Maximal Regularity

We give some results about the optimal regularity of a solution to a parabolic equation, set in non cylindrical domains U=⋃t∈]0,1[{t}× It with It={x:0<x<ϕ(t)}. Two models are studied. In the first, the function ϕ(t)=tα with α>1/2 is considered and the optimal regularity is obtained when the second member is regular. We use Labbas and Terreni's results [7]. This study is generalized

Problems of scheduling n jobs on a single machine to maximize regular objective functions are studied. Precedence constraints may be given on the set of jobs and the jobs may have different release times. Schedules of interest are only those for which the jobs cannot be shifted to start earlier without changing job sequence or violating release times or precedence constraints. Solutions to the maximization problems provide an information about how poorly such schedules can perform. The most general problem of maximizing maximum cost is shown to be reducible to n similar problems of scheduling n−1 jobs available at the same time. It is solved in O(mn+n 2) time, where m is the number of arcs in the precedence graph. When all release times are equal to zero, the problem of maximizing the total weighted completion time or the weighted number of late jobs is equivalent to its minimization counterpart with precedence constraints reversed with respect to the original ones. If there are no precedence constraints, the problem of maximizing arbitrary regular function reduces to n similar problems of scheduling n−1 jobs available at the same time.

Let Ω be a smooth open bounded set in RN, let ϱ be the (smoothed in the interior) distance function from ∂Ω, let (aij) be a uniformly elliptic matrix with continuous entries in Ω and A the associated second order elliptic operator. Under suitable conditions, we prove that the operator L=−ϱA+B, with B a first order operator with continuous coefficients,

Let Mμ be the set of all probability densities equivalent to a given reference probability measure μ. This set is thought of as the maximal regular (ie, with strictly positive densities) μ-dominated statistical model. For each f∈ Mμ we define (1) a Banach space Lf with unit ball Vf and (2) a mapping sf from a subset Uf of Mμ onto Vf, in such a way that the system (sf, Uf, f∈ Mμ) is an affine atlas on Mμ. Moreover each parametric exponential model dominated by μ is a finite-dimensional affine submanifold and each parametric statistical model ...

We study the generation of an analytic semigroup in Lp(Rd) and the de- termination of the domain for a class of second order elliptic operators with unbounded coecien ts in Rd. We also establish the maximal regularity of type Lq{Lp for the corre- sponding inhomogeneous parabolic equation. In contrast to the previous literature the coecien ts of the second derivatives

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in Lp and Cs for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the

The maximal regularity and continuation of the solution (and its fractional-order derivative) of the Cauchy problem of the non-homogeneous fractional order evolution equation D uğtŞ ¼ AuğtŞ ş fğtŞ, 2 ğ0, 1Ş have been studied in [AMA El-Sayed and Mohamed. AE Herzallah (2004). ...

For any proper polynomial mapf: C k →C k define the function α as $$\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$$ Letf=(P 1,...,P k ) be a proper polynomial map. We define a notion ofs-regularity using the extension off to Pk. Whenf is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 andd 1,...,d k , whered i :=degP i . We then describe dynamically the sets {α≤d i }. We give a concrete description of regular maps. Ifd i >1, this allows us to construct the equilibrium measure μ associated withf as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π,s)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.

The presence of surfactants, ubiquitous at most fluid/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface — so-called Marangoni effects. The mathematical model governing the dynamics of such systems is studied for the case in which the surfactant is soluble in one of the adjacent bulk phases. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk fluids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. Employing Lp-maximal regularity we obtain well-posedness of this model for a certain initial configuration. The proof is based on recent Lp-theory for two-phase flows without surfactant.