Volume 13, Issue 1

The paper deals with the maximal regularity of solutions of some semi-linear equations in function spaces of type and . A new method, based on quarkonial decompositions of functions, is presented.

Let Y be an integral scheme and fix locally free 𝒪 Y – sheaves ℰ, 𝒜 and ℬ of ranks 5, 3 and 3 respectively. Consider the projective bundle and a morphism δ : π *𝒜 → π *ℬ(1) X ≔ D 1 ( δ ) be the locus of points x ∈ ℙ where rk( δ x ) ≤ 1. Then the map ϱ ≔ π | X : X → Y is a cover of degree d = 6 if dim( X ∩ π –1 ( y )) = 0 for each y ∈ Y . We call such a cover scandinavian . We prove a Bertini – type theorem and we give some examples of scandinavian and non scandinavian covers of degree 6.

We propose a new asymptotic Dirichlet problem for harmonic maps via rough isometry on a certain class of Riemannian manifolds. We also prove that this problem is solvable for naturally defined class of data maps. This result generalizes those of Avilés, Choi and Micallef and of Choi and the present authors.

The question of extending L p -sub-Markovian semigroups to the spaces L q , q > p , and the interpolation of L p -sub-Markovian semigroups with Feller semigroups is investigated. The structure of generators of L p -sub-Markovian semigroups is studied. Subordination in the sense of Bochner is used to discuss the construction of refinements of L p -sub-Markovian semigroups. The rôle played by some function spaces which are domains of definition for L p -generators is pointed out. The problem of regularising powers of generators as well as some perturbation results are discussed.

We give conditions on a strongly continuous semigroup 𝒯 and an unbounded perturbation B in the class of Miyadera-Voigt such that the perturbed semigroup 𝒮 inherits asymptotic properties of 𝒯 as boundedness, asymptotic almost periodicity, uniform ergodicity and total uniform ergodicity. A systematic application of the abstract result to partial differential equations with delay is made.