order p-Laplacian operator. AIMS Electron. Res. Arch.
**2022**, 30, 3527–3547. [Google Scholar] [CrossRef] - Dong, X.; Bai, Z.; Zhang, S. Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Probl.
**2017**, 2017, 5. [Google Scholar] [CrossRef] [Green Version] - Sankar, R.R.; Sreedhar, N.; Prasad, K.R. Existence of positive solutions for 3n
^{th}order boundary value problems involving p-Laplacian. Creat. Math. Inform.**2022**, 31, 101–108. [Google Scholar] [CrossRef] - Wei, L.; Ma, R. Global continuum and multiple positive solutions to one-dimensional p-Laplacian boundary value problem. Adv. Differ. Equ.
**2020**, 2020, 204. [Google Scholar] [CrossRef] - Nhang, L.C.; Truong, L.X. Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl.
**2017**, 73, 2076–2091. [Google Scholar] - He, Y.; Gao, H.; Wang, H. Blow-up and decay of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity. Comput. Math. Appl.
**2018**, 75, 459–469. [Google Scholar] [CrossRef] - Jayachandran, S.; Soundararajan, G. p-biharmonic pseudo-parabolic equation with logarithmic nonlinearity. 3C TIC
**2022**, 11, 108–122. [Google Scholar] [CrossRef] - Chu, Y.; Wu, Y.; Cheng, L. Blow up and decay for a class of p-Laplacian hyperbolic equation with logarithmic nonlinearity. Taiwan J. Math.
**2022**, 26, 741–763. [Google Scholar] [CrossRef] - Xin, Y.; Liu, H. Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities. J. Inequal. Appl.
**2018**, 2018, 259. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hou, W.; Zhang, L.; Agarwal, R.P.; Wang, G. Radial symmetry for a generalized nonlinear fractional p-Laplacian problem. Nonlinear Anal. Model. Control
**2021**, 26, 349–362. [Google Scholar] [CrossRef] - Wu, L.; Niu, P. Symmetry and nonexistence of positive solution to fractional p-Laplacian equations. D.C.D.S.
**2018**, 39, 1573–1583. [Google Scholar] [CrossRef] [Green Version] - Asso, O.; Cuesta, M.; Doumaté, J.T.; Leadi, L. Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights. Electron. J. Differ. Equ.
**2023**, 2023, 1–29. [Google Scholar] [CrossRef] - Iannizzotto, A.; Mosconi, S.; Squassina, M. Sobolev versus Hölder minimizers for the degenerate fractional p-Laplacian. Nonlinear Anal.
**2020**, 191, 111635. [Google Scholar] [CrossRef] [Green Version] - Dwivedi, G. Generalized Picone identity for the Finsler p-Laplacian and its applications. Ukr. Math. J.
**2022**, 73, 1674–1685. [Google Scholar] [CrossRef] - Feng, T.; Yu, M. Nonlinear Picone identities to pseudo p-Lapalce operator and applications. Bull. Iranian Math. Soc.
**2017**, 43, 2517–2530. [Google Scholar] - Goel, D.; Kumar, D.; Sreenadh, K. Regularity and multiplicity results for fractional (p, q)-Laplacian equations. Commun. Contemp. Math.
**2020**, 22, 1950065. [Google Scholar] [CrossRef] [Green Version] - Meghea, I. Solutions for some mathematical physics problems issued from modeling real phenomena: Part 1. Axioms
**2023**, 12, 532. [Google Scholar] [CrossRef] - Meghea, I. Application of a Variant of Mountain Pass Theorem in Modeling Real Phenomena. Mathematics
**2022**, 10, 3476. [Google Scholar] [CrossRef] - Meghea, I. Applications of a perturbed variational principle via p-Laplacian. U.P.B. Sci. Bull. Ser. A
**2022**, 84, 141–152. [Google Scholar] - Chiappinelli, R.; Edmunds, D. Remarks on Surjectivity of Gradient Operators. Mathematics
**2020**, 8, 1538. [Google Scholar] [CrossRef] - Lliboutry, L. Traité de Glaciologie; (I); Masson & Cie: Paris, France, 1964. [Google Scholar]
- Lliboutry, L. Traité de Glaciologie; (II); Masson & Cie: Paris, France, 1965. [Google Scholar]
- Pélissier, M.C. Sur Quelques Problèmes non Linéaires en Glaciologie; Publications Mathèmatiques d’Orsay, no. 110, U.E.R. Mathématique; Université Paris IX: Paris, France, 1975. [Google Scholar]
- Lindquist, P. Stability for the solutions of div(|∇u|
^{p−2}∇u) = f with varying p. J. Math. Anal. Appl.**1987**, 127, 93–102. [Google Scholar] [CrossRef] [Green Version] - de Diego, G.; Farrell, P.; Hewitt, I. On the finite element approximation of a semicoercive stokes variational inequality arising in glaciology. SIAM J. Numer. Anal.
**2023**, 61, 1–25. [Google Scholar] [CrossRef] - Cuccu, F.; Emamizadeh, B.; Porru, G. Nonlinear elastic membranes involving the p-Laplacian operator. Electron. J. Diff. Equ.
**2006**, 2006, 1–10. [Google Scholar] - Silva, M.A.J. On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type. IMA J. Appl. Math. Adv. Access
**2012**, 78, 1130–1146. [Google Scholar] - Cuccu, F.; Emamizadeh, B.; Porru, G. Optimization or the best eigenvalue in problems involving the p-Laplacian. Proc. Am. Math. Soc.
**2009**, 137, 1677–1687. [Google Scholar] [CrossRef] [Green Version] - Nair, V.; Sharma, I. Equilibria of liquid drops on pre-stretched nonlinear elastic membranes through a variational approach. Phys. Fluids
**2023**, 35, 047111. [Google Scholar] [CrossRef] - Yousfi, Y.; Hadi, I.; Benbrik, A. Optimal Form for Compliance of Membrane Boundary Shift in Nonlinear Case. Int. J. Math. Math. Sci.
**2018**, 2018, 1689269. [Google Scholar] [CrossRef] - Zhu, L. Complete quenching phenomenon for a parabolic p-Laplacian equation with a weighted absorption. J. Inequal. Appl.
**2018**, 2018, 248. [Google Scholar] [CrossRef] [Green Version] - Merah, A.; Mesloub, F. Elastic Membrane Equation with Dynamic Boundary Conditions and Infinite Memory. Bol. Soc. Parana. Matemática
**2022**, 40, 1–15. [Google Scholar] [CrossRef] - Kawohl, B. A family of torsional creep problems. J. Reine Angew. Math.
**1990**, 410, 1–22. [Google Scholar] - Della Pietra, F.; di Blasio, G.; Gavitone, N. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle. Adv. Nonlinear Anal.
**2020**, 9, 278–291. [Google Scholar] [CrossRef] - Iannizzotto, A.; Mosconi, S.; Squassina, M. Fine boundary regularity for the degenerate fractional p-Laplacian. J. Funct. Anal.
**2020**, 279, 108659. [Google Scholar] [CrossRef] - Mihăilescu, M.; Pérez-Llanos, M. Inhomogeneous torsional creep problems in anisotropic Orlicz Sobolev spaces. J. Math. Phys.
**2018**, 59, 071513. [Google Scholar] [CrossRef] - Belloni, M.; Kawohl, B. The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞. ESAIM Control Optim. Calc. Var.
**2004**, 10, 28–52. [Google Scholar] [CrossRef] [Green Version] - Balogh, A.; Varlamov, V. Analysis of Nonlinear Elastic Membrane Oscillations by Eigenfunction Expansion. WSEAS Trans. Syst.
**2004**, 4, 1430–1435. [Google Scholar] - Wang, J.R.; Zhou, Y.; Fečkan, M. Alternative Results and Robustness for Fractional Evolution Equations with Periodic Boundary Conditions. Electron. J. Qual. Theory Differ. Equ.
**2011**, 97, 1–15. [Google Scholar] [CrossRef] - Rak, J.; Tucek, J. Solving magnetic induction heating problem with multidimensional Fredholm integral equation methods: Alternative approach for optimization and evaluation of the process performance. AIP Adv.
**2022**, 12, 105110. [Google Scholar] [CrossRef] - Cholewa, J.; Rodriguez-Bernal, A. Self-similarity in homogeneous stationary and evolution problems. J. Evol. Equ.
**2023**, 23, 42. [Google Scholar] [CrossRef] - Jiao, F.; Zhou, Y. Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl.
**2011**, 62, 1181–1199. [Google Scholar] - Atangana, A. Mathematical model of survival of fractional calculus, critics and their impact: How singular is our world? Adv. Differ. Equ.
**2021**, 2021, 403. [Google Scholar] [CrossRef] - Lanchon-Ducauquois, H.; Tulita, C.; Meuris, C. Modélisation du Transfert Thermique Dans l’He II; Congrès Français du Thermique: Lyon, France, 2000. [Google Scholar]
- Aronsson, G. On p-hrmonic functions, convex duality and an asymptotic formula for injection mould filing. Eur. J. Appl. Math.
**1996**, 7, 417–437. [Google Scholar] [CrossRef] - Emamizadeh, B.; Liu, Y. Constrained and unconstrained rearrangement minimization problems related to p-Laplace operator. Isr. J. Mat.
**2014**, 206, 281–298. [Google Scholar] [CrossRef] - Takahashi, K. Mean-field theory of turbulence from the variational principle and its application to the rotation of a thin fluid disk. Prog. Theor. Exp. Phys.
**2017**, 2017, 083J01. [Google Scholar] [CrossRef] [Green Version] - Lee, C.; Folgar, F.; Tucker, C.L. Simulation of compression molding for fiber-reinforced thermosetting polymers. Trans. ASME
**1984**, 106, 114–125. [Google Scholar] [CrossRef] - Bergwall, A. A geometric evolution problem. Q. Appl. Math.
**2002**, LX, 37–73. [Google Scholar] [CrossRef] [Green Version] - Janfalk, U. On a minimization problem for vector fields in L
^{1}. Bull. Lond. Math. Soc.**1996**, 28, 165–176. [Google Scholar] [CrossRef]

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Meghea, I.
Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2. *Axioms* **2023**, *12*, 726.
https://doi.org/10.3390/axioms12080726

**AMA Style**

Meghea I.
Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2. *Axioms*. 2023; 12(8):726.
https://doi.org/10.3390/axioms12080726

**Chicago/Turabian Style**

Meghea, Irina.
2023. "Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2" *Axioms* 12, no. 8: 726.
https://doi.org/10.3390/axioms12080726