Oscillation criteria for boundary value problems of high-order partial functional differential equations

In this paper, a class of boundary value problems associated with high-order partial functional differential equations with distributed deviating arguments is investigated. Some oscillation criteria of solutions to the problem are developed. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional oscillation one by employing some integral means of solutions and introducing some parameter functions. One illustrative example is considered.     

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.     

Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

Elliptic equations with measurable coefficients in Reifenberg domains

We prove W1,p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in (n−1) directions. We assume the boundary to be δ-Reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

Systems of partial differential equations of mixed hyperbolic-parabolic type

We prove the global existence and uniqueness of solutions of certain mixed hyperbolic-parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier-Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Quasineutral limit of a time-dependent drift-diffusion-Poisson model for p-n junction semiconductor devices

In this paper the vanishing Debye length limit of the bipolar time-dependent drift-diffusion-Poisson equations modelling insulated semiconductor devices with p-n junctions (i.e., with a fixed bipolar background charge) is studied. For sign-changing and smooth doping profile with ‘good’ boundary conditions the quasineutral limit (zero-Debye-length limit) is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods. The key point in the proof is to introduce a ‘density’ transform and two λ-weighted Liapunov-type functionals first, and then to establish the entropy production integration inequality, which yields the uniform estimate with respect to the scaled Debye length. The basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation.     

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior boundary value problems of the steady-state oscillation theory for anisotropic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved.     

Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.     

Interval oscillation criteria for second order partial differential equations with delays

In this paper, we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [_t0,∞)$[t_0,\infty )$, rather than on whole half-line. Our results are sharper than some previous results and handles the cases which are not covered by known criteria.     

On the Palais–Smale condition

We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes condition [8] to obtain Cerami condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth condition, while the strong superlinear condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.     

Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion

In this paper, we provide a blow-up mechanism to the modified Camassa–Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa–Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa–Holm equation on the blow-up phenomena is rather delicate.     

Completeness Theorems for a Non-Standard Two-Parameter Eigenvalue Problem

We consider two simultaneous Sturm-Liouville systems coupled by two spectral parameters. However, unlike the standard multiparameter problem, we now suppose that the principal part of each of the differential operators is multiplied by a different parameter. In a recent paper, Faierman and Mennicken derived various results concerning the eigenvalues and eigenfunctions, and in particular, they established the oscillation theory for this system. Here we continue this investigation focusing on the completeness of the set of eigenfunctions in a suitable function space. If either one of the potentials is identically zero, the completeness of the eigenfunctions is established, whereas, if this condition fails, then we show the existence of an essential spectrum having non-zero points. The completeness problem for this latter case will be left for a later work. M?ller and Watson supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory.     

Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.     

A bound for minimal graphs with a normal at infinity

It is shown that a minimal graph with a normal at infinity is in a-priori bounded vertical distance from its approximating halfcatenoid. This is used to show that the exterior contact angle problem is wellposed under natural geometric conditions on the domain, while the exterior Dirichlet problem can be solvable only for data which satisfy an oscillation bound.This paper was written under the support of the Deutsche Forschungsgemeinschaft while the author was visiting the department of mathematics at Stanford University.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD²u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D²u∈VMO.     

The best Sobolev trace constant in a domain with oscillating boundary

In this paper we study homogenization problems for the best constant for the Sobolev trace embedding W^1,p(Ω)?L^q(∂Ω)$W^{1,p}\left(\Omega \right)?L^q\left(\partial \Omega \right)$ in a bounded smooth domain when the boundary is perturbed by adding an oscillation. We find that there exists a critical size of the amplitude of the oscillations for which the limit problem has a weight on the boundary. For sizes larger than critical the best trace constant goes to zero and for sizes smaller than critical it converges to the best constant in the domain without perturbations.     

On the singular set of the parabolic obstacle problem

This paper is devoted to regularity results and geometric properties of the singular set of the parabolic obstacle problem with variable right-hand side. Making use of a monotonicity formula for singular points, we prove the uniqueness of blow-up limits at singular points. These results apply to parabolic obstacle problem with variable coefficients.     

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].