Recovery of the heat equation from a single boundary measurement

We prove that we can uniquely recover the coefficient of a one-dimensional heat equation from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well-known Gelfand–Levitan–Gasymov inverse spectral theory of Sturm–Liouville operators.     

The hilbert boundary value problem for elliptic differential equation in the plane

The solution w to the Hilbert boundary value problem ?w/?z = F(z, w,?w/?z) in D Re(a+ib) w = g on ?D has so far been solved in the space of Holder-continuously differentiable functions C₁α(D). It is shown here that theproblem has a unique solution in the more general Sobolev space W_1,p (D), 2 < p < ∞, provided that the boundaryfunction g is allowed to belong to the Slobodecky space W_s,p (?D), S = 1 ? 1/P.     

Positive solutions of an elliptic partial differential equation on R

In this paper, we discuss the existence, nonexistence and uniqueness of positive solutions of a one-parameter family of elliptic partial differential equations on R^N (N>2). These equations are of interests in mathematical biology and Riemannian geometry. Our approach are based on variational arguments and comparison principles.     

An interpolation matched interface and boundary method for elliptic interface problems

An interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228-246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method.     

Solvability for nonlinear elliptic equation with boundary perturbation

The solvability of nonlinear elliptic equation with boundary perturbation is consid- ered.The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.     

Spectrum of an elliptic equation

It is shown that the spectrum for the first boundary-value problem for a second-order elliptic equation always lies in the half-plane ₀ Re z, where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from ₀, there are no other points of the spectrum on the straight line Re z=₀.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 495–502, April, 1970.The authors are grateful to V. S. Vladimirov for discussing the results of the present paper and for pointing out the proposition in [3], which made it possible to shorten the proof that the leading eigenvalue of the problem (1) is simple.     

A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

Numerical Algorithms - Let Ω be an open region in ?d, d ≥ 2, that is diffeomorphic to ??d. Consider solving ?Δu + γu = 0 on Ω with the Neumann...     

Suboptimal boundary controls for elliptic equation in critically perforated domain

In this paper we study the asymptotic behaviour, as ε   tends to zero, of a class of boundary optimal control problems Pε${\mathbb{P}}_{\epsilon }$, set in ε-periodically perforated domain. The holes have a critical size with respect to ε  -sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type; on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problems Pε${\mathbb{P}}_{\epsilon }$.     

On the solvability of the first nonlocal boundary value problem for an elliptic equation

We consider problems of the existence, uniqueness, and sign-definiteness of the classical solutions of the problem

$(Lu)(x) = f(x)(x \in D),u(x) - \beta (x)u(\sigma x) = \psi (x)(x \in S),$

where L is a linear second-order operator elliptic in the closure of a domain D ? R ⁿ and σ is a single-valued continuous mapping of S ≡ ?D into \(\bar D\).

We show that, under natural assumptions on the smoothness of β, σ, and the coefficients of L, this problem is Fredholm provided that either σ has no attractors on S or σ generates an attractor Θ on S and the spectral radius of the operator A defined on η(x) ∈ C(Θ) by the formula (Aη)(x) = |β(x)|η(σx) is less than unity.We obtain semieffective (in terms of a test function) conditions for the unique solvability of the problem.