Second-order Inequalities Research Articles

A Liouville comparison principle for semilinear elliptic second-order inequalities

We consider semilinear elliptic second-order partial differential inequalities of the form (*) in the whole space , where , q>0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are measurable and locally bounded such that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of capacities associated with the operator L for solutions of (*) which are measurable and belong locally in to a Sobolev-type function space also associated with the operator L.

Nonclassical stationary and nonstationary problems with weight Neumann conditions for singular equations with KPZ-nonlinearities

ABSTRACT From a unique viewpoint, singular elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms are investigated in cylindrical domains. The weight Neumann condition is set on the lateral area of the cylinder; no condition is set on the base of the cylinder (regardless the type of the equation). Results of two kinds are established: the existence of a limit of each solution (if it exists) along the axis of the cylinder and sufficient conditions of a blow-up (including instant or complete one).

Examining the weak cosmic censorship conjecture of RN-AdS black holes via the new version of the gedanken experiment

Based on the new version of the gedanken experiment proposed by Sorce and Wald, we investigate the weak cosmic censorship conjecture (WCCC) for Reissner-Nordström-AdS (RN-AdS) black holes under the second-order approximation of the perturbation that comes from the matter fields. First of all, we propose that the cosmological constant is a portion of the matter fields. It means that the cosmological constant can be seen as a variable, and its value will vary with the process of the perturbation. Moreover, we assume that the perturbation satisfies the stability condition, which states that after the perturbation, the spacetime geometry also belongs to the class of RN-AdS solutions. Based on the stability condition and the null energy condition, while using the method of the off-shell variation, the first two order perturbation inequalities are derived. In the two inequalities, the term which contains the thermodynamic pressure and volume is involved firstly. Furthermore, based on the first-order optimal option and the second-order inequality, the WCCC for nearly extremal RN-AdS black holes is examined. It is shown that considering the cosmological constant varying with the perturbation, the WCCC for RN-AdS black holes cannot be violated under the second-order approximation of the matter fields perturbation.

A Paneitz–Branson type equation with Neumann boundary conditions

Abstract We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

Oscillatory Behavior of Third-Order Nonlinear Difference Equations with a Nonlinear-Nonpositive Neutral Term

We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form: $$\begin{aligned} \Delta \left( \left( a(t)\left( \Delta ^{2}\left( x(t)-p(t)x^{\alpha }(t-k) \right) \right) ^{\gamma } \right) +q(t)x^{\beta }(t-m+1)=0,\right. \end{aligned}$$ with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with $$\hbox {p (t)} = 0$$ . Examples are given to illustrate the main results.

On absence of global solutions of quasilinear differential-convolutional inequalities

ABSTRACTFrom a unique viewpoint, elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms and nonlocal convolutional terms arising in the description of reaction–diffusion processes, neural networks, and nonlocal phase transitions are investigated. Sufficient conditions of the absence of their global solutions, i.e. necessary conditions of their global solvability, are found.

Sharp Constant in Third-order Hardy–Sobolev–Maz’ya Inequality in the Half Space of Dimension Seven

Abstract This paper is concerned with the sharp constant in higher-order Hardy–Sobolev–Maz’ya inequalities in the half space $\mathbb{R}^{n}_{+}$. These inequalities were recently established by Lu and Yang [26]. We prove that, in the case of dimension seven, the sharp constant in the third-order Hardy–Sobolev–Maz’ya inequality coincides with the sharp third-order Sobolev constant. This provides an analogue of the corresponding results of Benguria, Frank, and Loss [3] and Lu and Yang [26], in which the sharp constant in the first-order inequality in dimension three and the sharp constant in the second-order inequality in dimension five are addressed respectively.

Second-Order Inequality: Clarifying and Modeling the Gini Coefficient with Measures of Internal Asymmetry

This paper presents a physical model of the Gini coefficient and its corresponding Lorenz curve. If the Lorenz curve is scaled to 1, then 1 represents gross domestic income, gross domestic product, or societal wealth. The value 1 also represents total population. On these assumptions, the value of x ∈ [0, 1] where the first derivative of the Gini coefficient equals one represents the population quantile that enjoys per capita income or wealth. This paper also describes methods for evaluating the internal asymmetry of any distribution corresponding to a particular Gini coefficient. It concludes with worked examples from Oxfam’s survey of global inequality and from French data on wealth inequality.

Some properties of solutions for an isothermal viscous Cahn\u2013Hilliard equation with inertial term

In this paper, we study the global existence and blow-up of solutions for an isothermal viscous Cahn–Hilliard equation with inertial term, which arises in isothermal fast phase separation processes. Based on the Galerkin method and the compactness theorem, we establish the existence of the global generalized solution. Using a lemma on the ordinary differential inequality of second order, we prove the blow-up of the solution for the initial-boundary problem.

Non-divergence parabolic equations of second order with critical drift in Lebesgue spaces

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift−ut+Lu=−ut+∑ijaijDiju+∑biDiu=0(≥0,≤0) in some domain Q⊂Rn+1. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift b is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is a “universal” spectral gap for the associated elliptic operator. The counterpart for uniformly elliptic equations of second order in non-divergence form is shown in [19].

A new Picone’s dynamic inequality on time scales with applications

In this paper, we will derive a new dynamic Picone-type inequality for half-linear dynamic equations and dynamic inequalities of second order on an arbitrary time scale T. As a consequence, we will apply this new Picone inequality to get a new Wirtinger-type inequality on time scales with two different weighted functions. The results contain the Wirtinger inequalities formulated by Beesack, Lee and Jaroš for the continuous case. For the discrete case our results are also new.

The equivalence of viscosity and distributional subsolutions for convex subequations — a strong Bellman principle

There are two useful ways to extend nonlinear partial differential inequalities of second order beyond the class of C 2 functions: one uses viscosity theory and the other uses the theory of distributions. This paper considers the convex situation where both extensions can be applied. The main result is that under a natural “second-order completeness” hypothesis, the two sets of extensons are isomorphic, in a sense that is made precise.

Regularity for nonlinear variational inequalities of hyperbolic type

In this paper, we deal with the regularity for nonlinear variational inequalities of second order in Hilbert spaces with more general conditions on the nonlinear terms and without condition of the compactness of the principal operators. We also obtain the norm estimate of a solution of the given nonlinear equation on C ( [ 0 , T ] ; V ) ∩ C 1 ( ( 0 , T ] ; H ) ∩ C 2 ( ( 0 , T ] ; V ∗ ) by using the results of its corresponding hyperbolic semilinear part.

On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation

As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-∞). Our results hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity, which at the moment have only been established in certain particular cases. Among the most important steps in our proof are 1) quantitative variants of Boltzmann’s H-theorem, as proven in [52,60], based on symmetry features, hypercontractivity and information-theoretical tools; 2) a new, quantitative version of the instability of the hydrodynamic description for non-small Knudsen number; 3) some functional inequalities with geometrical content, in particular the Korn-type inequality which we established in [22]; and 4) the study of a system of coupled differential inequalities of second order, by a treatment inspired from [21]. We also briefly point out the particular role of conformal velocity fields, when they are allowed by the geometry of the problem.

Blow-up results for some nonlinear hyperbolic problems

We study the non-existence of global solutions for equations and inequalities of the following type ∂ttu ≥ ΔA(x, t, u) + B(x, t, u) on RN+1 + with nonlinearity A and B of the form A(x, t, u) = ρ1(x, t)|u|p , B(x, t, u) = ρ2(x, t)|u|q. We also study the nonexistence for some more general inequalities of second order but without any assumption about the type of operator. The method relies on a suitable choice of test functions, resealing techniques and a dimensional analysis.

Almost Sure Stabilizability of Controlled Degenerate Diffusions

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton--Jacobi--Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory.

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.

ON NONLINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES OF SECOND ORDER

In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, an existence theorem for the Dirichlet boundary value problem is proved.

Nonlinear hemivariational inequalities of second order using the method of upper-lower solutions

In this paper we examine a nonlinear hemivariational inequality of second order. The differential operator is set-valued, nonlinear and depends on both x and its gradient Dx. The same is true for the zero order term f, while the right-hand side nonlinearity satisfies a one-sided Lipschitz condition. We use the method of upper and lower solutions, coupled with truncation and penalization techniques and the fixed point theory for multifunctions in an ordered Banach space.

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and 2 2 (M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H 2 2 (M), $$\left\| u \right\|_{2^\sharp }^2 \leqslant A\left\| {\Delta _g u} \right\|_2^2 + B\left\| u \right\|_{H_1^2 }^2 $$ where 2#=2n/(n−4) is critical, and\(\left\| \cdot \right\|_{H_1^2 } \) is the usual norm on the Sobolev space H 1 2 (M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K 0 2 where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality\(\left\| u \right\|_{2^\sharp } \leqslant K\left\| {\nabla u} \right\|_2 \). We prove in this article that for any compact Riemannian manifold, A=K 0 2 is attained in the above inequality.