Homogeneous Neumann Conditions Research Articles

Dynamics for a Diffusive Epidemic Model With a Free Boundary: Spreading Speed

ABSTRACTWe study the spreading speed of a diffusive epidemic model proposed by Li et al. [Dynamics for a diffusive epidemic model with a free boundary: spreading‐vanishing dichotomy, Zeitschrift für Angewandte Mathematik und Physik 75 (2024): Article No. 202], where the Stefan boundary condition is imposed at the right boundary, and the left boundary is subject to the homogeneous Dirichlet and Neumann condition, respectively. A spreading‐vanishing dichotomy and some sharp criteria were obtained in Li et al. In this paper, when spreading happens, we not only obtain the exact spreading speed of the spreading front described by the right boundary, but derive some sharp estimates on the asymptotical behavior of solution component . Our arguments depend crucially on some detailed understandings for a corresponding semi‐wave problem and a steady‐state problem.

Soap bubbles and convex cones: optimal quantitative rigidity

We consider a class of recent rigidity results in a convex cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ \Sigma , Alexandrov’s soap bubble-type results relative to Σ \Sigma , and Heintze-Karcher’s inequality relative to Σ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ = R N \Sigma = \mathbb {R}^N . Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in ∂ Σ \partial \Sigma . We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).

Homogenization of a nonlinear reaction‐diffusion problem in a perforated domain with different size obstacles

AbstractIn this paper we consider a nonlinear reaction‐diffusion problem with low diffusion scales of order in a porous domain which periodically perforated by two different size obstacles. A nonlinear Neumann condition is prescribed on the boundaries of smaller obstacles, while a homogeneous Neumann condition is imposed on the other boundaries. Our aim is to derive the homogenized model as the small parameters tend to zero by using the method of three‐scale convergence and periodic reiterated unfolding operator. The final homogenized macroscopic model depends on three scales.

Spectral optimization for weighted anisotropic problems with Robin conditions

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains Ω⊂RN, N≥1, under Robin boundary conditions, proving the existence of two positive principal eigenvalues λ± respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of λ± with respect to the sign-changing weight, showing that the optimal eigenvalues Λ± are equal if the domain has a center of symmetry and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of Ω satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optimization problem for λ+ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in an heterogeneous habitat.

Optimal control problem stated in a locally periodic rough domain: a homogenization study

We study the asymptotic behaviour of a linear optimal control problem posed on a locally periodic rapidly oscillating domain. We consider an L 2 -cost functional constrained by a Poisson problem having a mixed boundary condition: we assume a homogeneous Neumann condition on the oscillating part of the boundary and a homogeneous Dirichlet condition on the remaining part.

Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment

Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived.

An alternative expression of the time domain free surface Green function for deep water

An alternative expression of the time domain free surface Green’s function is proposed. Its source and image source contribution satisfies the homogeneous Neumann condition on the mean free surface, therefore it can be useful for the boundary element method based on the double-body flow linearization. Furthermore, it is proved that it is also the solution of Clement’s 4th order Ordinary Differential Equation (Clément, 1998). The Frobenius method with time step movement proposed by Chuang et al. (2007) is adopted for accurate and efficient evaluation.

Second-Kind Equilibrium States of the Kuramoto–Sivashinsky Equation with Homogeneous Neumann Boundary Conditions

In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated.

Eigenvalues of the Laplacian with moving mixed boundary conditions: The case of disappearing Neumann region

We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren-type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.

A dynamically consistent computational method to solve numerically a mathematical model of polio propagation with spatial diffusion

Background and objective: In this work, a mathematical model based on differential equations is proposed to describe the propagation of polio in a human population. The motivating system is a compartmental nonlinear model which is based on the use of ordinary differential equations and four compartments, namely, susceptible, exposed, infected and vaccinated individuals. Methods: In this manuscript, the mathematical model is extended in order to account for spatial diffusion in one dimension. Nonnegative initial conditions are used, and we impose homogeneous Neumann conditions at the boundary. We determine analytically the disease-free and the endemic equilibria of the system along with the basic reproductive number. Results: We establish thoroughly the nonnegativity and the boundedness of the solutions of this problem, and the stability analysis of the equilibrium solutions is carried out rigorously. In order to confirm the validity of these results, we propose an implicit and linear finite-difference method to approximate the solutions of the continuous model. Conclusions: The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and that it is capable of preserving unconditionally the positivity of the approximations. For illustration purposes, we provide some computer simulations that confirm some theoretical results derived in the present manuscript.

The Meyers Estimates for Domains Perforated along the Boundary

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

Local asymptotics for nonlocal convective Cahn-Hilliard equations with W1,1 kernel and singular potential

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W1,1 convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.

An overdetermined problem associated to the Finsler Laplacian

<p style="text-indent:20px;">We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

&lt;p style='text-indent:20px;'&gt;We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.&lt;/p&gt;

Nonlinear wave transmission in harmonically driven hamiltonian sine-Gordon regimes with memory effects

Abstract In this work, for the first time in the literature, we provide solid computational evidence that the phenomenon of nonlinear supratransmission is present in time-fractional sine-Gordon equations with damping. The model under study considers a Caputo-type partial derivative of order α ∈ ( 1 , 2 ] with respect to the temporal variable, and it possesses an associated continuous energy density function which is dissipated throughout time. In the present work, we determine the dispersion relation of the physical model and employ a reliable numerical technique to approximate the solutions. Associated with the finite-difference scheme, we also propose a discrete energy density function. This methodology was implemented computationally to approximate the solution of the model on a closed and bounded interval, considering sinusoidal perturbations on one end, and homogeneous Neumann conditions on the other. As the most relevant physical conclusions of our experiments, we have confirmed the presence of nonlinear supratransmission in nonlinear wave equations with time-fractional derivatives. Additionally, we have found out that nonlinear supratransmission gradually ceases to be present in the system as α tends from 2 to 1. Moreover, we provide some bifurcation diagrams depicting the relation between the driving amplitude at which supratransmission is triggered versus driving frequency. To that end, we follow a systematic approach which is relatively standard nowadays. For the sake of convenience, we provide a Matlab implementation of our code as an appendix at the end of this manuscript.

Isomorphic Well-Posedness of the Final Value Problem for the Heat Equation with the Homogeneous Neumann Condition

This paper concerns the final value problem for the heat equation subjected to the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the same problem was proven well-posed in the original sense of Hadamard under an additional assumption of Holder continuity of the source term. Like for its predecessor, the point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax–Milgram operators, now yielding isomorphic well-posedness for such problems. Hereby the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, resulting in a non-local compatibility condition on the data. As a novelty, a stronger version of the compatibility condition is introduced with the purpose of characterising the data that yield solutions having the regularity property of being square integrable in the generator’s graph norm (instead of in the form domain norm). This result allows a direct application to the class 2 boundary condition in the considered inverse Neumann heat problem.

An immersed boundary method for geometrical shock dynamics

A new algorithm is presented for the resolution of the Geometrical Shock Dynamics model in presence of obstacles in an Eulerian framework. The numerical method relies on a Fast-Marching like algorithm on a Cartesian grid, which allows dealing with complex geometrical configurations and shock waves interactions at a reduced computational cost. The application of an homogeneous Neumann condition at the border of rigid obstacles, not aligned with the mesh, is based on the Immersed Boundary Method. For a given obstacle, a set of ghost points is defined in the solid area and the corresponding unknown flow parameters are updated using a compatibility condition. A good agreement is observed between numerical results, experimental data and CFD simulations, both in 2D and 3D, which demonstrates the validity and the capabilities of this method.

A remark on nonlocal Neumann conditions for the fractional Laplacian

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.