On quantitative uniqueness for elliptic equations

Mathematische Zeitschrift - We address the question of quantitative uniqueness for the equation $$\Delta u=V u$$ with either periodic or Dirichlet boundary conditions in a disk. We construct...     

On a weak uniqueness for some elliptic equations

Uniqueness is proved to the Dirichlet problem for second order uniformly elliptic nondivergence equations. The coefficients of second derivatives are supposed to be continous except for a closed set having zero Hausdorff measure corresponding to the function h(s)=s ^α for some α>     

On nonuniqueness of solutions to elliptic equations for probability measures

We consider the set of all probability measures μ on Rd satisfying an elliptic equation L^∗μ=0 in the weak worm. We give sufficient conditions in order that this set contains at least two different elements. We also construct new examples of nonuniqueness for such equations.     

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.     

A uniqueness result for monotone elliptic problems

We give here a short proof of the uniqueness of entropy solutions for nonlinear monotone elliptic problems with L¹-data. To cite this article: M.M. Porzio, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Existence and uniqueness for nonlinear anisotropic elliptic equations

We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.     

On some inverse problems for elliptic equations and systems

Under study are the inverse problems of determining the right-hand side of a particular form and the solution for elliptic systems, including a series of elasticity systems. (On the boundary of the domain the solution satisfies either the Dirichlet conditions or mixed Dirichlet-Neumann conditions.) We assume that on a system of planes the normal derivatives of the solution can have discontinuities of the first kind. The conjugating boundary conditions on the discontinuity surface are analogous to the continuity conditions for the fields of displacements and stresses for a horizontally laminated medium. The overdetermination conditions are integral (the average of the solution over some domain is specified) or local (the values of the solution on some lines are specified). We study the solvability conditions for these problems and their Fredholm property.     

Nondegeneracy and uniqueness for boundary blow-up elliptic problems

In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Δu=λa(x)u^p+g(x,u) in Ω, with u=+∞ on ∂Ω, has a unique positive solution for large enough λ, and determine its asymptotic behavior as λ→+∞. Here p>1, a(x) is a continuous function which can be singular near ∂Ω and g(x,u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u^p by e^u or a “logistic type” function f(u).     

Discrete problems associated to elliptic equations

This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Quantitative uniqueness estimates for the general second order elliptic equations

In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and bootstrapping arguments.