一类次P-Laplace方程非负解的不存在性

该文得到了在Ω上以下问题 {L_p,_ku+f(u)=0, , u|∂Ω=0 非负解的不存在性结果. 其中Ω为Heisenberg型群G中的区域(有界或无界), L_{p, k}u=divX (| _X u|^p-2 _X u)为对应于Greiner型向量场 X 的一类次P-Laplace算子.     

一类非线性分数阶多点边值问题的可解性

研究下面一类非线性分数阶微分方程多点边值问题■通过应用Mawhin重合度理论得到解的存在性结果.此结论拓展了在分数阶多点边值问题这个领域的以前的结果.     

GlobalL 2-bifurcation of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem $$ - u ''(x) = f(u(x)) - \mu u(x), 0< x< 1, u(0) = u(1) = 0.$$ Let (u _n,(μ,x), μ), (n ωN) be a solution pair and α₂ (μ)=∥μ_μ∥2. The purpose of this paper is to study the globalL ²-bifurcation, that is, to establish an asymptotic formula of α_n(υ) as μ → ∞. Furthermore, we give an asymptotic formula ofu _n(μ,x) as μ → ∞.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

On solution sets of multivalued differential equations

Let X be a Banach space, 2^x\? the nonempty subsets of X,J = [o,a]?R and F:J×X→2^x\? a multivalued map. We consider U′ ? F(t,u) a.e. on J, u(o) = X_p ? X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X₀ + ∫^{0 t} u′(s)ds on J and u′(t)?F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ? in C_X (J), the space of continuous v : J → X with ∣v∣₀ = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact R_δ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.     

Unbeschränkte Randwerte bei parabolischen Differentialgleichungen

In this paper we will prove the existence of classical solutions u for a quasilinear parabolic differential equation of type $$(1)u_t = \sum\limits_{i = 1}^n {\tfrac{d}{{dx_i }}a_i (x,t,u,u_x )} + a(x,t,u,u_x )$$ in a cylindrical domain G, whereby unbounded boundary values g may be given on the parabolic boundary R_p of G. In general, u is unbounded and u_x?L₂(G). Under certain conditions (see Satz 3) we nevertheless get a reasonable boundary-behavior of u, that is: \(u(Q) \to g(\bar Q)\) when g is continuous in \(\bar Q \in {\text{R(}}Q \to \bar Q{\text{)}}\) , and u(Γ)→g in the L₂-sense for Γ→R_p where Γ means a parallel surface to R_p.     

An identification problem for a semilinear parabolic equation

Sunto Si considera un problema sovradeterminato per l'operatore parabolico semilineare D(u)=D_tu – D _x ² – a(u) contenente un termine incognito a(u) e si prova l'esistenza di almeno una soluzione (u, a).

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Gli autori sono membri del Gruppo Nazionale per l'Analisi Funzionale e le Applicazioni del C.N.R.

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Lavoro parzialmente finanziato dal Ministero della Pubblica Istruzione.     

R~N上一类p(x)-Laplacian方程的无穷多解问题

该文主要讨论了如下p(x)-Laplacian算子方程的解.其中1P-≤p(x)≤P+N.得到了上述方程在变指数Sobolev空间W~(1,p(x))(R~N)中的一列能量值趋向正无穷的解.     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete

An expression of the form $$l(u) = ( - 1)^m \sum\nolimits_{j = 1}^m {D_j^{2m} } u + [q(x)] + ir(x)]u$$ is considered. Sufficient conditions are found such that the minimum operator, formally conjugate tol(u), generated by the expression and the maximum operator generated by the expressionl(u) in ?₂ (E_n) should coincide. It is proved that if q(x)→∞ or q(x)+ r(x)→∞, ¦x¦→∞, then the operator generated byl(u) in ?₂ (E_n) has a discrete spectrum.     

On two-dimensional quasi-linear elliptic systems

The author shows the existence of a Hölder continuous solution for a class of two-dimensional non-linear elliptic systems of the type $$ - \Sigma _{i = 1}^2 \partial _i a_i (x,u,\triangledown u) + a_o (x,u,\triangledown u) = 0.$$ The principal part of the equation is required to satisfy a condition of uniform ellipticity and need not be in diagonal form. The lower order term a_o has at most quadratic growth in ?u and satisfies a one-sided condition a_o(x,u,βu) u≥?K or appropriate generalizations.     

Existence locale de solutions c∞ pour des equations de monge-ampere changeant de type

We prove the C∞ local solvability of the n-dimensional Monge-A mpé:ere equation det (u_ij + a_ij(x,u,▽u))= K(x) f(x, u, ▽u), f> 0, in a neighborhood of any point x₀ where K(x₀)= 0 but dK(x_o) ¦0.     

THE POINCARÉ INEQUALITY FOR A VECTOR FIELD WITH ZERO TANGENTIAL OR NORMAL COMPONENT ON THE BOUNDARY

ABSTRACT

Poincaré's inequality is well known: given a bounded domain G, ∥u∥_p ? c∥?u∥_p provided u(x) vanishes on the boundary ?G. The case where u(x) is a vector field u(x) that does not vanish on the boundary ?G is considered. It is shown that when either the tangential component or the normal component vanishes on the boundary ?G, then the Poincaré inequality is satisfied.     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

Phénomène d’explosion totale pour l’équation de la chaleur avec non-linéarité du type puissance

In this Note, we are interested in the possible continuation after the blow-up time T_m of radially symmetric positive classical solutions u of the heat equation with nonlinearity f(u) = u^p, where p > 1. We say that u blows up completely after T_m if u can not be extended beyond T_m (even in the weak sense). We obtain a complete blow up criterion which relies on the asymptotic behaviour of u around the blow-up singularity x = 0.     

Inner-shell photoabsorption spectra

Based on multi-scattering self-consistent field method, the photoabsorption spectra near C 1s threshold of CH₄(near-threshold structure) have been studied with the broken symmetry from To. The most possible geometry of the core excited CH₄** is determined as the C_3v symmetry; the core excited CH₄** molecules have the same bond angle (109.5^o) as that of the ground state CH₄ and consist of one shorter bond length (1.91 a. u.) and three longer bond lengths (2.04 a. u.). The three longer bond lengths are almost the same as that of the ground state CH₄, and the averaged bond length (2.01 a. u.) of core excited CH₄** is shorter than that of the ground state CH₄(2.06 a. u.).     

Comportement asymptotique des solutions d'un système elliptique conservatif non linéaire

Let M be a compact riemannian manifold; in a previous article we show that every non- negative solution of u_tt + Δ_g u=f(u) on M ×R ⁺, satisfying Dirichlet or Neumann boundary conditions, converges to a (stationary) solution Φ of Δ_g Φ=f(Φ) with exponential decay of ∥u - Φ∥c²(M), if we assume that f behaves like r? r^p - λr. We extend this result to a system in the following form $$\left\{ {\begin{array}{*{20}c} {u_{tt} + \Delta _g u + \alpha u - G_x (u,\upsilon ) = 0,} \\ {u_{tt} + \Delta _g \upsilon + \beta \upsilon - G_x (u,\upsilon ) = 0,} \\ \end{array} } \right.$$ . where G satisfies some growth and convexity properties.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

Long time classical solutions of a nonlinear wave equation

The existence and uniqueness of long time classical solutions of the Cauchy problem u_{t t}+μu_t = div(a(u)▽u), where a(u) = 1+u and μ ≥ 0, are studied for the case of two space dimensions. Let the initial data u(0,.) = φ and u_t(0,.) = ψ be supported compactly on R². Then for every T > 0, such a solution exists on [0,T] whenever (φ,ψ) is small enough in H⁴ (R²) x H³(R²). A result on the asymptotic relation between the maximal T and the size of the initial data is given.     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.