Angelo Favini

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal Lp-regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces Wp,q2,2.

An L^{p}-approach to singular linear parabolic equations in bounded domains

Singular means here that the parabolic equation is not in normal form neither can it be reduced to such a form. For this class of problems, following the operator approach used in [1], we prove global in time existence and uniqueness theorems related to (spatial) -spaces. Various improvements to [2], [3] are given.

Parabolicity of second order differrential equations in hilbert space

We study conditions entailing that the differnetial equation u″+Bu′+Au=0 in a Hilbert space generates a corresponding equation in a larger extrapolation space which is parabolic. Subsequently we prove some relations between solutions of the original equation and the extrapolated equation. An example of applications to initial boundary value problems for partial differential equations is given.

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

We use Sadovskiis fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

Degenerate Second Order Differential Operators Generating Analytic Semigroups inLp andW1,p

We deal with the problem of analyticity for the semigroup generated by the second order differential operator Au ≔ αu″ + βu′ (or by some restrictions of it) in the spaces Lp(0, 1), with or without weight, and in W1,p(0, 1), 1 0 in (0, 1), and the domain of A is determined by the generalized Neumann boundary conditions and by Wentzell boundary conditions.

A Product Formula Approach to a Nonhomogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System Part I: A Phase-field Model

The paper establishes existence, uniqueness and regularity for the solution of a phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions. Moreover, some estimates for the solution are given. The non-linear parabolic problem considered here can be used to modeling the solidification (liquidification) process to a matter. The non-homogeneous boundary conditions (depending both on time and space variables) can be understood as a boundary control.

Generalized Wentzell Boundary Conditions and Analytic Semigroups in C [0, 1]

In [4], we introduced for the first time the so-called generalized Wentzell boundary conditions for some classes of linear, or nonlinear, second order differential operator with domain in the space C[0, 1] of all real-valued continuous functions on [0, 1]. There we proved generation results which extended substantially those referred to Dirichlet, Neumann, Robin and Wentzell boundary conditions.

Analytic Semigroups on C [0,1] Generated by Some Classes of Second Order Differential Operators

C [0,1], α > 0 in (0,1) and α(1), we consider the second order differential operator on C[0,1] defined by Au: = αu″ + βu′, where D(A) may include Wentzell boundary conditions. Under integrability conditions involving √α and β/√α, we prove the analyticity of the semigroup generated by (A,D(A)) on Co[0,1], Cπ[0,1] and on C[0,1], where Co[0,1]: {u∈ C[0,1]|u (1)} and Cπ[0,1]: = {u∈ C[0,1]| u (0) = u (1)}. We also prove different characterizations of D(A) related to some results in [1], where β≡ 0, exhibiting peculiarities of Wentzell boundary conditions. Applications can be derived for the case αx = xk (1 - x )kγ(x )(k≥j/2, x∈ [0,1], γ∈ C{0,1}).

Degenerate Nonlinear Diffusion Equations

1 Parameter identification in a parabolic-elliptic degenerate problem.- 2 Existence for diffusion degenerate problems.- 3 Existence for nonautonomous parabolic-elliptic degenerate diffusion Equations.- 4 Parameter identification in a parabolic-elliptic degenerate problem.

Abstract potential operator and spectral methods for a class of degenerate evolution problems

Abstract The spectral theory of operators in Banach spaces is employed to treat a class of degenerate evolution equations. A basic role is played by the assumption that the Banach space under consideration may be expressed as a direct sum of two suitable subspaces. Two methods for solving the problem are studied. The first method is based on the expansion of the resolvent of a closed operator into Laurent series in a neighbourhood of 0. The second one makes use of the theory of abstract potential operators. In particular, an extension of the Hille-Yosida theorem on infinitesimal generators of (C0) semigroups of linear operators is obtained. Some examples relative to operators appearing in many applications to partial differential equations are given.