Arbitrary Subdomain Research Articles

Fractional anisotropic Calderón problem on complete Riemannian manifolds

We prove that the metric tensor [Formula: see text] of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator [Formula: see text]. Our result holds under the condition that the metric tensor [Formula: see text] is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].

Scaled boundary finite element method for calculating the J-integral based on LEFM

The present study develops the scaled boundary finite element method (SBFEM) to derive the J-integral directly based on defining the rectangular contour. To achieve this, firstly the J-integral is calculated for each element with different path integrals of radial direction in an arbitrary subdomain. Then, the computed values of elements are summed in the entire domain. Finally, to validate and displaying the efficiency of the SBFEM solution, three two-dimensional (2D) numerical examples are solved for different integration paths. The results indicate path-independent property and fast convergency of this semi-analytical method in the term of J-integral computation.

Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

Intrinsic winding of braided vector fields in tubular subdomains

Braided vector fields on spatial subdomains which are homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field’s topology—the entanglement of its field lines—is non-trivial, and can play a significant role in the vector field’s evolution. We present a complete topological characterisation of such vector fields (up to isotopy) using a quantity called field line winding. This measures the entanglement of each field line with all other field lines of the vector field, and may be defined for an arbitrary tubular subdomain by prescribing a minimally distorted coordinate system. We propose how to define such coordinates, and prove that the resulting field line winding distribution uniquely classifies the topology of a braided vector field. The field line winding is similar to the field line helicity considered previously for magnetic (solenoidal) fields, but is a more fundamental measure of the field line topology because it does not conflate linking information with field strength.

An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

Abstract We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form “identity + non-expansive”.

Carleman estimates for the structurally damped plate equations with Robin boundary conditions and applications

In this paper, we consider Carleman estimates for the damped fourth-order plate operators in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different from that of the usual plate equations without damping. As applications, we consider two types of inverse problems in determining source terms for the plate equation with structural damping. The time-dependent measurements can be restricted to an arbitrary small sub-domain or arbitrary sub-boundary, respectively. Under some assumptions on the regularity of the solutions and coefficients, we prove the global stability results for these inverse problems.

Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

This article presents an inverse source problem for a cascade system of \begin{document}$ n $\end{document} coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the \begin{document}$ n $\end{document} components at a fixed positive time \begin{document}$ T' $\end{document} over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

Stabilization and Control for the Biharmonic Schrödinger Equation

The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schr\"odinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=\lambda |u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability ensures that fourth order nonlinear Schr\"odinger is globally exactly controllable.

Localized peaking regimes for quasilinear parabolic equations

AbstractThis paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: urn:x-wiley:0025584X:media:mana201700436:mana201700436-math-0002with prescribed global energy function urn:x-wiley:0025584X:media:mana201700436:mana201700436-math-0003Here , , , Ω is a bounded smooth domain, . Particularly, in the case urn:x-wiley:0025584X:media:mana201700436:mana201700436-math-0008it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.

On the identification of multiple space dependent ionic parameters in cardiac electrophysiology modelling

In this paper, we consider the inverse problem of space dependent multiple ionic parameters identification in cardiac electrophysiology modelling from a set of observations. We use the monodomain system known as a state-of-the-art model in cardiac electrophysiology and we consider a general Hodgkin–Huxley formalism to describe the ionic exchanges at the microscopic level. This formalism covers many physiological transmembrane potential models including those in cardiac electrophysiology. Our main result is the proof of the uniqueness and a Lipschitz stability estimate of ion channels conductance parameters based on some observations on an arbitrary subdomain. The key idea is a Carleman estimate for a parabolic operator with multiple coefficients and an ordinary differential equation system.

Internal controllability of the korteweg–de vries equation on a bounded domain

This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval with a distributed control. When the control region is an arbitrary open subdomain, we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, the state function being controlled on the left part of the complement of the control region. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L2 space is also established.

Control and stabilization of the Benjamin-Ono equation on a periodic domain

It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain T \mathbb {T} with a distributed control supported on an arbitrary subdomain is exactly controllable and exponentially stabilizable. The aim of this paper is to extend those results to the full Benjamin-Ono equation. A feedback law in the form of a localized damping is incorporated into the equation. A smoothing effect established with the aid of a propagation of regularity property is used to prove the semi-global stabilization in L 2 ( T ) L^2(\mathbb {T}) of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and the local exponential stability in H s ( T ) H^s(\mathbb {T}) are also established for s > 1 / 2 s>1/2 by using the contraction mapping theorem. Finally, the local exact controllability is derived in H s ( T ) H^s(\mathbb {T}) for s > 1 / 2 s>1/2 by combining the above feedback law with some open loop control.

A flux preserving immersed nonconforming finite element method for elliptic problems

An immersed nonconforming finite element method based on the flux continuity on intercell boundaries is introduced. The direct application of flux continuity across the support of basis functions yields a nonsymmetric stiffness system for interface elements. To overcome non-symmetry of the stiffness system we introduce a modification based on the Riesz representation and a local postprocessing to recover local fluxes. This approach yields a P1 immersed nonconforming finite element method with a slightly different source term from the standard nonconforming finite element method. The recovered numerical flux conserves total flux in arbitrary sub-domain. An optimal rate of convergence in the energy norm is obtained and numerical examples are provided to confirm our analysis.

Inverse source problem for linearized Navier–Stokes equations with data in arbitrary sub-domain

We consider an inverse problem of determining a spatially varying factor in a source term in the non-stationary linearized Navier–Stokes equations by observation data in an arbitrarily fixed sub-domain over some time interval. We prove the Lipschitz stability provided that the t-dependent factor satisfies a non-degeneracy condition. Our proof is based on a new Carleman estimate for the linearized Navier–Stokes equations.

Simultaneous identification of parameters and initial datum of reaction diffusion system by optimization method

In this paper, we study an inverse problem of reconstructing two time independent coefficients and the initial data in the linear reaction diffusion system from the arbitrary subdomain measurement and final measurement. First the given problem is transformed into an optimization problem by using optimal control framework and we establish the existence of the minimizer for the control functional. Further the necessary optimality condition is established which is the key ingredient to establish the stability estimate.

Exotic baker and wandering domains for Ahlfors islands maps

Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere or a torus. We construct a variety of examples of analytic functions g:W->X, where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands condition". In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the omega-limit set of a Baker domain of such a function. As a corollary of our construction, we show that there are entire functions with Baker domains in which the iterates converge to infinity arbitrarily slowly. We also construct Ahlfors islands maps with wandering domains and logarithmic singularities, as well as examples where X is a compact hyperbolic surface.

On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn

AbstractWe establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

Inverse problems for the Boussinesq system

We obtain two results on inverse problems for a 2D Boussinesq system. One is that we prove the Lipschitz stability for the inverse source problem of identifying a time-independent external force in the system with observation data in an arbitrary sub-domain over a time interval of the velocity and the data of velocity and temperature at a fixed positive time t0 > 0 over the whole spatial domain. The other one is that we prove a conditional stability estimate for an inverse problem of identifying the two initial conditions with a single observation on a sub-domain.

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

In [38], Russell and Zhang showed that the Korteweg-de Vries equation posed on a periodic domain 𝕋 with an internal control is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of 𝕋. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability is established with the aid of certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system. With Slemrod's feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrarily large decay rate.