A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

A stability estimate of an inverse problem in financial prospection

In many option pricing models, there is a need to derive a stability estimate of their solutions. This paper is concerned with the stability estimate of the regularized solution arising in the inverse problem of option pricing. This kind of inverse problems, where one looks for causes for observed or desired effects, are usually “ill-posed”, i.e., their solution is not unique or unstable with respect to data perturbations. Stability estimate is needed for stable solution of these ill-posed problems. If one cannot guarantee the stability of a solution to these problems, then the problem does not make sense and there is no hope of handling them numerically. It is shown here that one can achieve stability by using so called stabilizing functional suggested by Tikhonov [Linear Integral Equation, Applied Mathematical Sciences, Springer, New york, 1989].     

A posterior error estimate for finite volume methods of the second order elliptic problem

We establish a posteriori error analysis for finite volume methods of a second‐order elliptic problem based on the framework developed by Chou and Ye [SIAM Numer Anal, 45 (2007), 1639–1653]. This residual type estimators can be applied to different finite volume methods associated with different trial functions including conforming, nonconforming and totally discontinuous trial functions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1165–1178, 2011     

Multi-layer solutions for an elliptic problem

In this paper, we prove the existence of solutions with multiple interior layers for an elliptic Neumann problem.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Recovering nonlinear terms in an inverse boundary value problem for Laplace's equation: A stability estimate

Stationary thermography can be used for investigating the functional form of a nonlinear cooling law that describes heat exchanges through an inaccessible part of the boundary of a conductor. In this paper, we obtain a logarithmic stability estimate for the associated nonlinear inverse problem. This stability estimate is obtained from the convergence and sensitivity analysis of a finite difference method for the numerical solution of the Cauchy problem for Laplace's equation, based on the Störmer–Verlet scheme.     

Existence and estimate of large solutions for an elliptic system

In this paper, an elliptic system with boundary blow-up is considered in a smooth bounded domain. By constructing certain upper solution and subsolution, we show the existence of positive solutions and give a global estimate. Furthermore, the boundary behavior of positive solutions is also discussed.     

A residual-type a posteriori error estimate of finite volume element method for a quasi-linear elliptic problem

In this paper, we analyze a residual-type a posteriori error estimator of the finite volume element method for a quasi-linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the H ¹-norm. Numerical experiments are provided to illustrate the performance of the proposed estimator.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

A surprising linear type estimate for nonlinear elliptic equations

Pointwise gradient bounds via Riesz potentials, like those available for the Poisson equation, actually hold for p-Laplacian type equations.     

The cauchy problem for an elliptic parabolic operator

Summary Necessary and sufficient conditions are established for the existence of a solution of a Cauchy problem which is not well posed in the sense of Hadamard. This research was supported in part by National Science Foundation Grant No. GP 5882 and in part by Air Force Contract AF OSR 396-63     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

A critical elliptic problem for polyharmonic operators

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existence of solutions in domains perforated with the small holes. Some unstable solutions are obtained at higher level sets by Coron's topological method, provided that the minimizing solution does not exist.     

A domain perturbation problem for elliptic operators

Summary Eigenvalue problems with null boundary data are considered for second order symmetric linear elliptic operators in bounded domains of Euclidean n-space. The main theorems give estimates for the perturbation of the eigenvalues and eigenfunctions under the deformation of attaching an ɛ-handle to the domain. In particular it is proved that the variation of every eigenvalue under such a deformation is of order ϕ(ɛ), where ϕ denotes the reciprocal of the parametrix. Entrata in Redazione il 23 giugno 1976.