Mild solutions of semilinear elliptic equations in Hilbert spaces

This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into $G$-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

We consider a utility maximization problem for an investment-consumption portfolio when the current utility depends also on the wealth process. Such kind of problems arise, e.g., in portfolio optimization with random horizon or with random trading times. To overcome the difficulties of the problem we use the dual approach. We define a dual problem and treat it by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth functions. This allows to define a smooth solution of the primal Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique in a suitable class and coincides with the value function of the primal problem. Some financial applications of the results are provided

Advanced electrospun matrices based on polysaccharide derivatives for applications in regenerative medicine

The complete regeneration of damaged human tissues and organs is still a significant challenge. The integrative use of biomaterials, cells and bioactive factors in all-in-one devices exploits all current knowledge of materials science, nanotechnology and stem cell biology to best mimic the complex hierarchical architecture of native tissues. The artificial microenvironment design must be properly tuned to match the physicochemical features of the target, offering adequate nanoscale patterns and biological domains for cellular interactions. Scaffolds must promote and guide the regeneration route by mimicking host signalling pathways through the controlled release and retention of drugs or growth factors. For these purposes, polysaccharides have been often used and proposed to manufacture bioengineered scaffolds for regenerative medicine applications due to high biomimetic characteristics. The efforts made by researchers have highlighted how the cellular interactions with electrospun biomaterials offer excellent performances to achieve the desired differentiation and integration with the surrounding tissue. Alkyl derivatives of gellan gum (GG) and hyaluronic acid (HA) have been investigated as novel bioactive electrospun membranes for wound healing and periodontal regeneration. In this thesis, all the derivatives of these two polysaccharides have been produced via activation of the primary hydroxyl groups of β-glucose of gellan gum or N-acetyl-D-glucosamine of hyaluronic acid with bis(4-nitrophenyl) carbonate and the grafts of aliphatic chains at a different length. For hyaluronic derivatives, small moieties with free amino groups have been additionally inserted with the same chemistry. Membranes based on the octyl- and dodecyl-derivative of gellan gum (GG-C8 and GG-C12) have been produced by electrospinning and characterized in terms of fiber distribution and orientation verifying the improved processability of the polymers compared to native gellan gum. Rheological analyses have studied the influence of alkyl derivatization on the spinnability of blends. The feasibility of the process regarded the octyl-derivative one, so the swelling ability of such scaffold has been analyzed under physiological conditions after crosslinking with calcium chloride at different concentrations. To treat partial-thickness wounds, the membrane has been proposed to improve the cell homing at the damaged site, the adhesion of cells, and to encourage the regeneration of the extracellular matrix lost. Similar instrumental settings are used to incorporate the growth factor FGF-2 in the octyl-based membrane (GG-C8). The study has investigated physical interactions between the active and the polymer and how the ionotropic sensitivity of the GG-C8 could be exploited to assess a suitable FGF-2 releasing profile. The fabrication of the bilayer biodevice has involved a hydrophilic layer covered by a synthetic polyurethane layer loaded with ciprofloxacin. Considering the crosslinking degree of the membrane, the dissolution rate and the releases of FGF-2 and ciprofloxacin have been valued with Franz cells. The antibacterial effect of the bilayer has been investigated against the inhibition of units forming colonies of Staphylococcus aureus in the timeframe compatible with the complete regeneration of the tissue. The chemoattraction ability of the scaffold and the cytocompatibility have been tested using cultures of human fibroblasts (NIH3T3) by a specific migration assay. Electrospun membranes based on four chemical derivatives of hyaluronic acid at increasing hydrophobic character have been prepared and loaded with dexamethasone (a known osteoinductive drug) to induce osteogenic differentiation in pre-osteoblasts (MC3T3). The fibrillar supports have been characterized with a scanning electron microscope, and the hydrolytic and enzymatic degradation, as well as dexamethasone releases, have been tested after an autocrosslinking procedure. HA membranes have been designed as guided-bone regeneration barriers for periodontal regeneration; thus, wettability properties and biological performances have been investigated. The proliferation of cells above membranes is valued for up to seven days, and in vitro osteogenic induction is followed by quantifying the activity of alkaline phosphatases and evaluating the calcium content after one month of MC3T3 cultures. The final aim of the thesis is to fabricate a hyaluronan based membrane with high antibacterial properties for the healing of chronic wounds. The proposed membrane consists of incorporating graphene oxide in an electrospun membrane of a hydrophilic derivative of HA loaded with ciprofloxacin. According to an external laser stimulation in the near-infrared, the drug-releasing profile and hyperthermal features have been studied and related to the ability to eradicate bacterial infections and inhibit the generation of biofilm. The cytocompatibility has been valued culturing fibroblasts for up to three days, and the membrane was valued as a wound dressing system

Path-dependent equations and viscosity solutions in infinite dimension

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit

HJB equations for the optimal control of differential equations with delay in the control variable

We study a class of optimal control problems with state constraint, where the state equation is a differential equation with delays in the control variable. This class of problems arises in some economic applications, in particular in optimal advertising problems. The optimal control problem is embedded in a suitable Hilbert space and the associated Hamilton- Jacobi-Bellman (HJB) equation is considered in this space. It is proved that the value function is continuous with respect to a weak norm and that it solves in the viscosity sense the associated HJB equation. The main result is the proof of a directional C1 regularity for the value function. This result represents the starting point to define a feedback map in classical sense going towards a verification theorem and the construction of optimal feedback controls for the problem

Public Debt And Financial Sustainability Of The Italian Public Finances

The analysis presented in this paper deals with two main issues: the one of debt sustainability, meant in the particular acceptation proposed in this article, and the one of the effects of debt decumulation for the various territorial communities, in particular for the weak areas of Italy (Mezzogiorno). The proposed analysis aims at showing some possible outcomes of the current economic crisis. Four hypotheses, concerning various kinds of constraints regulating the variation over time of the debt amount, are proposed: the zero (or constant) debt hypothesis, the invariance of GDP-debt ratio, the hypothesis of a ceiling on public debt and, lastly, the case of a programmed path of public debt reduction (the Fiscal Compact). In the best case proposed (the zero debt hypothesis), the results prefigure a prognosis of stagnation, which is more serious for the enterprises and the families of the Mezzogiorno than for the rest of Italy