Prescribed Boundary Values Research Articles

Fully nonlinear singularly perturbed models with non-homogeneous degeneracy

This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In its simplest form, for each \varepsilon>0 fixed, we seek a non-negative function u^{\varepsilon} satisfying \begin{cases} \left[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \Delta u^{\varepsilon} = \zeta_{\varepsilon}(x, u^{\varepsilon}) & \operatorname{in } \Omega,\\ u^{\varepsilon}(x) = g(x) & \operatorname{on } \partial \Omega, \end{cases} in the viscosity sense for suitable data p, q \in (0, \infty) , \mathfrak{a} and g , where \zeta_{\varepsilon} behaves singularly as \text{O} (\varepsilon^{-1}) near \varepsilon -level surfaces. In such a context, we establish the existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. In particular, for a restricted class of nonlinearities, we prove the finiteness of the (N-1) -dimensional Hausdorff measure of level sets. We also address a complete and in-deep analysis concerning the asymptotic limit as \varepsilon \to 0^{+} , which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory. Finally, we present some fundamental regularity tools in the theory of doubly degenerate fully nonlinear elliptic PDEs, which may have their own mathematical interest.

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.

A relative Sierpinski theorem: Erratum to “Nonhyperbolic Coxeter groups with Menger boundary”

The purpose of this erratum is to correct the proof of Proposition 2.3 of [HHS]. A classical theorem of Sierpinski states that every subspace of dimension at most one in the 2-dimensional disc D^2 can be topologically embedded in the Sierpinski carpet. The proof of Proposition 2.3 of [HHS] implicitly provides a relative version of Sierpinski’s theorem. Unfortunately the proof of Proposition 2.3 given in [HHS] is incorrect. We provide two brief proofs that each fill this gap. One is a self-contained argument suited specifically for the needs of [HHS], and in the other we explicitly prove a relative embedding theorem that produces embeddings in the Sierpinski carpet with certain prescribed boundary values.

Complex geodesics and complex Monge–Ampère equations with boundary singularity

We study complex geodesics and complex Monge–Ampère equations on bounded strongly linearly convex domains in \(\mathbb C^n\). More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge–Ampère equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in \({\mathbb {C}}^n\).

Optimal thermal performance of magneto-nanofluid flow in expanding/contracting channel

In a thermodynamical system, loss of energy takes the center of attention. According to laws of thermodynamics, energy of the system remains conserve, but all energy is employed to add work, and some of it is converted to rise the temperature and eventually increases entropy. Entropy of the system only increases and can never decrease. For the optimal thermal performance of a system, we need entropy generated to be minimum. This paper explores the enhancement of entropy in magneto-hydrodynamics nanofluid flowing inside the parallel plate which has a capacity of expansion or contraction. The lower plate is heated from outside. The mathematical constitution for the mass, momentum and transport of heat is described by a set of PDEs, which in turn generates a series solution by adopting the homotopy analysis method under prescribed boundary values. Different nano-sized particles, i.e., copper, silver and alumina are uniformly mixed in water. The impact of various factors on temperature, Nusselt number and velocity are elaborated pictorially and tabularly.

The Perron solution for vector-valued equations

Given a continuous function on the boundary of a bounded open set in $$\mathbb {R}^d$$ there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron’s classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

ABSTRACT This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem.

Nonlinear fractional elliptic systems with boundary measure data: Existence and a priori estimates

We are concerned with positive solutions of the fractional elliptic system(E){(−Δ)su=f(v)in Ω,(−Δ)sv=g(u)in Ω, where Ω is an arbitrary domain in RN (N>2s), s∈(12,1) and f,g∈Clocβ(R), for some β∈(0,1). We establish universal a priori estimate for positive solutions of (E). Then for C2 bounded domain Ω, we prove the existence of positive solutions of (E) with prescribed boundary value u=μ and v=ν where μ,ν are positive bounded measure on ∂Ω and discuss regularity property of the solutions.

Generalized boundary equations for conservative Navier–Stokes equations

CFD simulations in finite computational domains demand proper boundary treatment due to the absence of flow solutions beyond the boundaries, and accurate boundary residuals are critical to improve the computational stability and convergence rate, especially for implicit time integrations. Among the numerous boundary treatments in previous publications, the characteristic boundary conditions are the particularly successful ones that enjoy both the computational stability and accuracy for general CFD simulations. However, the underlying locally one-dimensional inviscid approximation inevitably causes the numerical drifting of boundary solutions and flow distortions, and the implementation for three-dimensional problems with transverse corrections is cumbersome. Inspired by the success and drawbacks of the characteristic boundary conditions, a set of generalized boundary equations are proposed in this study. It is shown that application of the physical boundary conditions effectively introduces corrections on the residuals that are computed using one-sided schemes, and a novel approach is proposed to determine the correction terms. The resulting boundary equations are in the conservative form, and independent of the time and spatial schemes. Therefore, their implementation in computer software is made more convenient, particularly for multi-dimensional problems on a direction-by-direction basis. The boundary equations preserve the specified boundary conditions accurately if solved exactly. Numerical experiments showed almost negligible deviations from the prescribed boundary values during the explicit integrations without using the relaxation terms that are indispensable for the characteristic boundary conditions.

Globally Lipschitz minimizers for variational problems with linear growth

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev spaceW1,1with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler–Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values,viathe construction of appropriate barrier functions in the tradition of Serrin’s paper [J. Serrin,Philos. Trans. R. Soc. Lond., Ser. A264(1969) 413–496].

Boundary Singularities of Solutions to Semilinear Fractional Equations

Abstract We prove the existence of a solution of ( - Δ ) s ⁢ u + f ⁢ ( u ) = 0 {(-\Delta)^{s}u+f(u)=0} in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f ⁢ ( u ) = u p {f(u)=u^{p}} and μ is a Dirac mass, we show the existence of several critical exponents p. We also demonstrate the existence of several types of separable solutions of the equation ( - Δ ) s ⁢ u + u p = 0 {(-\Delta)^{s}u+u^{p}=0} in ℝ + N {\mathbb{R}^{N}_{+}} .

Cavity type problems ruled by infinity Laplacian operator

We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly non-degenerate and have porous level surfaces. Moreover, for some restricted cases we show the finiteness of the (n−1)-dimensional Hausdorff measure of level sets. The analysis of the asymptotic limits is carried out as well.

Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy–Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.

Green's function modeling of response of two-dimensional materials to point probes for scanning probe microscopy

A Green's function (GF) method is developed for interpreting scanning probe microscopy (SPM) measurements on new two-dimensional (2D) materials. GFs for the Laplace/Poisson equations are calculated by using a virtual source method for two separate cases of a finite material containing a rectangular defect and a hexagonal defect. The prescribed boundary values are reproduced almost exactly by the calculated GFs. It is suggested that the GF is not just a mathematical artefact but a basic physical characteristic of material systems, which can be measured directly by SPM for 2D solids. This should make SPM an even more powerful technique for characterization of 2D materials.

Flow of a nanofluid in the microspacing within co-rotating discs of a Tesla turbine

This article presents, for the first time, the fluid dynamics of the rotating flow of a nanofluid through the narrow spacings within co-rotating discs of a Tesla turbine. The inter-disc-spacing of multiple concentric discs of a Tesla disc turbine is usually of the order of 100μm. The study is conducted with the help of both mathematical analysis and computational fluid dynamic simulations. Numerical values are reported for a specific nanofluid which is a dilute solution of ferro-particles in water (maximum volume fraction considered is 0.05). The velocity field, pressure field and fluid pathlines are calculated in the three-dimensional, axi-symmetric flow domain for prescribed boundary values for the velocity components at inlet. Detailed comparisons between the analytical and computational solutions are provided. It is explained how the fluid dynamics of rotating flow within a Tesla disc turbine is influenced by the change in volume fraction of nanoparticles. The present study reveals that, with an increase in the volume fraction of nanoparticles, the pressure-drop in the radial direction increases; the tangential velocity at any point inside the computational domain tends to increase (even though its boundary value at inlet is kept fixed for each set of computation); however, the radial velocity field remains almost invariant. The present analysis shows that, with a suitable selection of the combination of geometric and flow parameters, the use of nanofluid leads to a significant improvement in the power output (the magnitude of increase would depend on the choice of nanofluid; the sample calculations show more than 30% increase in power output when the volume fraction of nanoparticles increases from 0 to 0.05). Moreover, the gain in power output is achieved without appreciably affecting the efficiency of the turbine. Indeed the present study shows that it is possible to achieve a high efficiency (a figure of 56% is included in the paper as a sample case), revealing the potential of the Tesla disc turbine to emerge as an attractive engineering product in the field of micro-turbines.

Time-fractional thermoelasticity problem for a sphere subjected to the heat flux

The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative is used to study central symmetric thermal stresses in a sphere. The solution is obtained using the Laplace transform with respect to time and the finite sin-Fourier integral transform with respect to the radial coordinate. The physical Neumann problem with the prescribed boundary value of the heat flux is considered. Numerical results are illustrated graphically.

A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts

Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

A method for implementing Dirichlet and third-type boundary conditions in PTRW simulations

We present an efficient and accurate numerical method for implementing Dirichlet boundary conditions in particle tracking random walk (PTRW) simulations of advective-dispersive transport. This is a challenge, because defining concentrations for Dirichlet boundary conditions requires invoking control volumes of some kind, which are not natural to the Lagrangian-based PTRW concept. Our method performs a Galerkin projection of PTRW-based particle densities onto control volumes that discretize the boundary. Thus, we obtain concentration values at the boundary condition and can control the particle release rates such that the prescribed boundary values are met. This allows for complex-shaped internal and external boundaries, where concentration values are fixed to prescribed values. Third-type boundary conditions can be addressed as well. We test and illustrate the properties and behavior of our method in a series of test cases. The results are benchmarked against the conceptually related semianalytical method MASST (multiple analytical source superposition technique) and to those of a finite element method (FEM). While MASST is restricted to uniform velocity fields due to the underlying analytical solutions, FEM is limited in heterogeneous velocity fields at large Peclet numbers by numerical dispersion in the feasible discretization range. The results demonstrate that our proposed method performs better than the other methods in both regimes.

HARMONIC FUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.

Median values, 1-harmonic functions, and functions of least gradient

Motivated by the mean value property of harmonic functions, we introduce the local andglobal median value properties for continuous functions of two variables. We show that theDirichlet problem associated with the local median value property is either easy or impossibleto solve, and we prove that continuous functions with this property are $1$-harmonic in theviscosity sense. We then close with the following conjecture: a continuous function havingthe global median value property and prescribed boundary values coincides with the functionof least gradient having those same boundary values.