Luca Lorenzi

Analytical Methods for Markov Semigroups

Introduction MARKOV SEMIGROUPS IN RN The Elliptic Equation and the Cauchy Problem in Cb(RN): The Uniformly Elliptic Case One Dimensional Theory Uniqueness Results, Conservation of Probability and Maximum Principles Properties of T(t) in Spaces of Continuous Functions Uniform Estimates for the Derivatives of T(t)f Pointwise Estimates for the Derivatives of T(t)f Invariant Measures and the Semigroup in LP(RN, ) The Ornstein-Uhlenbeck Operator A Class of Nonanalytic Markov Semigroups in Cb(RN) and in Lp(RN, ) MARKOV SEMIGROUPS IN UNBOUNDED OPEN SETS The Cauchy-Dirichlet Problem The Cauchy-Neumann Problem: The Convex Case The Cauchy-Neumann Problem: The Nonconvex Case A CLASS OF MARKOV SEMIGROUPS IN RN ASSOCIATED WITH DEGENERATE ELLIPTIC OPERATORS The Cauchy Problem in Cb(RN) APPENDICES Basic Notions of Functional Analysis in Banach Spaces An Overview on Strongly Continuous and Analytic Semigroups PDEs and Analytic Semigroups Some Properties of the Distance Function Function Spaces: Definitions and Main Properties References Index

Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

We study a class of elliptic operators A with unbounded coeffi- cients defined in I × R d for some unbounded interval IR. We prove that, for any s 2 I, the Cauchy problem u(s,·) = f 2 Cb(R d ) for the parabolic equation Dtu = Au admits a unique bounded classical solution u. This allows to associate an evolution family {G(t, s)} with A, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function G(t, s)f. Under suitable assumptions, we show that there exists an evolution system of measures for {G(t, s)} and we study the first properties of the extension of G(t, s) to the L p -spaces with respect to such measures.

Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators A with unbounded coefficients in R N . Using a Bernstein approach we provide several uniform estimates for the semigroup T(t) generated by the realization of the operator A in the space of all bounded and continuous or Holder continuous functions in R N . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation λu - Au = f (λ > 0) and the nonhomogeneous Dirichlet Cauchy problem D t u = Au + g. Then, we prove two different kinds of pointwise estimates of T(t) that can be used to prove a. Liouville-type theorem. Finally, we provide sharp estimates of the semigroup T(t) in weighted L p -spaces related to the invariant measure associated with the semigroup.

Hypercontractivity and Asymptotic Behavior in Nonautonomous Kolmogorov Equations

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in I × ℝ d , where I is a right-halfline. We prove logarithmic Sobolev and Poincaré inequalities with respect to an associated evolution system of measures {μ t : t ∈ I}, and we deduce hypercontractivity and asymptotic behavior results for the evolution operator G(t, s).

Optimal Schauder Estimates for Parabolic Problems with Data Measurable with Respect to Time

We prove some optimal Schauder estimates for the solution to second-order parabolic equations with coefficients which are measurable with respect to time and Holder continuous with respect to space variables in the strip [0,T] times mathbb{R}^{n}

A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results

. We allow also polynomially or exponentially weighted Holder norms.

Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains

We deal with a free boundary problem, depending on a real parameter λ, in a infinite strip in R2, providing stability, instability and bifurcation.

A free boundary problem stemmed from combustion theory. Part I: Existence, uniqueness and regularity results

Abstract We consider a class of uniformly elliptic operators 𝒜 with unbounded coefficients in unbounded domains . Under suitable assumptions on the geometry of and on the coefficients, we prove that the Cauchy-Neumann problem associated with the operator 𝒜 admits a unique bounded classical solution u for any initial datum f which is bounded and continuous in . Moreover, we prove uniform and pointwise gradient estimates for u. Finally, we give some applications of the so obtained estimates.

Rigorous derivation of the Kuramoto–Sivashinsky equation from a 2D weakly nonlinear Stefan problem

Abstract We deal with a free boundary problem, depending on a real parameter λ , in a infinite strip in R 2 , which admits a planar travelling wave solution for every λ∈ R . We prove existence, uniqueness and regularity results for the solutions near the travelling wave.

stability in a two-dimensional free boundary combustion model

In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter