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.     

Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.     

Twist periodic solutions of second order singular differential equations

We study the twist periodic solutions of second order singular differential equations. Such twist periodic solutions are stable in the sense of Lyapunov and present much interesting dynamical features around them. The proof is based on the third-order approximation method. The estimates of periodic solutions of Ermakov-Pinney equations and the estimates on rotation numbers of Hill equations play an important role in the analysis.     

Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.     

二阶奇异椭圆方程的Dirichlet问题

对二阶奇异椭圆方程-△u 1/uα-λuP=0的Dirichlet问题(λ＞0,0＜P＜1)证明了当α≥1时无解存在,当0＜α＜1时存在极小解;并对较一般的奇异方程给出了一个存在性结果.     

Periodic solutions for second order singular damped differential equations

We study the existence of positive periodic solutions for second order singular damped differential equations by combining the analysis of the sign of Green?s functions for the linear damped equation, together with a nonlinear alternative principle of Leray–Schauder. Recent results in the literature are generalized and significantly improved.     

一类半正奇异二阶脉冲微分方程的正解

利用Schauder不动点理论和上下解方法,讨论了一类半正奇异二阶微分方程,在Neumann边值条件下受脉冲影响的正解存在性.     

Existence and uniqueness of global strong solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions \({u: \mathbb{R}^n \longrightarrow \mathbb{R}^N}\) to the fully nonlinear PDE system

$$\label{equ1}F(\cdot,D^2u ) = f, \quad \text{ a.e. on }\mathbb{R}^n,\quad \quad(1)$$

when \({ f\in L^2(\mathbb{R}^n)^N}\) and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1).

     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

Representations of solutions of abstract second order differential equations with applications to stiff equations and singular perturbations

The purpose of this paper is to demonstrate the applicability of the theory of operator equations to the theory of abstract differential equations. Attention is focused on the representation and approximation of solutions. Particular topics considered are existence-uniqueness, stiff equations, and singular perturbations.     

Removable singularities of solutions of linear uniformly elliptic second order equations

Let L be a uniformly elliptic linear second order differential operator in divergence form with bounded measurable real coefficients in a bounded domain G ? ?ⁿ (n ? 2). We define classes of continuous functions in G that contain generalized solutions of the equation L? = 0 and have the property that the compact sets removable for such solutions in these classes can be completely described in terms of Hausdorff measures.     

L p -estimates for the solutions of second order elliptic equations

Summary We investigate the homogeneous Dirichlet problem in H^2,p for a second order elliptic partial differential equation in nondivergence form Lu=f in the case in which the leading coefficients of L belong to H^1,n(), Rⁿ. We prove that if p belongs to a suitable neighbourhood of 2, then the above problem, has a unique solution u satisfying D²u_p Cf_p; furthermore, if f H^k,p, k=1,2, ..., and the coefficients of L satisfy some natural conditions, then the solution satisfies .Lavoro eseguito nell'ambito del gruppi 40% e 60% del M.P.I.     

Very weak solutions to elliptic equations with singular convection term

We study Dirichlet problem for a nonlinear equation with a drift term. Despite the presence of the singular convection term, we establish existence and uniqueness of a solution in spaces larger than the natural one.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|². Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday Received: May 4, 2004     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in \mathbbRⁿ ,\mathbb{R}^{n} , where the linear term is given by Schr?dinger operators H =  − Δ  +  V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|².