Comparison theorems for second-order elliptic inequalities

We study non-negative solutions of the elliptic inequalities$\sum _{i,j=1}^n{a}_{\mathit{ij}}(x)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+\sum _{i=1}^n{b}_i(x)\frac{\partial u}{\partial x_i}⩾F(x,u)\mathrm{in}\Omega ,$where $\Omega \ne 0$ is an open subset of ${\mathbb{R}}^n,n⩾2$, and $F:\Omega \times [0,\infty )\to [0,\infty )$ is some function. In particular, the influence of the geometric properties of $\Omega$ in the behavior of solutions is investigated.     

Nonexistence theorems for some nonlinear systems of elliptic inequalities

We study the blow-up of solutions to two classes of nonlinear elliptic systems by using test function method.     

On doubling inequalities for elliptic systems

We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lamé system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions.     

Comparison theorems for elliptic inequalities with a non-linearity in the principal part

We obtain estimates for non-negative solutions of the elliptic inequality divA(x,Du)?F(x,u) in unbounded domains.     

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $Lu+{|u|}^{q?1}u?Lv+{|v|}^{q?1}v$, where $q>0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

$L=\sum _{i,j=1}^n\frac{?}{?x_i}[a_{ij}(x)\frac{?}{?x_j}].$

We assume that $n?2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots ,n$, are measurable bounded functions on ${\mathbb{R}}^n$ such that $a_{ij}(x)=a_{ji}(x)$, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On a Liouville comparison principle for entire weak solutions of quasilinear elliptic partial differential inequalities

We establish a new Liouville-type comparison principle for entire weak solutions of quasilinear elliptic partial differential inequalities of the form $A(u)?A(v)$ on ${\mathbb{R}}^n$, $n?2$. Typical examples of the operator $A(w)$ are the p-Laplacian and its well-known modifications for $1<p?2$. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.     

On two-coefficient identification in elliptic variational inequalities

Abstract

This paper is concerned with elliptic variational inequalities that depend on two parameters. First, we investigate the dependence of the solution of the forward problem on these parameters and prove a Lipschitz estimate. Then, we study the inverse problem of identification of these two parameters and formulate two optimization approaches to this parameter identification problem. We extend the output least-squares approach, provide an existence result and establish a convergence result for finite-dimensional approximation. Further, we investigate the modified output least-squares approach which is based on energy functionals. This latter approach can be related to vector approximation.     

二阶非线性椭圆型微分方程解的振动比较定理

建立几个微分不等式,讨论了一类二阶非线性椭圆型微分方程解的振动性,得到几个新的振动比较定理．