Abstract This paper establishes lower and upper bounds for the radial eigenvalues of nonlinear elliptic systems defined in appropriate annular domains of $$\mathbb {R}^N$$ R N . For the problem, we will prove a result in the sense of a conjecture proposed by Nápoli and Pinasco in the paper Estimates for eigenvalues of quasilinear elliptic systems , J. Diff. Eq. 227 (2006), 102–115, when we are in certain domains of $$\mathbb {R}^N$$ R N . Moreover, we obtain a hyperbolic type function defining a region that contains all the generalized eigenvalues (whether variational or not), and with general hypotheses, the proof is carried out via ABP estimate for the p-Laplacian operator.

Clustering type solutions for critical elliptic system in dimension two

The article examines a block cryptographic algorithm using a two-component shared secret key obtained according to the Diffie-Hellman key exchange principle on elliptic curve points over the field Z p . The goal is to eliminate shortcom­ings of individual classical algorithms and, through their combination, increase overall system strength. Key generation and exchange between users are carried out using elliptic curve cryptographic systems with public key. Two methods are proposed for generating shared secret keys for interacting users: applying the Diffie-Hellman cryptographic protocol on multiple elliptic curve points or additionally using a recurrence formula. Encryption elements are represented by blocks as square matrices constructed on elliptic curve point coordinates. Encryption proceeds in two stages: the first uses stream cipher with scalar multiplication of elliptic curve points, and the second involves forming matrix blocks and performing Hill matrix transformation with feedback. Each encryption stage utilizes its corresponding component of the users’ shared secret key: a numerical gamma sequence or a square key matrix. The cryptographic strength is based on the computational complexity of solving the discrete logarithm problem on elliptic curves and the security of the sharing service with secure authentication of interacting users. The block implementation of the second encryption stage ensures the system’s resistance to frequency analysis. As an illustration of the presented algorithm’s operation, the article provides a step-by-step example of encrypting/decrypting a text message.

Elliptical gears are widely used in the fields of aerospace and automation equipment due to their excellent variable speed characteristics and compact structure. To reveal the control mechanism of phase angle on the motion characteristics of two-stage elliptical gear transmission system (TSEGTS) and improve its stability under variable speed loading, this paper proposes a modeling method that integrates two-dimensional tooth profile envelope method and image processing technology to extract tooth profile and analyze system motion characteristics. By introducing the fluctuation coefficient as a stability indicator, the system studied the effects of phase angle, input speed, and load torque on angular velocity, angular acceleration, and torque. The results indicate that the phase angle has a significant periodic modulation effect on the motion characteristics of the system. The angular velocity and acceleration of intermediate shaft B and output shaft C, as well as the torque of output shaft C, exhibit a distribution with 90° as the axis of symmetry within the range of 0°–180°. The average torque fluctuation coefficients of input shaft A and intermediate shaft B in the range of 0°–90° are 4.02 and 3.03, respectively, which are significantly higher than the 3.46 and 1.99 coefficients in the range of 90°–180°. As the input speed increases from 5π to 20π rad/s, the torque fluctuation coefficients of shafts A, B, and C increase from 2.87, 1.37, and 0.53 to 22.54, 14.19, and 8.43, respectively. When the load torque increases from 2 to 8 N·m, the fluctuation coefficients of the three axes decrease from 11.33, 7.12, and 4.22 to 3.51, 1.91, and 1.05, respectively. The increase in load torque effectively suppresses fluctuations and improves meshing stability. The experimental and simulation results are consistent, verifying the effectiveness of the model. The research results can provide theoretical basis for parameter optimization and dynamic performance control of TSEGTS.

A Poincaré–Hopf formula for functionals associated to quasilinear elliptic systems

Liouville type theorems for higher order elliptic systems

Critical and singular nonlinearities in quasilinear elliptic systems: Existence and concentration of sign-changing solutions

A system of two singular semi–linear elliptic equations, patterned after the Schrödinger-Maxwell system, is considered. If the reaction term of the first equation contains a datum $f\in L^m$, existence of positive solutions with finite energy is established for suitable ranges of m. In particular, the results from the theory of single, singular equations are improved. At the same time, thanks to an approach based on approximation schemes and a priori estimates on the approximated sequences of solutions, it is shown that the integrability assumptions on the datum produce higher integrability of the solutions.

We investigate the existence and decay properties of solutions to the following elliptic systems, which arise in the context of Bose–Einstein condensation: [Formula: see text] To analyze the decay behavior of solutions, we use a variant of Moser’s iteration technique, with general conditions imposed on the potentials [Formula: see text] and [Formula: see text]. Our results extend recent findings by Angeles, Clapp, and Saldaña (2023). We pay particular attention to potential classes that either vanish or are unbounded at infinity. The existence of ground states is established under these assumptions, and the derived decay estimates are also used to show that weak solutions exhibit either exponential or polynomial decay.

Accumulation of amyloid beta proteins is a defining feature of Alzheimer's disease, and is usually accompanied by cerebrovascular pathology. Evidence suggests that amyloid beta and cerebrovascular pathology are mutually reinforcing; in particular, amyloid beta suppresses perfusion by constricting capillaries, and hypoperfusion promotes the production of amyloid beta. Here, we propose a whole-brain model coupling amyloid beta and blood vessel through a hybrid model consisting of a reaction-diffusion system for the protein dynamics and porous-medium model of blood flow within and between vascular networks: arterial, capillary and venous. We discretize the resulting parabolic--elliptic system of PDEs by means of a high-order discontinuous Galerkin method in space and an implicit Euler scheme in time. Simulations in realistic brain geometries demonstrate the emergence of multistability, implying that a sufficiently large pathogenic protein seeds is necessary to trigger disease outbreak. Motivated by the "two-hit vascular hypothesis" of Alzheimer's disease that hypoperfusive vascular damage triggers amyloid beta pathology, we also demonstrate that localized hypoperfusion, in response to injury, can destabilize the healthy steady state and trigger brain-wide disease outbreak.

We prove the existence of local solutions of any second order quasilinear elliptic system with prescribed 1-jet and present some applications in Riemannian geometry.

This paper is concerned with the Neumann initial-boundary problem for the degenerate parabolic–elliptic–elliptic attraction-repulsion chemotaxis system $ \begin{align*} \begin{cases} u_t = \nabla\cdot(u^{m-1}\nabla u-\chi u^{p-1}\nabla v+\xi u^{q-1}\nabla w),\\ 0 = \Delta v +\alpha u-\beta v,\\ 0 = \Delta w + \gamma u-\delta w \end{cases} \end{align*} $ in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n \in \mathbb{N}) $ with smooth boundary, where $ m \ge1 $, $ p, q \ge 2 $, $ \chi, \xi, \alpha, \beta, \gamma, \delta > 0 $ are constants. This paper proves that if $ p<q $ or if $ p = q $ and $ \chi\alpha-\xi\gamma<0 $, then a global weak solution $ (u,v,w) $ exists and $ \sup\limits_{t>0}(\|u(\cdot,t)\|_{L^\infty(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|w(\cdot,t)\|_{W^{1,\infty}(\Omega)})<\infty. $ Also, by assuming further a smallness condition on the mass of initial data $ u_0 $, it is shown that$ (u(\cdot,t), v(\cdot,t), w(\cdot,t)) \to \Big(\overline{u_0}, \frac{\alpha}{\beta}\overline{u_0}, \frac{\gamma}{\delta}\overline{u_0}\Big) $weakly* in $ L^\infty(\Omega)\times W^{1,\infty}(\Omega) \times W^{1,\infty}(\Omega) $ as $ t \to \infty $, where $ \overline{u_0}: = \frac{1}{|\Omega|}\int_\Omega u_0\,{\rm d}x $.

This paper concerns quasi-periodic perturbations with small parameters of 2-dimensional degenerate systems. By the Leray-Schauder Continuation Theorem and the KAM technique, it is proved that if the equilibrium point of the unperturbed system is elliptic-type degenerate, then the system has a small response solution for many sufficiently small parameters.

The existence and location of solutions are established for an elliptic system with full gradient dependence and intrinsic operators. The abstract results are applied to a system with convolution products.

The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.

Abstract This paper is concerned with weak solutions of elliptic equations and of systems of the form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\sum _{i=1}^{n}D_i \left( A^\alpha _i\left( x,Du(x)\right) \right) = 0, \ \ & \textrm{in }\ \Omega , \ \ \forall \ \alpha \in \left\{ 1, \cdots , m\right\} ,\\ u(x)=u_*(x), & \textrm{on }\ \partial \Omega . \end{array}\right. } \end{aligned}$$ - ∑ i = 1 n D i A i α x , D u ( x ) = 0 , in Ω , ∀ α ∈ 1 , ⋯ , m , u ( x ) = u ∗ ( x ) , on ∂ Ω . We show that, assuming some high degree of integrability of $$Du_*$$ D u ∗ the gradient of the boundary datum, we can obtain bounds of the difference $$ |u^\gamma -u^\gamma _* |_\infty $$ | u γ - u ∗ γ | ∞ , $$\gamma \in \left\{ 1, \cdots , m\right\} $$ γ ∈ 1 , ⋯ , m . In particular, when $$m=1$$ m = 1 , things are easier and we can get a better result.

Abstract In this paper, we establish some new Liouville‐type theorems for nonnegative weak solutions to fractional elliptic systems with different orders. To prove our result, we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet‐to‐Neumann operator of a degenerate elliptic equation by the extension technique. Our proof is based on Alexandrov–Serrin method of moving planes based on some maximum principles that obtained by establishing some key integral inequalities.

The well-known example by A.V. Bitsadze (1948) demonstrates that, unlike a single scalar equation, there exist elliptic systems for which the Dirichlet problem is not Fredholm. In the first part of the paper, it is established that for systems of two equations, a similar situation persists for general local boundary value problems of Poincare type. In contrast, in the second part of the paper, a nonlocal boundary value problem of the Carleman type is proposed, which is well-posed (in the sense of unconditional solvability) for all second-order elliptic systems.

Synchronization and Stability Analysis of a Four-Exciter Elliptical Vibration System with Coaxial Elastic Coupling Three Vibrators

Abstract In this paper, we present an ultra-fast technique for brain tumor detection in microwave brain imaging systems based on compressive sensing (CS). To achieve this, we designed an elliptical array-based microwave imaging system by simulating sixteen elements of modified bowtie antennas in the CST medium around a multi-layer head phantom. Additionally, we designed an appropriate matching medium to radiate in the desired band from 1 to 4 GHz. The algorithm section of our technique involves pre-processing steps for calibration, a processing step to create a two-dimensional image of the received signals, and a post-processing step for CS. In the processing section, we used a confocal image-reconstructing method based on delay and sum and delay, multiply, and sum beam-forming algorithms. Finally, we applied a new CS technique that includes an L1-norm convex optimization method to reconstruct low-dimension images from the original reconstructed images. We present simulated results to validate the effectiveness of our proposed method for precisely localizing the tumor target in a human full head phantom. The simulated results demonstrate that by using our proposed CS method, the image reconstruction processing time decreased to 63% and the compressed image size reduced to 25% of the original image.