Dirichlet Boundary Data Research Articles

Boundary determination for hybrid imaging from a single measurement

Abstract We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} -Laplacian as a forward model with the interior density data σ ⁢ | ∇ ⁡ u | q {\sigma|\nabla u|^{q}} , and find out that single measurement specifies the boundary conductivity when p - q ≥ 1 {p-q\geq 1} , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.

Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations

In this paper, we consider k k -curvature equations σ k ( κ [ M u ] ) = f ( x , u , ∇ u ) \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to ( k + 1 ) (k+1) -convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k k -curvature equations.

Generalized second order vectorial ∞-eigenvalue problems

We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

Utilization of the Modified Adomian Decomposition Method on the Bagley-Torvik Equation Amidst Dirichlet Boundary Conditions

The Bagley-Torvik equation is an imperative differential equation that considerably arises in various branches of mathematical physics and mechanics. However, very few methods exist for the treatment of the model analytically; in fact, researchers frequently shop for semi-analytical and numerical methods in their studies. Therefore, the main goal of this research is to find theexact analytical solution for the fractional Bagley-Torvik equation fitted with Dirichlet boundary data, as well as a system of fractional Bagley-Torvik equations. Thus, this research aims to show that the modified Adomian decomposition method (MADM) via the proposed two algorithms is a very effective method for treating a class of Bagley-Torvik equations endowed with Dirichletboundary data. Certainly, MADM is a very powerful approach for solving dissimilar functional equations without the need for either linearization, discretization, perturbation, or even unnecessary restraining postulations. Additionally, the method reveals exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible. Lastly, some illustrative test problems of the governing model are examined to demonstrate the superiority of the proposed algorithms.

Weak limit of homeomorphisms in W1,n−1: Invertibility and lower semicontinuity of energy

Let Ω, Ω′ ⊂ ℝn be bounded domains and let fm: Ω → Ω′ be a sequence of homeomorphisms with positive Jacobians Jfm > 0 a.e. and prescribed Dirichlet boundary data. Let all fm satisfy the Lusin (N) condition and supm ∫Ω( |D fm|n - 1 + A( |cof D fm|) + φ(Jf)) < ∞, where A and φ are positive convex functions. Let f be a weak limit of fm in W1,n−1. Provided certain growth behaviour of A and φ, we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

Boundary regularity results for minimisers of convex functionals with (p, q)-growth

Abstract We prove improved differentiability results for relaxed minimisers of vectorial convex functionals with ( p , q ) \left(p,q) -growth, satisfying a Hölder-growth condition in x x . We consider both Dirichlet and Neumann boundary data. In addition, we obtain a characterisation of regular boundary points for such minimisers. In particular, in case of homogeneous boundary conditions, this allows us to deduce partial boundary regularity of relaxed minimisers on smooth domains for radial integrands. We also obtain some partial boundary regularity results for non-homogeneous Neumann boundary conditions.

Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

AbstractThe asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.

Curvature estimates for spacelike graphic hypersurfaces in Lorentz–Minkowski space R1n+1$\mathbb {R}^{n+1}_{1}$

AbstractIn this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the ‐dimensional Lorentz–Minkowski space , and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed 2‐nd Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane of center at origin and radius 1, can be proven.

Singular behavior for a multi-parameter periodic Dirichlet problem

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0, proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple ( ϵ , ϕ , g , f ). Our aim is to study how the solution depends on ( ϵ , ϕ , g , f ), especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple ( ϵ , ϕ , g , f ) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2, we have to add log ϵ times the integral of f / 2 π.

An error estimate for finite element approximation to elliptic PDEs with discontinuous Dirichlet boundary data

This note provides an error estimate for finite element approximation to elliptic partial differential equations (PDEs) with discontinuous Dirichlet boundary data. Solutions of problems of this type are not in H1 and, hence, the standard variational formulation is not valid. To circumvent this difficulty, an error estimate of a finite element approximation in the W1,r(Ω) (0<r<2) norm is obtained through a regularization by constructing a continuous approximation of the Dirichlet boundary data.

Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $$\mathcal {D}_t^\alpha (\partial _{t}u)$$ and $$\mathcal {D}_t^\alpha u $$ (with $$\alpha \in (0,1)$$ ), where $$\mathcal {D}_t^\alpha $$ denotes the Caputo fractional differential operator (in time) of order $$\alpha $$ . We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe’s method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multivariate Mittag-Leffler function.

The Majda-Biello system on the half-line

The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter α as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for 0<α<1 or 1<α<4 all these problems are well-posed for initial data in Sobolev spaces Hs, s≥0. For α=1 or α>4 well-posedness holds for Dirichlet data if s>−3/4, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For α=4 well-posedness of all problems holds for s≥3/4. The Robin and Neumann boundary data are in Hs/3 while the Dirichlet boundary data are in H(s+1)/3. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained her e are optimal.

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of implicit constitutive relations we establish the existence and uniqueness of a global-in-time large-data weak solution. We then focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises, which is that the Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.

Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes

Abstract This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ ( Ω ) {H^{2}(\Omega)} but only to H r ⁢ ( Ω ) {H^{r}(\Omega)} for some r ∈ ( 1 , 2 ) {r\in(1,2)} . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} and H 1 ⁢ ( Ω ) {H^{1}(\Omega)} . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results.

Remote trajectory tracking of a rigid body in an incompressible fluid at low Reynolds number

In this paper we study the motion of a rigid body driven by Newton’s law immersed in a stationary incompressible Stokes flow occupying a bounded simply connected domain. The aim is that of trajectory tracking of the solid by the means of a control in the form of Dirichlet boundary data on the outside boundary of the fluid domain. We show that it is possible to exactly achieve any smooth trajectory for the solid that stays away from the external boundary, by the means of such a remote control. The proof relies on some density methods for the Stokes system, as well as a reformulation of the solid equations into an ODE.

Analysis of a Full Discretization for a Fractional/Normal Diffusion Equation with Rough Dirichlet Boundary Data

Analysis of a Full Discretization for a Fractional/Normal Diffusion Equation with Rough Dirichlet Boundary Data

Finite-time blow-up in a repulsive chemotaxis-consumption system

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system \[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$. It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0&lt; D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi &gt;0$ with some ${K_D}&gt;0$ and $\alpha &gt;0$, then for all initial data from a considerably large set of radial functions on $\Omega$, the corresponding initial-boundary value problem admits a solution blowing up in finite time.

The critical fractional Ambrosetti–Prodi problem

In this paper we focus on the following nonlocal problem with critical growth: (-Δ)su=λu+u+2s∗-1+f(x)inΩ,u=0inRN\\Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} (-\\Delta )^{s} u = \\lambda u + u_{+}^{2^{*}_{s}-1} + f(x) &{} \ ext{ in } \\Omega ,\\\\ u=0 &{} \ ext{ in } \\mathbb {R}^{N}\\setminus \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where sin (0, 1), N>2s, Omega subset mathbb {R}^{N} is a smooth bounded domain, lambda >0, (-Delta )^{s} is the fractional Laplacian, f= te_{1}+h where tin mathbb {R}, e_{1} is the first eigenfunction of (-Delta )^{s} with homogeneous Dirichlet boundary datum, and hin L^{infty }(Omega ) is such that int _{Omega } h e_{1}, dx=0. According to the interaction of the nonlinear term with the spectrum of (-Delta )^{s}, we establish some existence and multiplicity results for the above problem by means of variational methods.

Boundary behavior for the heat equation on the half‐line

The initial‐boundary value problem for the heat equation on with nonzero Dirichlet boundary data is studied rigorously, with emphasis on the behavior of the solution along the boundary of the spatiotemporal domain, that is, in the limits or , including the case . Our results specify how the regularity of the solution at the boundary is dictated by the level of smoothness of the relevant initial and boundary data. To the best of our knowledge, for unbounded domains like the half‐line , such results are not available in the literature. The corresponding analysis for the derivatives of the solution is also performed and yields additional compatibility conditions on the initial and boundary data which guarantee the extension of the solution to a function up to the boundary, that is, on . The analysis relies on the explicit solution formula provided by the unified transform, also known as the Fokas method, whose rigorous derivation is presented. The main advantage of this formula, namely, its uniform convergence at the boundary, largely reduces the proof of our results to fundamental tools from real and complex analysis such as Lebesgue's dominated convergence theorem and Cauchy's theorem. Moreover, thanks to uniform convergence, the unified transform solution formula can actually be directly evaluated at the boundary to yield the prescribed initial and boundary data.

A Connection Between Symmetry Breaking for Sobolev Minimizers and Stationary Navier–Stokes Flows Past a Circular Obstacle

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier–Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.