Related papers: Enclosure method for the p-Laplace equation
We study a bulk-surface coupled Laplace system involving an embedded open boundary. The problem is reformulated as an integro-differential equation using boundary integral representations, for which we establish existence and uniqueness of…
Exact solutions describing trapped modes in a plane quantum waveguide with a small rigid obstacle are constructed in the form of convergent series in powers of the small parameter characterizing the smallness of the obstacle. The terms of…
This paper shows how a class of non-convex optimization problems constrained by discretized nonlinear partial differential equations may be solved to global optimality using an interior point continuation method. The solution procedure…
We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions…
This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard…
The fractional Calder\'on problem asks to determine the unknown coefficients in a nonlocal, elliptic equation of fractional order from exterior measurements of its solutions. There has been substantial work on many aspects of this inverse…
In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $\epsilon$ between the inclusions, which tends…
We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calder\'on's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be…
The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation $-\Delta u=|u|^{p-1} u$ with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution $u$…
We deal with the problem of reconstructing interfaces using complex geometrical optics solutions for the Maxwell system. The contributions are twofold. First, we justify the enclosure method for the impenetrable obstacle case avoiding any…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
In this paper, we address a particular case of Calder\'on's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high…
In this paper, we study the stability of the inverse conductivity problem of determining a convex polyhedral inclusion embedded in a homogeneous isotropic medium from a single boundary measurement. The main tools in our analysis are…
In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved.…
The Calder\'on formulas (i.e., the combination of single-layer and hyper-singular boundary integral operators) have been widely utilized in the process of constructing valid boundary integral equation systems which could possess highly…
It is well known that the Laplace cascade method is an effective tool for constructing solutions to linear equations of hyperbolic type, as well as nonlinear equations of the Liouville type. The connection between the Laplace method and…
The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere ${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a transport problem for…
We use Ikehata's enclosure method to reconstruct penetrable unknown inclusions in a plane elastic body in time-harmonic waves. Complex geometrical optics solutions with complex polynomial phases are adopted as the probing utility. In a…
We linearize and solve the Van der Pol equation (with additional nonlinear terms) by the application of a generalized form of Cole-Hopf transformation. We classify also Lienard equations with low order polynomial coefficients which can be…
In this work, we will prove a uniqueness result for Calder\'on's inverse problem via some integral representation formulas for solutions of the Vekua equation in the framework of Clifford analysis.