Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

We study the subcritical problemsP :–u=u ^p–,u>0 on;u=0 on , being a smooth and bounded domain in ^N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P₀).

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Résumé Nous étudions les problèmes sous-critiquesP :–u=u ^p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de ^N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P₀).

     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.

     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG ⁿ (n3) with boundary of class C². Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.     

Elliptic boundary-value problems with discontinuous nonlinearities

For a class of elliptic boundary value problems with discontinuous nonlinearities, the existence of strong solutions is established. Two applications are then developed. In particular, one of them is devoted to implicit elliptic equations of the form (–u)=(u), where is a continuous function and has a set of discontinuity points of Lebesgue measure zero. The abstract framework where these problems are studied is that of set-valued analysis.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

On Non-Linear Rayleigh quotients

The stability with respect top of the non-linear eigenvalue problem div(|u| ^p–2u)+|u| ^p–2 u=0 is studied.     

Perturbation of the Laplacian by the Coulomb potential and a point interaction in L p (ℝ)3

Perturbations of -+/|x| (with >0) by a point interaction centered at zero are defined in L ^p(³). This is done for 3/20 (³{0}), such that the extension is the negative generator of an analytic semigroup on L ^p(³).     

A degenerate and strongly coupled quasilinear parabolic system not in divergence form

This paper deals with positive solutions of degenerate and strongly coupled quasi-linear parabolic system not in divergence form: u_t=v^p(u+au), v_t=u^q (v+bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min {a, b} ₁ then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (we can not prove the uniqueness result in general); (ii) When min {a, b} > ₁, there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u₀v₀) satisfies u₀ + au₀ 0, v₀+bv₀ 0 in , then the positive classical solution is unique and blows up in finite time, where ₁ is the first eigenvalue of – in with homogeneous Dirichlet boundary condition.This project was supported by PRC grant NSFC 19831060 and 333 Project of JiangSu Province.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators

We consider the equation (pu)-qu+wu = f in (0,1) subject to homogenous boundary conditions at x = 0 and x = 1, e.g., u(0) = u(1) = 0. Let ₁ be the first eigenvalue of the corresponding Sturm-Liouville problem. If f 0 but 0 then it is known that there exists > 0 (independent on f) such that for (₁, ₁ + ] any solution u must be negative. This so-called uniform anti-maximum principle (UAMP) goes back to Clément, Peletier [4]. In this paper we establish the sharp values of for which (UAMP) holds. The same phenomenon, including sharp values of , can be shown for the radially symmetric p-Laplacian on balls and annuli in ⁿ provided 1 n < p. The results are illustrated by explicitly computed examples.     

Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L^m-norm, m [1,], of non-negative solutions to the Cauchy problem _t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established.     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

Sobolev inequalities on homogeneous spaces

We consider a homogeneous spaceX=(X, d, m) of dimension 1 and a local regular Dirichlet forma inL ² (X, m). We prove that if a Poincaré inequality of exponent 1p< holds on every pseudo-ballB(x, R) ofX, then Sobolev and Nash inequalities of any exponentq[p, ), as well as Poincaré inequalities of any exponentq[p, +), also hold onB(x, R).Lavoro eseguito nell'ambito del Contratto CNR Strutture variazionali irregolari.     

Existence and convergence of positive weak solutions of −Δu=u^{\frac{n}{{n - 2}}} in bounded domains of ℝ n,n≥3

For the equation –u= in ⁿ we prove the existence of positive weak solutions with prescribed singular set. We also prove the existence of sequences of weak solutions which converge weakly but not strongly inL ^n/n–2()     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.