Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

A nonlinear descent method for a variational inequality on a nonconvex set

In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.     

The isoperimetric inequality on a surface

We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its Euler characteristic, the length of its boundary, and its Gaussian curvature. Received: 31 July 1998     

On a nonlinear inequality on the time scale

In the present paper, we consider a dynamic nonlinear integral inequality with a power-law nonlinearity. We obtain a solution of this inequality for arbitrary nonlinearity exponents exceeding unity. These results can be constructively used in the analysis of stability properties (including nonclassical stability properties) of quasilinear dynamic equations.     

A nonlinear inequality of Moser-Trudinger type

In this note, we will prove an inequality for almost plurisubharmonic functions on any K?hler-Einstein manifolds with positive scalar curvature. This inequality generalizes the stronger version of the so called Moser-Trudinger-Onofri inequality on , which was proved in [Au], and also refines a weaker inequality found by the first author in [T2]. Received: May 27, 1997 / Accepted: June 11, 1999     

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.     

On the isoperimetric inequality on a minimal surface

We derive an explicit formula for the isoperimetric defect of an arbitrary minimal surface ,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the correct sign. As a by-product of these computations we show that the best known universal isoperimetric estimate, that for any minimal surface (due to L. Simon), may be improved to the universal estimate .Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002     

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved.

     

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i -1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1~(**)(Σ, g)), we have■When b b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.     

A Blichfeldt-type inequality for the surface area

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that #(K?Bbb Zⁿ) ￡ n! vol(K)+n#(Kcap{Bbb Z}^n)le n! {rm vol}(K)+n , whenever K ì Bbb RⁿKsubset{Bbb R}^n is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely #(K?Bbb Zⁿ) < vol(K) + ((?n+1)/2) (n-1)! F(K)#(Kcap{Bbb Z}^n) < {rm vol}(K) + ((sqrt{n}+1)/2) (n-1)! {rm F}(K) . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where t ? Bbb RⁿBbb Zⁿtin{Bbb R}^nsetminus{Bbb Z}^n and P is an n-dimensional polytope with integral vertices. Then we have #((t+P)?Bbb Zⁿ) ￡ n! vol(P)#((t+P)cap{Bbb Z}^n)le n! {rm vol}(P) . Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K?Bbb Zⁿ) < vol(K) + 2 F(K)#(Kcap{Bbb Z}^n)< {rm vol}(K) + 2 {rm F}(K) .     

A fully nonlinear version of the Yamabe problem and a Harnack type inequality

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Convergence analysis of a nonlinear Lagrangian algorithm for nonlinear programming with inequality constraints

In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an unconstrained programming, convergents locally to a Kuhn-Tucker point of the primal nonlinear programming problem.