Stationary twists and energy minimizers on a space of measure preserving maps

Let be a bounded Lipschitz domain, a suitably quasiconvex integrand and consider the energy functional

over the space of measure preserving maps

In this paper we discuss the question of existence of multiple strong local minimizers for over . Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called p-Dirichlet energy, i.e., when .     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Periodic solutions of sublinear Liénard differential equations

In this paper, we study the existence of periodic solutions of the second order differential equations x^″+f(x)x^′+g(x)=e(t). Using continuation lemma, we obtain the existence of periodic solutions provided that F(x) () is sublinear when x tends to positive infinity and g(x) satisfies a new condition

where M, d are two positive constants.     

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation ut/|x|s-△pu=uq,(x,t)∈Ω×(0,T), u(x,t)=0,(x,t)∈(?)Ω×(0,T), u(x,0)=u0(x),u0(x)≥0,u0(x)(?)0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, whereΩis a smooth bounded domain in RN(N>p),0∈Ω,△pu=div(|▽u|p-2▽u),0≤s≤2,p≥2,p-1相似文献   

Eigenvalue comparisons for boundary value problems for second order difference equations

We consider the eigenvalue problems for boundary value problems of second order difference equations

(1)

and

(2)

Comparison results for the eigenvalues of the problem (1) and the problem (2) are established.     

Homogenization of degenerate nonlinear parabolic equation in divergence form

In this paper the homogenization of degenerate nonlinear parabolic equations

where a(t,y,λ) is periodic in (t,y), is studied via a weighted compensated compactness result.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the form

where , , f is a nonnegative function and satisfies the Carathéodory conditions on . Here, f can have a time singularity at t=0 and/or t=T and a space singularity at x=0 and/or y=0. We present conditions for the existence of solutions positive on [0,T) and having continuous first derivatives on [0,T].     

Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

Weakly sequentially continuous differentiable mappings

It is well known that a (linear) operator between Banach spaces is completely continuous if and only if its adjoint takes bounded subsets of Y^* into uniformly completely continuous subsets, often called (L)-subsets, of X^*. We give similar results for differentiable mappings. More precisely, if UX is an open convex subset, let be a differentiable mapping whose derivative is uniformly continuous on U-bounded subsets. We prove that f takes weak Cauchy U-bounded sequences into convergent sequences if and only if f^′ takes Rosenthal U-bounded subsets of U into uniformly completely continuous subsets of . As a consequence, we extend a result of P. Hájek and answer a question raised by R. Deville and E. Matheron. We derive differentiable characterizations of Banach spaces not containing ℓ₁ and of Banach spaces without the Schur property containing a copy of ℓ₁. Analogous results are given for differentiable mappings taking weakly convergent U-bounded sequences into convergent sequences. Finally, we show that if X has the hereditary Dunford–Pettis property, then every differentiable function as above is locally weakly sequentially continuous.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations

Let H,V and K be separable Hilbert spaces. In this paper we consider the existence and uniqueness of energy solutions to the following stochastic evolution equation:

where is a linear bounded operator with coercivity, monotone condition and hemicontinuity, and are measurable functions and satisfy the local non-Lipschitz condition proposed by the author [T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992) 152–169].     

Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem

depending on the behaviour at the origin and at infinity of the potential . Besides solutions in W^2,1(0,1), we also consider solutions in which are possibly discontinuous at the endpoints of [0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Nonradial large solutions of sublinear elliptic problems

Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup_|x|=rp(x)−inf_|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)u^γ admits a positive solution in satisfying lim_|x|→∞u(x)=∞ if and only if

     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

In this article, we consider the following eigenvalue problems

('∗

λ' render=n">

where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ(0,λ^*), a unique positive solution if λ=λ^*, and no positive solution if λ>λ^*. We also obtain some further properties of the positive solutions of (*)_λ.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,