Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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Boundary regularity problems for some elliptic-parabolic equations
Full text linkIn this note we review some recent results in [64, 95, 96] concerning necessary and sufficient conditions for the regularity of boundary points relatively to the Dirichlet problem for linear degenerate-parabolic operators with well-behaved fundamental solutions. The main focus is on Wiener-type criteria for a class of operators whose degeneracy is controlled by Hormander vector fields
Invariant subring of the Cox ring of K3 surfaces
Full text linkIn this paper, we consider the invariant subring of the Cox ring by the automorphism group of the projective variety X under some assumption. We prove that the ring is finitely generated if X is a K3 surface. 
Horizontal Ga-actions on affine T-varieties of complexity one
Full text linkWe classify the G a -actions on normal affine varieties defined over a field that are hori-zontal with respect to a torus action of complexity one. This generalizes previous versions that wereavailable for perfect ground fields (cf. [7, 12, 11])
Quantitative truncation estimates for fractional Hardy-Sobolev optimizers
Full text linkThe general stability problem of truncations for a family of functions concentrating mass at the origin is described and a concrete example in the framework of entire optimizers for the fractional Hardy-Sobolev inequality is given. In this short note we point out some quantitative stability estimates, useful in dealing with critical p-q fractional equations. 
Anisotropic estimates of subelliptic type
Full text linkWe discuss some estimates of subelliptic type related with vector fields satisfying the Hormander condition. Our approach makes use of a class of approximate exponentials studied in our previous papers. Such kind of estimates arises naturally in the study of regularity theory of weak solutions of degenerate elliptic equations
A spectral theory for discontinuous Sturm-Liouville problems on the whole line
Full text linkIn this study, we consider the singular discontinuous Sturm-Liouville problem on the whole line with transmission conditions. For this problem the existence of a spectral matrix-valued function is proved. A Parseval equality and an expansion formula are given for such problem
Perfect essential graphs
Full text linkLet R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. Let EG(R) be a simple undirect graph associated with R whose vertex set is the set of all nonzero zero-divisors of R and and two distinct vertices x, y in this graph are joined by an edge if and only if AnnR(xy) is an essential ideal. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose EG(R) is perfect
The Bernstein problem in Heisenberg groups
Full text linkIn these notes, we collect the main and, to the best of our knowledge, most up-to-date achievements concerning the Bernstein problem in the Heisenberg group; that is, the problem of determining whether the only entire minimal graphs are hyperplanes. We analyze separately the problem for t-graphs and for intrinsic graphs: in the first case, the Bernstein Conjecture turns out to be false in any dimension, and a complete characterization of minimal graphs is available in H1 for the smooth case. A positive result is instead available for Lipschitz intrinsic graphs in H1; moreover, one can see that the conjecture is false in Hn with n at least 5, by adapting the Euclidean counterexample in high dimension; the problem is still open when n is 2, 3 or 4
Approximated fixed points via completion
Full text linkIn this paper, using the concept of completion, we establish the existence of approximate fixed points for contracting set-valued mappings defined in a metric space not necessarily assumed to be complete. As a consequence, we obtain approximate versions of the famous results of the global Contraction Mapping Principle and Nadler Fixed Point Theorem without completeness
On the Harmonic characterization of domains via mean value formulas
Full text linkThe Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x0 be a point of D. If
for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x0. On the other hand, on every sufficiently smooth domain D and for every point x0 in D there exist Radon measures μ such thatfor every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D