Mixed Boundary Value Conditions Research Articles

Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions

<p>In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u & = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u& = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu & = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $\end{document} </tex-math></disp-formula></p><p>First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.</p>

Dynamic response and liquefaction potential of porous seabed induced by partial standing ocean waves

The analysis of ocean wave-induced dynamic response of a porous seabed is particularly important for coastal and geotechnical engineers when designing and constructing maritime structures. In this study, an analytical solution is presented to analyze the dynamic response and liquefaction potential of a poro-elastic seabed induced by partial standing waves with arbitrary reflectivity. The porous seabed is modeled using Biot’s theory describing the propagation of elastic waves, and coupled deformation and water flow of porous media, whereas the ocean waves are described using linear ocean wave theory. Based on the mixed boundary-value conditions, explicit expressions of displacements, effective stresses and excess pore water pressure of seabed are derived with consideration of the effects of inertial forces, compressibility of solid and fluid, and arbitrary reflectivity of standing waves. The results of degenerated analytical solutions are compared with the existing ones to verify the correctness of the proposed method. The effects of several pertinent parameters of ocean wave-seabed system, including reflection coefficient, phase lag and period of standing waves, depth of water, permeability, degree of saturation, and shear modulus of seabed deposits, etc., on the dynamic response of seabed and liquefaction potential, are examined and discussed. It is found that the reflection of standing wave has a significant effect on the dynamic response and liquefaction potential of porous elastic seabed. Compared with that of no wave reflection, the liquefaction depth of seabed induced by fully-reflected standing waves increases 82.49% under certain conditions of wave-seabed system. In addition, phase lag, wave period, water depth and mechanical and physical properties of seabed soil such as saturation, permeability and shear modulus have different effects on the dynamic response and liquefaction potential of porous elastic seabed. The investigation of the dynamic response and liquefaction of the porous elastic seabed under partial standing ocean waves will help to better predict the influence of standing waves on breakwaters and seabed soil, and can provide some guidance for the design of offshore structures.

Approach for a floating pile embedded in poroelasitc soil under horizontal dynamic load

The deformation mechanism in the horizontal direction of a floating pile under dynamic load should be paid extensive attention. The closed-form solution of dynamic characteristics of a floating pile in poroelasitc soil subject to horizontal dynamic load is obtained by a newly developed framework in this paper. Firstly, based on the three-dimensional continuum model and Biot’s poroelastic theory, the governing equations of the pile surrounding soil and pile end soil are respectively established. Secondly, the pile shaft reaction is calculated by use of the conventional method used in many existing literatures, and the pile base resistance is obtained with the aid of the Hankel transform technique and mixed boundary-value conditions that came from the contact interaction between a rigid massless disk with its beneath soil layer of finite thickness. Thirdly, the Timoshenko beam theory is applied to depict the pile vibration considering its shear deformation, and thus the dynamic impedance functions at pile head are derived. Subsequently, the correctness of the proposed solution is verified through comparisons with the existing results from other scholars. Finally, a parameter analysis about the thickness of the pile end soil, pile slenderness ratio and soil permeability are performed. Results suggest that the soil layer thickness and pile slenderness ratio have a pronounced effect on the pile dynamic behavior, and considering the soil permeability is more realistic to pile vibration response. The obtained results can exert great significance to the serviceability of floating pile for engineering practice.

Existence theory and stability analysis to a coupled nonlinear fractional mixed boundary value problem

In this manuscript, we conclude a comprehensive approach to a class of nonlinear coupled system of fractional differential equations with mixed type boundary value conditions. Subsequently, the solution of coupled system exists and unique under mixed type boundary value conditions with the reference of Schaefer and Banach fixed-point theorems. Further, we developed the Hyers- Ulam stability for the considered problem. Finally, we set an example for the support of our results.

Existence of solutions for resonant double phase problems with mixed boundary value conditions

We study a double phase problem with mixed boundary value conditions with reaction terms that resonate at the first eigenvalue of the related eigenvalue problem. Based on the maximum principle and homological local linking, we are going to prove the existence of at least two bounded nontrivial solutions for this problem.

Double phase obstacle problems with multivalued convection and mixed boundary value conditions

<p style='text-indent:20px;'>In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by <inline-formula><tex-math id="M1">\begin{document}$ \mathcal S $\end{document}</tex-math></inline-formula> the solution set of the mixed boundary value problem and by <inline-formula><tex-math id="M2">\begin{document}$ \mathcal S_n $\end{document}</tex-math></inline-formula> the solution sets of the approximating problems, we establish the following convergence relation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ s $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M6">\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}</tex-math></inline-formula> stand for the weak and the strong Kuratowski upper limit of <inline-formula><tex-math id="M7">\begin{document}$ \mathcal S_n $\end{document}</tex-math></inline-formula>, respectively.</p>

A chemotaxis‐Stokes system with nonhomogeneous boundary condition: Global large‐datum solution and asymptotic behavior

AbstractIn this paper, we study the global solvability and stabilization to a chemotaxis‐Stokes model with porous medium diffusion and mixed nonhomogeneous boundary value conditions in three‐dimensional space. When m is slightly bigger than 1, we can get a solution with strong regularity, but when m is close to 1, the regularity of the solution becomes weak. Specifically, our results are divided into two cases: (i) and (ii) . For case (i), we obtain a global bounded weak existence with good regularity for any initial datum, and for decay incoming oxygen, we also prove that the bounded solution will converge to a constant steady state. But for case (ii), it is hard to obtain the boundedness of solutions, and a global “very” weak solution is obtained.

Coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic and layered poroelastic half‐space

The coupled horizontal and rocking vibrations of a rigid circular disc resting on a transversely isotropic (TI) and layered poroelastic half-space is solved by a newly developed semi-analytical method for the first time. The poroelastic medium is modelled via Biot's poroelastodynamic theory. The Green's function corresponding to the horizontal uniform and vertical linearly changing patch loads on the surface of the layered half-space are first derived by virtue of the cylindrical system of vector functions and dual variable and position method. To solve the dynamic interaction, the influence function by the ring load is then introduced and calculated via the method of superposition. Together with the mixed boundary-value conditions, the dynamic horizontal, rocking, and coupling compliances are finally solved by a novel discretization scheme and an integral least-square approach. The developed fundamental solutions are verified by comparing with existing solutions in the purely elastic and poroelastic media. Numerical examples are presented to investigate the influence of poroelastic properties on the coupled vibration characteristics, which could be useful to the dynamic design of foundation on the saturated soil. It should be emphasized that the effect of the coupling compliance must be considered since its contribution can be as large as 26%, compared to the diagonal compliances.

An efficient finite element iterative method for solving a nonuniform size modified Poisson-Boltzmann ion channel model

In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution of Model 1 collecting all the singular points and Models 2 and 3 much easier to solve numerically than the original nuSMPBIC model. A damped two-block iterative method is then presented to solve Model 3, along with a novel modified Newton iterative scheme for solving each related nonlinear algebraic system. To this end, an effective nuSMPBIC finite element solver is derived and then implemented as a program package that works for an ion channel protein with a three-dimensional molecular structure and a mixture of multiple ionic species. Numerical results for a voltage-dependent anion channel (VDAC) in a mixture of four ionic species demonstrate a fast convergence rate of the damped two-block iterative method, the high performance of the software package, and the importance of considering nonuniform ion sizes. Moreover, the nuSMPBIC model is validated by the anion selectivity property of VDAC.

Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.

Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

Vertical-rocking-horizontal vibrations of a rigid disk resting on multi-layered soils with groundwater level

The vertical, rocking and horizontal vibrations of a rigid disk on multi-layered soils are studied by using a coupled approach of stiffness matrix method and “integral equation method”. Considering groundwater level, the layered soils consist of a number of dry layers and saturated layers, and a bedrock at the bottom. To allow the assembly of a global equation of this layered system, the stiffness matrices of the top and bottom saturated layers are modified to match those of adjacent dry layer and bedrock using drainage conditions. Together with the mixed boundary-value conditions at the disk plane, three disk-soil interaction problem, i.e., vertical, rocking and horizontal vibration cases, can be expressed as corresponding dual integral equations for solving. The correctness of the proposed method is verified by existing solutions with good convergence. Parametric studies illustrate the effects of groundwater level, permeability of saturated soil and the hydraulic condition at dry-saturated soil interface on dynamic compliance coefficients of rigid risk and pore pressure in saturated soil layer.

On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators

We investigate the existence of solutions for a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions. In this way, we use a generalization of the hybrid Dhage’s fixed point result for sum of three fractional operators. Finally, we give an example to illustrate our main result.

Existence of solutions to impulsive fractional differential equations with mixed boundary value conditions

Existence of solutions to impulsive fractional differential equations with mixed boundary value conditions

RADIAL SOLUTION OF ASYMPTOTICALLY LINEAR ELLIPTIC EQUATION WITH MIXED BOUNDARY VALUE IN ANNULAR DOMAIN

In this paper, we study nonlinear elliptic equation with mixed boundary value condition in annular domain. It is assumed that the nonlinearity is asymptotically linear and depends on the derivative term. Some results on the existence of solution are established by nonlinear analysis methods.

Periodic pattern formation in the coupled chemotaxis-(Navier–)Stokes system with mixed nonhomogeneous boundary conditions

AbstractWe consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions

We are concerned with the existence and multiplicity of component-wise positive solutions for nonlinear system of Hammerstein integral equations with the weighted functions and the associated nonlinear eigenvalue problem. Our discussions are based on the product formula of fixed point index on product cones and the fixed point index theory. Moreover, we establish the existence and multiplicity of component-wise positive solutions for the associated nonlinear systems of second-order ordinary differential equations under the mixed boundary value conditions.

Liouville Results for m-Laplace Equations in Half-Space and Strips with Mixed Boundary Value Conditions and Finite Morse Index

We consider sign-changing solutions of the equation \(-\Delta _m u= |u|^{p-1}u\) where \(m\ge 2\) and \(p>1\) in half-space and strips with nonlinear mixed boundary value conditions. We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.

Stresses around a moving Griffith crack at an interface between a nonhomogeneous bonding layer and two dissimilar orthotropic half-spaces

In this study, stresses were estimated for a moving interface crack located on the lower interface between a nonhomogeneous elastic layer and a lower orthotropic elastic half-space while the upper interface between the layer and the upper orthotropic elastic half-space was not weakened by any cracks. It was assumed that the moving crack was open at one end and closed at the other and moved at a constant speed. The material properties of the bonding layer were assumed to vary continuously from the lower half-space to the upper half-space. A self-equilibrated system of pressure was applied to the crack surfaces. The mixed boundary value conditions with respect to the crack were reduced to dual integral equations. To solve the equations, the differences in the crack surface displacements were expanded to a series of functions that equal zero outside the crack. The unknown coefficients in the series were solved using the Schmidt method. The stress intensity factors were calculated numerically for the case in which the lower and upper half-spaces are made of unidirectional glass fiber-reinforced epoxy composites.