We provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the form where Hu denotes the Hessian matrix of u.

Let u be a bounded slowly oscillating mild solution of an inhomogeneous Cauchy problem, u̇ (t) = Au (t) + f (t), on R or R+, where A is a closed operator such that σap (A) ∩iR is countable, and the Carleman or Laplace transform of f has a continuous extension to an open subset of the imaginary axis with countable complement. It is shown that u is (asymptotically) almost periodic if u is totally ergodic (or if X does not contain c0 in the case of a problem on R). Similar results hold for second-order Cauchy problems and Volterra equations.

In this paper we give a local classification of the integral curves of implicit differential equations where F is a smooth function and p = dy/dx, at points where Fp = 0, Fpp ≠ 0 and where the discriminant {(x, y) : F = Fp = 0} has a Morse singularity. We also produce models for generic bifurcations of such equations and apply the results to the differential geometry of smooth surfaces. This completes the local classification of generic one-parameter families of binary differential equations (BDEs).

We consider the system of reaction-diffusion equations known as the Sel'kov model. This model has been applied to various problems in chemistry and biology. We obtain a priori bounds on the size of the positive steady-state solutions of the system defined on bounded domains in Rn, 1 ≤ n ≤ 3 (this is the physically relevant case). Previously, such bounds had been obtained in the case n = 1 under more restrictive hypotheses. We also obtain regularity results on the smoothness of such solutions and show that non-trivial solutions exist for a wide range of parameter values.

Focal point and conjugacy criteria for the half-linear second-order differential equation are obtained using the generalized Riccati transformation. An oscillation criterion is given in case when the function c(t) is periodic.

We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problem we prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

Sufficient conditions are obtained for a ring R, faithfully graded by a bisimple inverse semigroup S, to be (a) prime and (b) right primitive, these conditions being on the subring RG consisting of all elements of R with support contained in G, a maximal subgroup of S. Earlier results on semigroup rings arise as special cases.

It is known that the condition ‘either ∂L (F) ≠ Ø or there exist υ1,…,υq ∈ Rnsuch thatF ∈ int co {υ1,…,υq} characterizes solvability of the problem with f(·) = 〈F,·〉.

We extend this result to the case of lower semicontinuous integrands L : Rn → R.

We also show that validity of this condition for all F ∈ Rn is both a necessary and sufficient requirement for solvability of all minimization problems with sufficiently regular Ω and f. Moreover, the assumptions on Ω and f can be completely dropped if L has sufficiently fast growth at infinity.

The main result of this paper is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets. As an application, the polynomial vector fields admitting certain types of algebraic integrating factors are characterized.

The entire positive solutions of a conformally invariant biharmonic equation in Rn will be classified using the method of moving spheres. As a byproduct, one also shows that any entire non-negative solution of the equation Δ2u = up with 1 ≤ p < (n + 4)/(n−4) with n ≥ 2 is zero.