Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.

We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y / i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions.

This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L1 functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

Cette thèse concerne la géométrie asymptotique de variétés riemanniennes complètes non compactes, dont la courbure tend vers zéro à l'infini, assez vite. Afin de compléter des travaux déjà existants, on s'attache à comprendre le cas où la croissance du volume est non maximale, c'est-à-dire strictement moins rapide que dans l'espace euclidien de même dimension. Dans ce contexte, on prouve tout d'abord une inégalité de Sobolev à poids et une inégalité de Hardy, qui permettent de généraliser nombre de résultats établis quand la croissance du volume est maximale. On obtient en particulier des résultats de rigidité et de finitude de la topologie pour des variétés Ricci plates et asymptotiquement plates. On obtient ensuite un résultat de structure asymptotique pour une classe d'instantons gravitationnels : typiquement, ceux qui ont une croissance du volume cubique sont asymptotes à des fibrations en cercles au-dessus d'une variété asymptotiquement localement euclidienne .

Cette thèse est consacrée à l'étude de quelques équations aux dérivées partielles non linéaires de type Dirichlet ou Neumann sur un domaine borné régulier, qui sont à structure variationnelle, et<br />qui présentent un défaut de compacité.<br />Dans la première partie, nous étudions une EDP homogène avec un opérateur non linéaire faisant<br />intervenir un poids strictement positif, une non-linéarité critique au sens de Sobolev et un paramètre $\lambda$. Nous établissons des résultats d'existence et de non-existence de solutions qui dépendent du comportement du poids au voisinage de ses minima, du paramètre $\lambda$ et de la géométrie du domaine. Dans la seconde partie, nous nous intéressons à des EDP non homogènes avec poids et avec une non-linéarité critique au bord au sens de l'inclusion de trace. Nous montrons des résultats d'existence qui dépendent des différents<br />coefficients des EDP étudiées et de la courbure moyenne en un point minimum de poids.

Cette thèse est consacrée à l' étude des solutions globales régulières pour deux équations d'évolution semi-linéaire différentes.<br />Dans la première partie, nous étudions les solutions régulières globales d'une équation particulière semi-linéaire faiblement<br />hyperbolique d'ordre quatre . Les linéarisés de cette équation<br />vérifient une hypothèse du type de Levi.<br />Dans la seconde patie, nous donnons des exemples d'opérateurs d'évolution notés L = partial_{tt}^2 - p(t, D_x), faisant intervenir des opérateurs singuliers p pour lesquels une perturbation <br />quasi-linéaire donne des équations admettant des solutions régulières et globales.

Thèse doctorat : Mathématiques : La Rochelle : 2006. / Bibliogr. p. 157-161.