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Local Monotonicity and Isoperimetric Inequality on Hypersurfaces in Carnot groups
Full text linkLet G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument.We stress however that there are many dierences, due to our non-Euclidean setting.Some of the tools developed ad hoc are, in order, a \blow-up" theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1 H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups.Let G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument.We stress however that there are many dierences, due to our non-Euclidean setting.Some of the tools developed ad hoc are, in order, a \blow-up" theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups
Sulla Regolarità delle soluzioni di viscosità dell'equazione iconale
Full text linkWe study the regularity of a viscosity solution of equations of eikonal type in two dierent framework: we consider either the non-degenerate eikonal equation under low regularity assumptions on the data or a possibly degenerate eikonal equation with datahaving analytic regularity. Furthermore, we describe the structure of the singular set.We study the regularity of a viscosity solution of equations of eikonal type in two dierent framework: we consider either the non-degenerate eikonal equation under low regularity assumptions on the data or a possibly degenerate eikonal equation with data having analytic regularity. Furthermore, we describe the structure of the singular set
I Teoremi di Campbell, Baker, Hausdorff e Dynkin. Storia, Prove, Problemi Aperti
Full text linkThe aim of this lecture is to provide an overview of facts and references about past and recent results on the Theorem of Campbell, Baker, Hausdorff and Dynkin (shortcut as the CBHD Theorem), following the recent preprint monograph [13]. In particular, we shall give sketches of the following facts: A historical précis of the early proofs (see also [1]); the statement of the CBHD Theorem as usually given in Algebra and that employedin the Analysis of linear PDE’s; a review of proofs of the CBHD Theorem (as given by: Bourbaki; Hausdorff; Dynkin; Varadarajan) together with a unifying demonstrational approach; an application to the Third Theorem of Lie (in local form). Some new resultswill be also commented: The intertwinement of the CBHD Theorem with the Theorem of Poincaré-Birkhoff-Witt and with the free Lie algebras (see [12]); recent results on optimal domains of convergence
Capacità d'insiemi su alberi, grafi e spazi metrici Ahlfors-regolari
Full text linkWork in collaboration with R. Rochberg E. Sawyer and B. Wick [ARSW]. We show that the potential theory of Bessel-type kernels on Ahlfors-regular metric spaces is equivalent, in a precise sense, to potential theory on trees. The basis of this work was in [ARS2], where the relationship between discrete potential theory and some classical function theory was considered. Some applications [Ar] to estimation of sets' and condenser's capacity are discussed
Risultati di regolarità per il problema dell'ostacolo relativo ad equazioni di Kolmogorov degeneri
Full text linkWe prove optimal regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. This problem arises in nancial mathematics, when considering path-dependent derivative contracts with the early exercise feature
Sull’equazione det Du = f senza ipotesi di segno
Full text linkWe consider the nonlinear problemdet?u (x) = f (x) x ? u (x) = x x ? ?where k ? 1 is an integer, is a bounded smooth domain in Rn and f ? Ck ???? satisfiesZf (x) dx = meas.The positivity of f is a standard assumption in the literature. In a recent joint paper withB.Dacorogna and O.Kneuss (EPFL) we prove the existence of a solution u ? Ck ????;Rn with no assumptions on the sign of f. Here we state this theorem together with somerelated results and we outline the main features of the problem
Vector valued Fourier multipliers and applications
Full text linkIn questo seminario sono illustrati alcuni recenti sviluppi della teoria dei moltiplicatori di Fourier negli spazi L^p a valori in spazi di Banach. Seguono alcune applicazioni a problemi al contorno di tipo ellittico e a problemi misti di tipo parabolico
Risultati di perturbazione per operatori lineari multivoci ed applicazioni
Full text linkoai:journals.unibo.it:article/2226Pertubation results for generators of infinitely differentiable semigroups of linear operators are given. Some application to partial differential equations are described.Pertubation results for generators of infinitely differentiable semigroups of linear operators are given. Some application to partial differential equations are described