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We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack…

This work considers a nonlinear inverse source problem in a coupled diffusion equation from the terminal observation. Theoretically, under some conditions on problem data, we build the uniqueness theorem for this inverse problem and show…

We deduce a sufficient condition of the exponential (integral) turnpike property for infinite dimensional generalized linear-quadratic optimal control problems in terms of structural properties of the control system, such as exponential…

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on…

We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions…

This work belongs to the framework of inverse problems with linear model. The resolution of this type of problem consists in minimizing (possibly under constraints) a function of discrepancy between the measurements and a physical model of…

The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a…

In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G\^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we…

We revisit the global modes and instabilities of homogeneous rotating ellipsoidal fluid masses, which are the simplest global models of rotationally and tidally deformed gaseous planets or stars. The tidal flow in a short-period planet may…

In this paper we study the inverse boundary value problem of determining the potential in the Schr\"{o}dinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that,…

Corrigendum : An inverse problem in corrosion detection:stability estimates, J. Inv. Ill-posed Problems 12 (4) (2004), 349-367.

In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a…

Three papers describing different methods to solve the inverse scattering problem of the reconstruction of the shape and/or impedance of an obstacle have been chosen for analysis. This literature review consists of an evaluation of these…

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we…

This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the…

The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly…

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new…

The main thrust of our current work is to exploit very specific characteristics of a given problem in order to acquire improved compactness for supercritical problems and to prove existence of new types of solutions. To this end, we shall…

In this letter, by regarding finite-time stability as an inverse problem, we reveal the essence of finite-time stability and fixed-time stability. Some necessary and sufficient conditions are given. As application, we give a new approach…

In this paper we address the numerical solution of nonlinear ill-posed systems by iterative regularization methods in the classes of Levenberg-Marquardt, trust-region and adaptive quadratic regularization procedures. Both with exact and…