N-dimensional Domain Research Articles

Homogenization and corrector result of optimal control problem for Stokes system

This article introduces the boundary optimal control problem in an N-dimensional domain governed by the stationary Stokes equations. Controls are applied to the states through Neumann data on the boundary. The aim of this paper is to study the asymptotic behavior of optimal control and states and identify the limit optimal control problem in the framework of the two-scale convergence. Finally, strong convergence of L 2 -cost functional and a corrector result for velocities is proved.

Global and bounded solution to a quasilinear parabolic-elliptic pursuit-evasion system in N-dimensional domains

In this paper, we deal with the global existence and boundedness of solutions to a pursuit-evasion model{ut=Δu−χ∇⋅(u(u+1)r−1∇w)+u(λ1−μ1ur1−1+av),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z)+v(λ2−μ2vr2−1−bu),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0 in a bounded domain Ω⊂RN(N≥1) with smooth boundary, where r, λ1, λ2, μ1, μ2, χ, ξ, a, b are positive constants as well as ri>1(i=1,2). Under the Neumann boundary conditions, if min⁡{(r1−1)(r2−1),(r1−r)(r2−1)}>(N−2)+N, it is proved that the above system possesses a unique global-in-time and bounded classical solution for all appropriately regular nonnegative initial data. This extends the previous global boundedness result for r=1, ri=2(i=1,2) and N≤3 (see Li-Tao-Winkler [9]), as well as the result for r=1, ri>1(i=1,2) and N≥1 (see Zheng and Zhang [38]).

Stable discontinuous stationary solutions to reaction-diffusion-ODE systems

A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.

Limiting dynamics of stochastic heat equations with memory on thin domains

This paper is concerned with the limiting behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an -dimensional narrow domain. We also establish the upper-semicontinuity of these attractors when a family of -dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness.

Single-spike solutions to the 1D shadow Gierer–Meinhardt problem

A fundamental example of reaction–diffusion system exhibiting Turing type pattern formation is the Gierer–Meinhardt system, which reduces to the shadow Gierer–Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni, 1998 [1]; Wei, 1998 [2]), and therefore analytical expressions for single spike solutions are generally lacking.By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer–Meinhardt problem for any 1<p<∞, representing both inner and boundary spike solutions depending on the location of the peak. Our approach not only confirms numerical results existing in literature, but also provides guidance for tackling extensions of the shadow Gierer–Meinhardt problem based on different boundary conditions (e.g. mixed) and/or n-dimensional domains.

Neumann boundary optimal control problems governed by parabolic variational equalities

Abstract We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α &gt; 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα . We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

Boundary-value problems for singular p– and p(x)– Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p– and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

Localisation of the first eigenfunction of a convex domain

We study the first Dirichlet eigenfunction of the Laplacian in a n-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti [1,2], imply that the ratio of the L 2-norm and -norm of the eigenfunction is minimised when the domain is a ball. However, when the eccentricity of the domain is large, the eigenfunction should spread out at a certain scale and this ratio should increase. We make this precise by obtaining a lower bound on the L 2-norm of the eigenfunction and show that the eigenfunction cannot localise to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, [3], in the general n-dimensional case. The main feature of the proof is to obtain sufficiently sharp estimates on the first eigenvalue in order to estimate the first derivatives of the eigenfunction.

Regular random attractors for non-autonomous stochastic evolution equations with time-varying delays on thin domains

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction–diffusion equations with time-varying delays on thin domains. First, we prove the existence and uniqueness of the regular random attractor. Then, we prove the upper semicontinuity of the regular random attractors for the equations on a family of (n + 1)-dimensional thin domains that collapses onto an n-dimensional domain.

The bi-Helmholtz equation with Cauchy conditions: ill-posedness and regularization methods

In this paper, the bi-Helmholtz equation with Cauchy conditions is nominated in a n-dimensional strip domain. It is shown that this Cauchy problem may be ill-posed in the sense of Hadamard. In order to overcome the ill-posedness, a suitable regularization method has to be applied to the Cauchy problem. Hence, two well-known wavelet and Fourier regularization methods are used for solving this ill-posed problem. Regarding our experiences, wavelet and Fourier regularization methods act similarly, since these methods remove high frequencies in the frequency space which are the reason for ill-posedness. To explore abilities of the wavelet regularization method, stability of the solution with the Meyer wavelet regularization is investigated by obtaining some error bounds. It is demonstrated numerically that Shannon wavelet is an alternative to the Meyer wavelet in the regularization method. It is explored that the Fourier regularization method for solving the bi-Helmholtz equation with the Cauchy conditions is also applicable. Numerical algorithms of the desired regularization methods are proposed in detail based on the fast Fourier transform (FFT). Various numerical examples in two-dimensional strip domain are shown for the validation and verification of the regularization techniques.

Random dynamics for non-autonomous stochastic evolution equations without uniqueness on unbounded narrow domains

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a multivalued random dynamical system with the solution operators of the equation, we will prove the existence and upper semicontinuity of attractors by multivalued random dynamical system theory.

A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles

We study conditions of Hormander’s L2-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of the Hormander-type condition, we show a converse of Hormander L2-estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson (1998). We also de_ne new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.

On the Correct Determination of Flow of a Discontinuous Solenoidal Vector Field

We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(U)-norm (U is an n-dimensional domain that contains S ). On the basis of these inequalities, we propose a correct definition of the flow through the surface S of a discontinuous solenoidal vector field f ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).

Banach algebras of weakly differentiable functions

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an n-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.

Global existence and boundedness of classical solutions to a forager–exploiter model with volume-filling effects

This work is concerned with a chemotactic model for the dynamics of social interactions between two species — foragers u and exploiters v, as well as the dynamics of food resources w consumed by these two species. The foragers search for food directly, while the exploiters head for food by following the foragers. Specifically, the parabolic system in a smoothly bounded convex n-dimensional domain Ω, ut=Δu−∇⋅(S1(u)∇w),x∈Ω,t>0,vt=Δv−∇⋅(S2(v)∇u),x∈Ω,t>0,wt=dΔw−λ(u+v)w−μw+r(x,t),x∈Ω,t>0,is considered for the constants d,λ,μ>0 and r∈C0(Ω×[0,∞)) with a uniform bound. Volume-filling effects account for a simple version by taking S1(u)=u(1−u),S2(v)=v(1−v).We prove the global existence and boundedness of the unique classical solution to this forager–exploiter model associated with no-flux boundary conditions under the mild assumption that the initial data u0, v0, w0 satisfy 0≤u0,v0≤1 and w0≥0.

Homogenization of a Boundary Value Problem for the n-Laplace Operator on a n-Dimensional Domain with Rapidly Alternating Boundary Condition Type: The Critical Case

We study the asymptotic behavior of the solution of a boundary value problem for the p-Laplace operator with rapidly alternating nonlinear boundary conditions posed on e-periodically arranged subsets on the boundary of a domain Ω ⊂ ℝn. We assume that p = n, construct a homogenized problem, and prove the weak convergence as e → 0 of the solution of the original problem to the solution of the homogenized problem in the so-called critical case, which is characterized by the fact that the homogenization changes the character of nonlinearity of the boundary condition.

On some properties of functions from a Besov–Morrey type spaces

In this paper it is constructed a new Besov–Morrey type spaces. Utilizing integral representation of generalized derivatives of functions defined on n-dimensional domains satisfying flexible $$\varphi $$ -horn condition an embedding theorem is proved. Also, it is proved that the generalized derivatives of functions from this spaces satisfies the generalized Holder condition.

Mixed integer\u2013real least squares estimation for precise GNSS positioning using a modified ambiguity function approach

Mixed integer–real least squares (MIRLS) estimation still has two open scientific problems, i.e., the validation of results and computational efficiency for a large number of satellites. This paper presents and discusses a non-conventional approach to MIRLS estimation, which belongs to the ambiguity function method (AFM) class. Because the solution is searched for in the constant three-dimensional coordinate domain instead of the n-dimensional ambiguity domain, the computational efficiency does not depend as much on the number of satellites as it does in conventional MIRLS estimation. Simple numerical pretests have shown that the reliability and precision of results from the presented approach and the conventional MIRLS estimation are exactly the same. Hence, the presented approach, contrary to AFM, may be treated as MIRLS estimation. Furthermore, the presented approach is a few hundred times faster than AFM and may be considered in (near) real-time GNSS positioning. In light of the above, the new field of research on MIRLS estimation may be opened.