On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30

Lp Microlocal Properties for Multi-Quasi-Elliptic Pseudodifferential Operators

2010 Mathematics Subject Classification: Primary 35S05; Secondary 35A17.

Crack Theory with Singularities at the Boundary

2002 Mathematics Subject Classification: 35S15, 35J70, 35J40, 38J40

Anomalous Singularities for Hyperbolic Equations with Degeneracy of Infinite Order

2002 Mathematics Subject Classification: 35L80

Existence of Solutions for a Class of Quasi-Linear Singular Integro-Differential Equations

2000 Mathematics Subject Classification: 45F15, 45G10, 46B38.

A Note on Div-Curl Lemma

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.

Generalized Backscattering and the Lax-Phillips Transform

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.

Solutions Globales Régulières pour Quelques Équations Lineaires D'évolution du Type Pseudo-Différentiel Singulier

2000 Mathematics Subject Classification: 35C15, 35D05, 35D10, 35S10, 35S99.

Il concetto di funzione: storia, applicazioni, sviluppi

Il concetto di funzione è tra i più rilevanti, ma anche tra i più controversi concetti matematici. In questo lavoro di tesi si è esaminato questo concetto a partire dalle sue origini fino ad arrivare alla definizione bourbakista, che è quella insegnata a tutti gli studenti a partire dalla scuola secondaria fino ad arrivare all'università. Successivamente si è analizzato in che modo questo delicato concetto viene presentato agli studenti delle scuole secondarie di secondo grado, osservando come le recenti Indicazioni Nazionali e Linee Guida danno suggerimenti per affrontare questo argomento, anche esaminando alcuni libri di testo. Infine si è descritto come il concetto di funzione abbia preso, in tempi relativamente recenti, un respiro più ampio dando luogo all'analisi funzionale, laddove le funzioni non sono più viste come corrispondenza punto a punto ma come oggetti che vengono osservati globalmente. Si considereranno infatti nuovi spazi i cui elementi sono funzioni.

Chemical and mineral composition of rocks from the Philippine Sea

The book is devoted to geology of the Philippine Sea floor. This region is studied most extensively among other marginal seas of the Pacific Ocean. Rocks of the sedimentary and basalt layers within this sea have been studied during five legs of D/S Glomar Challenger. International geological expedition on board R/V Dmitry Mendeleev carried out according to the Project ''Ophiolites of Continents and Comparable Rocks of the Ocean Floor''obtained unique collection of rocks from the second and third layers of the ocean crust in the Philippine Sea. The book provides detailed petrographic and geochemical description of igneous and sedimentary formations from the Philippine Sea and compares them with rocks of the continental ophiolite association. An analysis of structure and history of the ocean crust formation in the region is based on all known geological information. The main periods of tectonic movement activation and nature of their manifestations within the sea are shown.

High-pressure experiments on a natural olivine gabbro from ODP Hole 176-735B

We have conducted high-pressure experiments on a natural oceanic gabbro composition (Gb108). Our aim was to test recent proposals that Sr-enrichment in rare primitive melt inclusions from Mauna Loa, Hawaii, may have resulted from melting of garnet pyroxenite formed in the magma source regions by reaction of peridotite with siliceous, Sr-enriched partial melts of eclogite of gabbroic composition. Gb108 is a natural, Sr-enriched olivine gabbro, which has a strong positive Sr anomaly superimposed on an overall depleted incompatible trace element pattern, reflecting its origin as a plagioclase-rich cumulate. At high pressures it crystallises as a coesite eclogite assemblage, with the solidus between 1,300 and 1,350°C at 3.5 GPa and 1,450 and 1,500°C at 4.5 GPa. Clinopyroxenes contain 4-9% Ca-eskolaite component, which varies systematically with pressure and temperature. Garnets are almandine and grossular-rich. Low degree partial melts are highly siliceous in composition, resembling dacites. Coesite is eliminated between 50 and 100°C above the solidus. The whole-rock Sr-enrichment is primarily hosted by clinopyroxene. This phase dominates the mode (>75 wt%) at all investigated PT conditions, and is the major contributor to partial melts of this eclogite composition. Hence the partial melts have trace element patterns sub-parallel to those of clinopyroxene with ~10* greater overall abundances and with strong positive Sr anomalies. Recent studies of primitive Hawaiian volcanics have suggested the incorporation into their source regions of eclogite, formerly gabbroic material recycled through the mantle at subduction zones. The models suggest that formerly gabbroic material, present as eclogite in the Hawaiian plume, partially melted earlier than surrounding peridotite (i.e. at higher pressure) because of the lower solidus temperature of eclogite compared with peridotite. This produced highly siliceous melts which reacted with surrounding peridotite producing hybrid pyroxene + garnet lithologies. The Sr-enriched nature of the formerly plagioclase-rich gabbro was present in the siliceous partial melts, as demonstrated by these experiments, and was transferred to the reactive pyroxenite. These in turn partially melted, producing Sr-enriched picritic liquids which mixed with normal picritic partial melts of peridotite before eruption. On rare occasions these mixed, relatively Sr-rich melts were trapped as melt inclusions in primitive olivine phenocrysts.Yaxley-Sobolev

Funzioni BV e disuguaglianze isoperimetriche

All'interno della mia tesi verrà introdotta la teoria delle funzioni in R^{n} a variazione limitata (BV), seguendo le presentazioni di Lawrence C.Evans e Ronald F.Gariepy nel libro Measure Theory and Fine Properties of Functions e di Enrico Giusti nell'opera Minimal Surfaces and Functions of Bounded Variation. Le funzioni BV sono funzioni le cui derivate prime deboli sono misure di Radon, ossia misure di Borel regolari finite sui compatti. In particolare verranno anche analizzati gli insiemi E che hanno perimetro finito, ossia tali che la funzione indicatrice dell’insieme E sia una funzione BV. Nello specifico, nel primo capitolo verranno date le definizioni di funzioni BV e insiemi di perimetro finito, sia in una versione globale che in una locale, verrà enunciato un primo importante teorema per le funzioni BV e verrà analizzata la relazione tra funzioni di Sobolev e funzioni BV. Nel secondo capitolo, invece, verranno analizzate la semicontinuità inferiore, l'approssimazione con funzioni lisce e la compattezza di funzioni BV, mentre nel terzo capitolo verranno elencati alcuni risultati sulle funzioni BV riguardanti la Traccia, l'Estensione e la formula di Coarea. Infine, nel quarto ed ultimo capitolo, verranno studiate le disuguaglianze di Sobolev e Poincaré e le disuguaglianze isoperimetriche per funzioni BV.

On the suboptimality of the p-version interior penalty discontinuous Galerkin method

We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p- IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.

Second-order elliptic PDE with discontinuous boundary data

We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive an L^2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h- and hp-version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.

Lipschitz regularity for weak solutions of parabolic p-Laplacian type equations in certain subriemannian structures

The main aim of the thesis is to prove the local Lipschitz regularity of the weak solutions to a class of parabolic PDEs modeled on the parabolic p-Laplacian. This result is well known in the Euclidean case and recently has been extended in the Heisenberg group, while higher regularity results are not known in subriemannian parabolic setting. In this thesis we will consider vector fields more general than those in the Heisenberg setting, introducing some technical difficulties. To obtain our main result we will use a Moser-like iteration. Due to the non linearity of the equation, we replace the usual parabolic cylinders with new ones, whose dimension also depends on the L^p norm of the solution. In addition, we deeply simplify the iterative procedure, using the standard Sobolev inequality, instead of the parabolic one.