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We study renormalized solutions to the problem $$\begin{align*} \begin{cases} -\Delta_pu=0 & \mbox{ in } \left\lbrace x \in \mathbb{R}^N : \left\lvert {x} \right\rvert>1\right\rbrace \\ \left\lvert {\nabla u} \right\rvert^{p-2}u_\nu + g(u) = \mu & \mbox{ on } \left\lbrace x \in \mathbb{R}^N : \left\lvert {x} \right\rvert=1\right\rbrace \end{cases} \end{align*} $$ { − Δ p u = 0 in { x ∈ R N : | x | > 1 } | ∇ u | p − 2 u ν + g (u) = μ on { x ∈ R N : | x | = 1 } where 1

This work, aims to study and examine the description of the gradient reduced order-strategic sensors of type boundary exponential (γΘ* EGRO-strategic sensors) for completion gradient reduced order-detectability of type boundary exponential (γΘ* EGRO-detectability). Thus, this concept is linked to an estimator in distributed parameter systems (DPSS) in Neumann problem. So, we present numerous consequences regarding to diverse kinds of information, region Θ* and conditions of boundary region to allow existence of γΘ* EGRO-detectable systems. In addition, we have estimated at the junction interface that the interior solution is harmonized with the exterior solution for γΘ* EGRO-detectable and, we give the relationship between this concept and sensors structures. Finally, we demonstrate some applications with many circumstances of sensor positions. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

In this document we will study and solve the nonlinear partial differential equation, with initial conditions for vehicle entry that serves to model the dynamics of traffic flow. To find a numerical solution to the dynamics that govern the behavior of traffic flow, the Finite Element Method in a spatial dimension was used. In accordance with the temporal dynamics, simulations were developed to know the flow in terms of time. The numerical solution is interesting for predicting the number of vehicles at the entrance to a high-flow road. Some theorems are enunciated that guarantee the existence of the solution and the uniqueness is given by the boundary conditions. [ABSTRACT FROM AUTHOR]

Several important problems in Mechanics can be efficiently solved using Raviart–Thomas mixed finite element methods. Whenever the domain of interest has a curved boundary the methods of this family for N-simplexes are the natural choice. But in this case the question arises on the best way to prescribe normal flux conditions across the boundary, if any. It is generally acknowledged that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polytope in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in Bertrand and Starke (2016). In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this work an alternative with straight-edged triangles for two-dimensional problems is examined. The key feature of this approach is a Petrov–Galerkin formulation, in which the test-flux space is a little different from the shape-flux space. Based on previous author’s experience with this technique, as applied to Lagrange finite elements, it would lead to an overall accuracy improvement here as well. The experimentation reported hereafter provides examples and counterexamples confirming or not such an expectation, depending on the unknown field of the mixed problem at hand.

This manuscript explores the flow features of the Modified Cross Model in a channel with symmetric trapezoidal cavities in the presence of a circular obstacle. The non-dimensional governing equations and model for different parameters are evaluated via a Galerkin Finite Element Method The system of non-linear algebraic equations is computed by adopting the Newton method. A space involving the quadratic polynomials (P2) has been selected to compute for the velocity profile while the pressure profile is approximated by a linear (P1) finite element space of functions. Simulations are performed for a wide range of physical parameters such as modified parameter (from 0.0 to 0.5), power-law index (from 0.5 to 1.5), relaxation parameter (from 1 to 3), and Reynolds number (from 10 to 40). For the case of a modified parameter (b) and relaxation parameter (λ), it is observed that the drag coefficient (CD) shows an increasing trend while the lift coefficient (CL) is changing sign at lower values of (λ), and then becomes positive at λ=3.

In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0.

In this article, we study the well-posedness of the Master Equation of Mean Field Games in a framework of Neumann boundary condition. The definition of solution is closely related to the classical one of the Mean Field Games system, but the boundary condition here leads to two Neumann conditions in the Master Equation formulation, for both space and measure. The global regularity of the linearized system, which is crucial in order to prove the existence of solutions, is obtained with a deep study of the boundary conditions and the global regularity at the boundary of a suitable class of parabolic equations. [ABSTRACT FROM AUTHOR]

This work reviews some basic features on the second (first nontrivial) eigenvalue λ2 to the Neumann problem ... where Ω is a bounded Lipschitz domain of RN, is the outer unit normal, and ∆pu = div(∇) is the p-Laplacian operator. We are mainly concerned with the variational characterization of &#955;2 and place emphasis on the range 1 < p < 2, where the nonlinearity |∆| becomes non smooth. We also address the corresponding result for the p-Laplacian in graphs. [ABSTRACT FROM AUTHOR]

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

From a unique viewpoint, singular elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms are investigated in cylindrical domains. The weight Neumann condition is set on the lateral area of the cylinder; no condition is set on the base of the cylinder (regardless the type of the equation). Results of two kinds are established: the existence of a limit of each solution (if it exists) along the axis of the cylinder and sufficient conditions of a blow-up (including instant or complete one). [ABSTRACT FROM AUTHOR]

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

We study the dynamics of a Neumann boundary value problem arising in fluid dynamics. We prove the nonexistence, existence and uniqueness of positive solutions under suitable conditions. At the same time, under stricter conditions, we also obtain the dynamic properties of the Neumann boundary value problem, such as the stability and instability of positive solutions. The methods of proof mainly involve the upper and lower solutions method, eigenvalue theory and some analysis techniques. [ABSTRACT FROM AUTHOR]

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with C 2 boundary, i.e. , positive solutions u to Δ u + f (u) = 0 in Ω ⊂ (M 2 , g) so that u = 0 and ∂ u ∂ η → = c t e along ∂Ω, η → the unit outward normal along ∂Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem. When (M 2 , g) is the standard sphere S 2 and f is a C 1 function so that f (x) > 0 and f (x) ≥ x f ′ (x) for any x ∈ R + ⁎ , we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in S 2 for this choice of f. More precisely, this shows that if u is a positive solution to Δ u + f (u) = 0 on a topological disk Ω ⊂ S 2 with C 2 boundary so that u = 0 and ∂ u ∂ η → = c t e along ∂Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33,35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28] , also called Serrin Problem) in S 2. [ABSTRACT FROM AUTHOR]

We investigate the geometry and topology of extremal domains in a Hadamard manifold, i.e., domains that support a positive solution to an overdetermined elliptic problem (OEP). First, we study narrow properties of such domains and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity and how the domain behaves at infinity. This shows interesting relations with the Singular Yamabe Problem. Later, we focus on extremal domains in the Hyperbolic Space. Symmetry and boundedness properties will be shown. In a certain sense, we extend Levitt–Rosenberg's Theorem to OEPs, which suggests a strong relation with constant mean curvature hypersurfaces. In particular, we are able to prove the Berestycki–Caffarelli–Nirenberg Conjecture under certain assumptions either on the boundary at infinity of the extremal domain or on the OEP itself. Also a height estimate for solutions on extremal domains in a Hyperbolic Space will be given. [ABSTRACT FROM AUTHOR]

Using boundary functions method in combination with the method of sewing connection proved the existence of a solution and constructed its asymptotic expansion for 2nd order quasi-linear system with Neumann boundary conditions and discontinuous term. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method based on Adomian decomposition method is applied for solving Burgers’ equation with Dirichlet and Neumann boundary conditions. This mathematical-physical model describes the propagation of waves in nonlinear optical media. Finally, some illustrative examples are presented to show the efficiency of the proposed method and the results obtained by this method are compared by the exact solution. [ABSTRACT FROM AUTHOR]

In this article, a time-optimal control problem for linear infinite order distributed parabolic systems involving multiple time-varying lags appearing both in the state equation and in the boundary condition is presented. Some particular properties of the optimal control are discussed. [ABSTRACT FROM AUTHOR]

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used. [ABSTRACT FROM PUBLISHER]

We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new point in the domain is chosen randomly by following a bouncing process with the same distribution as the original one. By a suitable definition outside the niche, the density of the population ends up solving a mixed local/nonlocal equation, in which the dispersion is given by the superposition of the classical and the fractional Laplacian. This density function satisfies two types of Neumann conditions, namely the classical Neumann condition on the boundary of the niche, and a nonlocal Neumann condition in the exterior of the niche. • An ecological model describing a biological niche is presented. • The egress and return to the niche of a biological population is described. • The combination of Lévy and Poisson dispersal strategies is confronted with optimal foraging hypotheses. • The mathematical formulation of the model is obtained in light of fractional calculus and probability. • The return to the niche is modeled by a new type of mixed Neumann condition. [ABSTRACT FROM AUTHOR]