Eugênia Viana Cerqueira

Benjaminmotte-Baumvol

Flexible work arrangements and household-related journeys. Who takes the lead in dual-earner heterosexual couples?

Keywords: flexible work arrangements, travel behavior, telework, dual-earner couples, sustainable transport

This research provides an exploration of the relationship between flexible work arrangements (FWA), CO2 emissions, and household travel allocation among dual-earner couples. Using data from the United Kingdom National Travel Survey for the period between 2002 and 2017, we develop a path analysis approach based on SEM that allows for a comprehensive study of travel behavior and trade-off effects. Results suggest that, although the partner who benefits from FWAs tends to carry out domestic provision and maintenance activities, the allocation of tasks within the household is still asymmetrical. Results also outline that the presence of children will intensely affect the allocation of activities and household arrangements, increasing gender inequalities.

Introduction

Flexible work arrangements (FWAs) [START_REF] Allen | Work-Family Conflict and Flexible Work Arrangements: Deconstructing Flexibility[END_REF] or alternative work designs [START_REF] Brewer | Work design, flexible work arrangements and travel behaviour: policy implications[END_REF] such as teleworking, working from home, part-time working, or selfemployment have become a hot topic during the COVID-19 outbreak, accelerating the debate about the way we work and its implications for commuting and travel. For example, homebased work has become the "new normal" for millions of workers the world over during lockdowns and scholars have promptly sought to assess not only the current shifts in people's daily activities [START_REF] Hensher | Working from Home and its implications for Strategic Transport Modelling based on the early days of the COVID-19 pandemic[END_REF][START_REF] Nguyen | Perception and Preference for Home-Based Telework in the COVID-19 Era: A Gender-Based Analysis in Hanoi, Vietnam, Sustainability[END_REF] but also the main impacts of a potential extended remote-working routine in the post-COVID era [START_REF] Jesus | A 'new normal' following COVID-19 and the economic crisis: Using systems thinking to identify challenges and opportunities in disability, telework, and rehabilitation[END_REF][START_REF] Schur | Telework After COVID: A "Silver Lining" for Workers with Disabilities?[END_REF][START_REF] Nguyen | Factors influencing home-based telework in Hanoi (Vietnam) during and after the COVID-19 era[END_REF].

FWAs are receiving attention in the field of transport for their capacity to transform many aspects of commuting, such as the number of trips, distances traveled, modal share, congestion at peak times, and pollutant emissions. The potential issues surrounding FWAs, in the event of their marked development after the COVID-19 outbreak, are therefore very significant. The energy and climate impacts of some forms of FWA such as teleworking were investigated in the pre-crisis period and have since been the subject of many studies. Findings demonstrate that the reduced distance travelled for commuting could positively impact in energy consumption, traffic congestion and CO2 emissions [START_REF] Shabanpour | Analysis of telecommuting behavior and impacts on travel demand and the environment[END_REF].

Nonetheless, many points of the capacity of FWAs to reduce commuting and CO2 emissions remain disputed, as the more rigorous studies that include non-work trips or home energy use find smaller savings or even an absence of reduction of GHG emissions [START_REF] Cerqueira | Does working from home reduce CO2 emissions? An analysis of travel patterns as dictated by workplaces[END_REF]. [START_REF] Hook | A systematic review of the energy and climate impacts of teleworking[END_REF] emphasize that despite the positive forecast on the climate impacts of FWA, there are still numerous uncertainties about its actual benefits, as they could lead to unpredictable increases in non-work travel.

Thus, although the relationship between FWA and the existence of rebound effects for non-work trips has been analyzed in the academic literature for entire households [START_REF] Cerqueira | Does working from home reduce CO2 emissions? An analysis of travel patterns as dictated by workplaces[END_REF], few studies have focused on the intra-household level. This issue must be considered in the light of the role of flexible work in dual-income couples who are the main ones concerned by these types of employment (Chung et al., 2020a). Studies show that FWAs are used in particular by workers who want to better balance work and family life, with a high prevalence of women. Household responsibilities and the travel that goes with them (such as children escorting trips, for example) tend to weigh more heavily on the spouse with flexible work, freeing up the other spouse and adjusting the couple's non-work mobility and the associated CO2 emissions. These arrangements could thus entail some intra-couple rebound effects, which would counterpoint the commonly established notion that FWA would reduce traveling and climate impacts. In the event of a surge in FWAs in a post-COVID period, it is important to analyze intra couple travel trade-offs and their impacts on CO2 emissions.

Thus, this research focuses on the how FWAs will affect household travel allocation among dual-income couples and consequently, its CO2 emissions. We argue that it is capital to grasp the trade-offs that operate in couples with one member who benefits from FWAs and their impacts on overall household CO2 emissions. Does FWA allows minimizing CO2 emissions at the couple scale or does it rather increase them? The data used were obtained from the United Kingdom National Travel Survey, which collects data from trip diaries for a seven-day observation period, between 2002 and 2017. The case is made for focusing exclusively on nonwork trips, as previous investigations demonstrate that FWAs can entail substantial trade-offs between work and non-work trips and that the latter account for a significant share of CO2 emissions [START_REF] Ravalet | Teleworking: Decreasing Mobility or Increasing Tolerance of Commuting Distances?[END_REF].

We use Structural Equation Modeling (SEM) implementing path analysis and notably highlight the implications of FWAs for household travel patterns. The results show that, although the partner who benefits from FWAs tends to carry out domestic provision and maintenance activities, the allocation of tasks within the household still involves gender inequalities. We also find that the presence of children, combined with FWAs, implies more escorting and caregiving-related trips, particularly for women.

The article is structured as follows. First, the literature review section summarizes previous findings on the effects of telework on intra-household arrangements and the influence of those arrangements on travel patterns and activity choices. The data and methods section presents the travel survey data and explains the SEM developed. The results are presented in Section 4 and the final section summarizes the main findings and highlights lessons that can be drawn for future studies.

2-Literature review

2.1-Flexible Work Arrangements

There is no established and widely used definition of FWAs in the literature. However, the literature points to two main dimensions by which this flexibility can be characterized: the temporal and spatial dimensions [START_REF] Burkinshaw | The Tenuous and Complex Relationship Between Flexible Working Practices and Travel Demand Reduction[END_REF][START_REF] Moen | Does a flexibility/support organizational initiative improve high-tech employees' wellbeing? Evidence from the work, family, and health network[END_REF]. For temporal flexibility, multiple characteristics can be taken into account, such as the possibility of arriving late or leaving early on certain occasions, of being able to change one's schedule depending on the day and/or week, or even of having control of the schedule [START_REF] Schieman | Is there a downside to schedule control for the work-family interface[END_REF].

Flexible working can also include control over the number of hours worked, as with part-time jobs. Of course, all of these characteristics are understood to determine the actual level of work flexibility in different degrees. Spatial flexibility means allowing work outside the usual place of work [START_REF] Schieman | Trouble at the border?: Gender, flexibility at work, and the workhome interface[END_REF]. The definition of the usual place of work is open to debate. In general, what is covered is telework at home or in any other third place within the framework allowed by the employer and the legislation.

Despite more favorable legislation, particularly in the UK, the number of workers adopting FWAs increased moderately in the pre-COVID-19 period, at least in the early 2010s [START_REF] Tipping | The fourth work-life balance employee survey[END_REF]. One reason is the prevalence of flexibility stigma, i.e., the perception that workers who use FWAs for care purposes are less productive and less committed to the workplace [START_REF] Williams | Cultural schemas, social class, and the flexibility stigma[END_REF]. Finally studies have also shown that as flexible working becomes more widespread, it is less stigmatized [START_REF] Moen | Does a flexibility/support organizational initiative improve high-tech employees' wellbeing? Evidence from the work, family, and health network[END_REF] and the COVID-19 crisis may have contributed to reducing the stigma of flexibility [START_REF] Chung | Working from home during the COVID-19 lockdown: Changing preferences and the future of work[END_REF].

2.2-FWAs, work-life balance and gender inequalities

In the transportation field, the effects on commuting have only been investigated for some FWAs, in particular those related to spatial flexibility: teleworking and working from home. The effects of temporal flexibility have not been studied directly, but may appear among other determinants in studies on commuting and/or non-work trips. For example, the highly developed part-time jobs for women in the Netherlands are associated with more shopping and escorting trips [START_REF] Schwanen | If You Pick up the Children, I'll Do the Groceries: Spatial Differences in between-Partner Interactions in out-of-Home Household Activities[END_REF]. In addition, part-time jobs are associated with shorter commuting distances, but these effects appear to be largely endogenous with those related to gender [START_REF] Mcquaid | Commuting times -The role of gender, children and part-time work[END_REF]. The effects of potentially flexible working hours for escorting children in the morning can also be observed particularly for women [START_REF] Motte-Baumvol | Who escort children: mum or dad? Exploring gender differences in escorting mobility among parisian dual-earner couples[END_REF].

From the perspective of spatial flexibility, the development of ICT has enabled more individuals to work from home either permanently or several times a week. Telework has gained significant attention as a potential way of reducing travel demand and its negative effects, such as congestion and CO2 emissions [START_REF] Hook | A systematic review of the energy and climate impacts of teleworking[END_REF]. Over the past two decades, numerous investigations have analyzed the effects of home-based work from different viewpoints, illustrating the profile of individuals who adopt these arrangements as well as showing its environmental, social, and economic implications [START_REF] Ory | Which Came First, the Telecommuting or the Residential Relocation? An Empirical Analysis of Causality[END_REF][START_REF] Zhu | The impact of telecommuting on personal vehicle usage and environmental sustainability[END_REF][START_REF] Abreu E Silva | Home telework, travel behavior, and land-use patterns: A path analysis of British single-worker households[END_REF]. While some of these studies highlight the positive effects of telework, such as the reduction in the number of trips and kilometers traveled [START_REF] Choo | Does telecommuting reduce vehicle-miles traveled? An aggregate time series analysis for the U[END_REF][START_REF] Helminen | Relationships between commuting distance, frequency and telework in Finland[END_REF] and the increase in free time for activity allocation [START_REF] Hilbrecht | I'm Home for the Kids': Contradictory Implications for Work-Life Balance of Teleworking Mothers[END_REF], findings also emphasize that working from home often entails significant rebound effects, since time saved on commuting trips may be converted into additional non-work trips or longer commutes [START_REF] Zhu | Are telecommuting and personal travel complements or substitutes?[END_REF][START_REF] He | Telecommuting, income, and out-of-home activities[END_REF]Author et al., 2020).

In terms of its contribution to the work-life balance, teleworking seems to be an attractive arrangement for time-pressured working couples seeking to avoid often long daily commutes to work. Home-based work may lead to a greater sense of control over time and allocation of household responsibilities, increasing individuals' satisfaction with their worklife balance. Although flexible work could be a useful tool for promoting gender equality [START_REF] Chung | Flexible Working, Work-Life Balance, and Gender Equality: Introduction[END_REF], a notable body of research demonstrates that the temporal flexibility created by home-based work arrangements is used differently by male and female teleworkers and it could potentially reinforce gender roles in the labor market and in domestic tasks [START_REF] Lott | Gender Discrepancies in the Outcomes of Schedule Control on Overtime Hours and Income in Germany[END_REF].

The literature is almost unequivocal in depicting that, when it is women who telework, teleworking contributes to the reproduction of gender inequality by reinforcing their responsibility for caregiving and the management of domestic tasks, especially when there are children in the family [START_REF] Stier | Work-family conflict in comparative perspective: The role of social policies[END_REF][START_REF] Chung | Flexible Working, Work-Life Balance, and Gender Equality: Introduction[END_REF]. [START_REF] Hilbrecht | I'm Home for the Kids': Contradictory Implications for Work-Life Balance of Teleworking Mothers[END_REF] investigate the impacts of teleworking on the work-life balance among married women with children in Canada, demonstrating that time saved on commuting trips is mostly reallocated to childcare, domestic work, or work-related tasks, rather than to time for personal leisure.

Moreover, the "stay-at-home mother" role contributes to the close association of personal work schedules with the rhythms of children's activities, which include escort trips to school and leisure activities. Thus, for some mothers, the combination of working from home and childcare responsibilities may result in increased stress, as time spent during the day with their children often involves the trade-off of their evening leisure time [START_REF] Hilbrecht | Remixing work, family and leisure: teleworkers' experiences of everyday life: Remixing work, family and leisure[END_REF].

On the other hand, evidence suggests that although male teleworkers may sometimes undertake a significant share of housework, they tend to maintain a greater degree of separation between the work and family spheres. They often follow conventional industrial working time patterns and also increase their working intensity when working from home [START_REF] Sullivan | Home-based Telework, Gender, and the Synchronization of Work and Family: Perspectives of Teleworkers and their Co-residents[END_REF][START_REF] Chung | Flexible Working, Work-Life Balance, and Gender Equality: Introduction[END_REF]. Men also appeared to convert the time saved on commuting into increased leisure opportunities more often than women [START_REF] Hilbrecht | Remixing work, family and leisure: teleworkers' experiences of everyday life: Remixing work, family and leisure[END_REF].

When comparing teleworking impacts on family management in Poland and Sweden, [START_REF] Kurowska | Gendered Effects of Home-Based Work on Parents' Capability to Balance Work with Non-work: Two Countries with Different Models of Division of Labour Compared[END_REF] shows that, in Poland, men are able to avoid the trap of the double burden of paid and domestic work, while in Sweden the negative effects of home-based work on the ability to balance work with non-work activities affects both men and women.

More recently, scholars have emphasized that dramatic changes in work arrangements brought about by the COVID-19 pandemic contribute to a greater disproportion in the allocation of household responsibilities between males and females [START_REF] Nguyen | Perception and Preference for Home-Based Telework in the COVID-19 Era: A Gender-Based Analysis in Hanoi, Vietnam, Sustainability[END_REF]. Most of the negative effects of teleworking are reported among couples with children, particularly among mothers with young children [START_REF] Andrew | How are mothers and fathers balancing work and family under lockdown[END_REF][START_REF] Seiz | Equality in Confinement: Nonnormative Divisions of Labor in Spanish Dual-Earner Families During the Covid-19 Lockdown[END_REF]. Nonetheless, higher rates of females compared with their male counterparts had a positive perception of telework within the lockdown period and a desire to continue with this type of work arrangement post-COVID-19 [START_REF] Bachelet | What If Working from Home Will Stick? Distributional and Climate Impacts for Germany[END_REF][START_REF] Nguyen | Perception and Preference for Home-Based Telework in the COVID-19 Era: A Gender-Based Analysis in Hanoi, Vietnam, Sustainability[END_REF]. This confirms previous findings showing that women consider teleworking as beneficial for achieving a better worklife balance [START_REF] Hilbrecht | I'm Home for the Kids': Contradictory Implications for Work-Life Balance of Teleworking Mothers[END_REF].

It is also important to note that telework does not in itself transform the gendered allocation of domestic tasks and caregiving [START_REF] Sullivan | Home-based Telework, Gender, and the Synchronization of Work and Family: Perspectives of Teleworkers and their Co-residents[END_REF]. Despite the overlap of the family and work spheres, the association between telework and work-life balance seems to be positively perceived in many cases [START_REF] Maruyama | A multivariate analysis of work-life balance outcomes from a large-scale telework programme[END_REF]. Increased time for leisure activities is considered to be a valued outcome of telework, as well as an opportunity to take advantage of non-peak periods at recreation facilities [START_REF] Hilbrecht | Remixing work, family and leisure: teleworkers' experiences of everyday life: Remixing work, family and leisure[END_REF]. As working from home directly impacts time management and the allocation of tasks within the household, it will also have significant consequences for travel behavior and choices.

2.3-Intra-household arrangements and travel-behavior

In transport research the individual has conventionally been regarded as the basic decision-making agent. However, it is well known that individual daily activity and travel patterns are often influenced by other household members and, in some cases, choices may involve a group of people [START_REF] Zhang | Modeling household discrete choice behavior incorporating heterogeneous group decision-making mechanisms[END_REF]. Thus, a growing avenue of research has integrated intra-household interactions on modeling activity and travel-related decisions, as evidence demonstrates that household resource allocation (such as car use) and task and time allocation all involve close coordination among household members at different levels in order to meet group needs under social, spatial, and resource constraints [START_REF] Ho | Intra-household interactions in transport research: a review[END_REF]. For instance, a family member may opt to use public transport in order to free the family car; household members may design joint travel arrangements to save time or reduce the number of trips. Thus, choices of activity and travel patterns are closely associated with household and individual characteristics such as the presence of children, car ownership, income, working hours, and travel patterns.

A set of studies examines the way individual choices of activity and travel patterns are substantially impacted by household decisions, particularly from a couple's perspective. [START_REF] Schwanen | If You Pick up the Children, I'll Do the Groceries: Spatial Differences in between-Partner Interactions in out-of-Home Household Activities[END_REF] demonstrate that, across a sample of over 700 couples in the Netherlands, individuals' participation in out-of-home activities is significantly influenced by their partner's travel behavior. [START_REF] Srinivasan | A multiple discrete-continuous model for independent-and joint-discretionary-activity participation decisions[END_REF] make the point that women who undertake serve-child trips, such as escorting children to and from school, are less likely to carry out discretionary activities. In these households, child escort responsibilities also decrease the possibility of parents pursuing joint activities. In that sense, investigations also provide evidence that men often have greater influence than their spouses on joint task allocation and household travel decisions [START_REF] Zhang | A model of household task allocation and time use[END_REF][START_REF] Zhang | Representing household time allocation behavior by endogenously incorporating diverse intra-household interactions: A case study in the context of elderly couples[END_REF].

The literature often distinguishes mandatory from non-mandatory activities [START_REF] Vovsha | Model for Allocation of Maintenance Activities to Household Members[END_REF][START_REF] Bradley | A model for joint choice of daily activity pattern types of household members[END_REF][START_REF] Srinivasan | A multiple discrete-continuous model for independent-and joint-discretionary-activity participation decisions[END_REF]. Despite being of an individual nature, mandatory activities such as attending work, school, university, and escorting children often orchestrate household travel decisions, entailing travel cooperation in the form of pick-ups and drop-offs on the way [START_REF] Vovsha | Explicit Modeling of Joint Travel by Household Members: Statistical Evidence and Applied Approach[END_REF]. Similarly, although many of the domestic provision and maintenance activities (such as grocery shopping, trips to the post office, etc.) are carried out individually, their allocation to a specific household member is an active part of the joint decision-making process. Thus maintenance activities are "characterized by a significant degree of intrahousehold coordination, substitution, and possibly sharing" (Vovsha et al., 2004, p. 171). In that sense, decisions behind the allocation of maintenance activities may not be completely assessed within a single one-day framework, which is why it is ideal to work with longer data cycles to analyze non-mandatory and less regular activities [START_REF] Bradley | A model for joint choice of daily activity pattern types of household members[END_REF].

Finally, [START_REF] Vovsha | Model for Allocation of Maintenance Activities to Household Members[END_REF] investigate the allocation of domestic maintenance activities among household members according to their employment status, demonstrating that household travel patterns and activity choices are closely associated with personal characteristics, time availability, and residential location. [START_REF] Srinivasan | A multiple discrete-continuous model for independent-and joint-discretionary-activity participation decisions[END_REF] find that individuals who work from home for fewer than 4 hours a day are most likely to travel for nonmandatory activities, suggesting that home-based telework could impact the allocation of discretionary activity within the household. They also find that couples are less likely to undertake non-mandatory activities jointly if one or both of them work outside the home.

Moreover, individual changes in travel behavior due to flexible working hours or earlier school hours reportedly also directly affect the travel patterns of other household members [START_REF] Ho | Intra-household interactions in transport research: a review[END_REF]. Therefore, we argue that the temporal flexibility resulting from telework arrangements impacts not only individual travel patterns, but also household activity patterns.

Data and methods

Data

The data used were obtained from the United Kingdom National Travel Survey (NTS) for the period between 2002 and 2017 in order to ensure a healthy sample size for FWAs, which are relatively rare. The data used refer to the travel of English residents within Great Britain. In the NTS, we selected heterosexual couples with two workers, allowing us to observe trade-offs between spouses and gender differences. In total, 25,658 couples were included in our analysis, 44% of whom have children. In order to calculate GHG emissions, the NTS provides CO2 emission factors (derived from the Driver & Vehicle Licensing Agency data set). Where the CO2 emission factor was missing, we applied the conversion factors from DECC & DEFRA1 (2020). We also assigned CO2 coefficients to public transport trips based on the type of transport mode used (bus, light rail). No coefficients were assigned to cycling and walking. Four categories of workplaces used in the analysis are pre-defined in the NTS survey on the basis of a main criterion which is the "usual" place of work (same place, two places, different places and homeworkers). However, to further explore the issue of FWAs, we also opted to create an additional "teleworkers" category. The referred category was created from the variable "How often do you work from home?" We selected individuals who do not fall into the "homeworkers" category and who declared they worked from home "three or more times a week" or "once or twice a week". Thus, the definition of teleworker adopted in this work is relatively restrictive compared to the definitions that can be found in the literature and includes only individuals with a usual place of work outside of home, i.e., whose teleworking does not exceed three days a week. Nonetheless, it is important to note that few variables in the National Travel Survey account for work flexibility.

Description of the variables used in the model

To address the effects of FWAs on travel behavior, the number of available variables is limited in the UK NTS. For spatial flexibility, we used the place of work variable with the addition of a telework category for individuals declaring at least one day of telework per week.

Thus we have two categories assimilated to FWAs: work-at-home and telework. These categories are more frequent for women with children (Table 1). Conversely, the different places category is a type of workplace with strong constraints, both in terms of time and space, since it involves, for example, a home care nurse making her rounds or an employee of a cleaning company working in different places. This place of work appears less frequently for women with children and to a lesser extent for men with children.

In order to address the temporal dimension of FWAs, only one variable relating to working time-part-time or full-time-is used. Part-time jobs are much more frequent for women with children, but not for men, on the contrary. The NTS data does not include any other variable indicating the flexibility of workers' schedules. Finally, we used the variable distinguishing employees from self-employed on the assumption that the latter potentially have greater spatial and temporal flexibility.

The travel indicators, i.e., the number of trips for a given reason, are calculated over a whole week, corresponding to the UK TS observation period. Women and men with children make more trips than their counterparts without children, with 4.7 and 1.5 more trips per week respectively. This corresponds almost exclusively to an increase in escort travel. As a result, the CO2 emissions of couples with children are higher than those of couples without children due to the greater number of escort trips. Non-work trips account for more than half of the emissions of couples (excluding the return home) and 63% for couples with children.

Model Framework

Although many studies focusing on CO2 emissions analyze the effects of commuting trips, in this paper we choose to focus exclusively on non-work trips, as we seek to understand the trade-off effects that operate between dual-earner couples in terms of travel arrangements and its potential impacts. As previous investigations have demonstrated, with the same dataset, the relationship between FWA and the compensations between work and non-work trips [START_REF] Cerqueira | Does working from home reduce CO2 emissions? An analysis of travel patterns as dictated by workplaces[END_REF], our goal is to explore this subject further at the intra-household level.

We argue that these trade-offs pertain exclusively to non-work-related travel, as couples would tend to reallocate household responsibilities to the partner benefiting from FWA, potentially generating additional trips and GHG emissions.

The model developed in this research aims to establish two points. First, certain characteristics of spouses' activity such as telework, working from home, and part-time work are linked to an increase in non-work activities (in particular escort and shopping) for the spouse concerned. Second, the increase in escort and shopping trips by one spouse, linked to his or her activity, results in a trade-off with the other spouse, who then reduces his or her participation in escorting and shopping.

To determine this, SEM allows us to test several hypotheses simultaneously by analyzing independent variables one by one while controlling for the others as covariates and also to evaluate mediation effects. The model (Figure 1) is structured in two stages:

-In the first stage, the model is built around six dependent variables that we seek to explain on the basis of women's employment characteristics, men's employment characteristics, and finally household characteristics. These variables correspond to the number of non-work trips made by women and men for three types of motives: shopping, leisure/personal, and escorting.

At this stage, it is already possible to highlight cross-effects between the employment characteristics of one spouse and the travel of the other spouse. These six variables are then used as independent variables in the structural model to explain CO2 emissions.

-

4.Results

The model obtained is considered a good model with respect to the indicators generally used [START_REF] Golob | Structural equation modeling for travel behavior research[END_REF]. The root mean square error of approximation (RMSEA) is less than 0.05 and the standardized root mean square residual (SRMR) less than 0.02, which is judged satisfactory. Moreover, the comparative fit index (CFI), which compares the proposed model with an unrestricted base model, exhibits a value of 0.97 (a good model should have a value of more than 0.90).

4.1-Couples with children

4.1.1-The effects of women's employment characteristics

For couples with children (Table 2), women's FWAs, part-time work, work from home, self-employment, or telework, have positive effects on non-work trips. There are also positive effects of car access as the main driver of a vehicle. Conversely a low-earner and a high-earner (compared to an average earner) are associated with a lower level of shopping and escorting.

In terms of cross-effects, there are negative effects of FWAs for women on their spouse's nonwork trips. This is consistent with the trade-off hypothesis that FWAs are associated with increased out-of-work travel for the spouse who benefits from them and restricted travel for the other spouse. This trade-off effect is, however, limited in terms of the levels and reasons for travel involved. It is observed for escorting in the case of working from home, and for shopping and escorting in the case of part-time work. A trade-off is also observed for shopping and escorting in the case of the woman being the main driver of a vehicle. For income, this tradeoff is only observed for escorting.

Concerning the CO2 emissions of the couple as a whole linked to non-work trips, we observe that the FWAs for the woman lead to an increased level of emissions for the couple.

This means that the trade-off, when there is one, is not sufficient to balance the emissions between women and men. Therefore, part-time work, working from home, self-employment, and finally telework, in decreasing order of effect, lead to higher levels of emissions for the couple's non-work trips. The decomposition of the effects between men's and women's trips

shows that the positive effects for women are only very partially offset by negative effects for men. The break-down by purpose shows that it is the increase in escorting that contributes most to these higher levels of CO2 emissions, followed by leisure/personal and shopping trips.

4.1.2-The effects of men's employment characteristics

For couples with children (Table 2), men's FWAs have similar effects to those observed for women. Part-time work, working from home, and telework have positive effects on their non-work trips. However, it should be mentioned that these positive effects are not as pronounced for escorting, reflecting the lower propensity of men to escort in general. Thus, self-employment for men does not lead them to escort more as is the case for women, but on the contrary to escort less. For men, this activity status is therefore not a way of freeing up time to escort children. Working in different places decreases men's out-of-work travel.

As for the cross-effects of men's job characteristics on women's off-work travel, we observe trade-off effects similar to those described for women's job characteristics. Thus, men's part-time work and working from home are associated with a lower probability for women to engage in escorting and shopping. Conversely, the lower propensity of men who are selfemployed or work in different places to make non-work trips does not lead to a trade-off with their spouse. The same is true for men's telework, which allows them more escorting and leisure/personal trips, but does not result in fewer trips for these motives for their spouses.

In the end, FWAs for men lead to more CO2 emissions for the couple's non-work mobility. This is simply because men's excess travel in these situations does not correspond to an equivalent trade-off or no trade-off at all on the part of their spouse. This is similar to what was observed for women. However, it should be noted that the effects of the characteristics of telework, working from home, and part-time work are much lower than for the characteristics of women. This corresponds to the fact that these employment situations for women are linked to a greater investment in non-work activities, which is not equivalent for men in the same situations. Data: National Travel Survey 2002[START_REF] Motte-Baumvol | Who escort children: mum or dad? Exploring gender differences in escorting mobility among parisian dual-earner couples[END_REF], Tools: R with the Lavaan package Signif. codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 All values of the estimates are standardized and allow comparisons between reasons, between women and men and between couples with and without children. The missing coefficients correspond to non-significant values, p < 0.1, and the categories/variables were removed from the final computation.

Results of the model for the household with child

Results of the model for the household without child

4.2-Couples without children

For couples without children (Table 3), the effects of women's characteristics on women's and men's travel are not very different from those observed for couples with children. Two types of differences can be highlighted. First, we notice the weaker effects on escorting, which are much less numerous for couples without children. Second, there is no significant effect of selfemployment. This activity status seems to allow women with children to be more flexible in terms of mobility outside of work, in particular for support. This is therefore not the same for women without children.

The same type of effect of women's characteristics can be observed for the CO2 emissions of the couple linked to non-work travel as for couples with children. That is, part-time jobs and working from home lead to higher levels of emissions for non-work trips. The higher number of women's trips is only partially offset by the lower number of men's trips. Also, there is no effect of telework and self-employment, which are not used by women without children to increase their non-work mobility, unlike women with children.

Regarding the effects of men's characteristics on trips, there is first of all a relative similarity in the mobility of men without children and men with children. Part-time jobs and working from home allow for greater mobility out of work. However, this greater mobility only concerns leisure/personal and shopping trips rather than escorting. Conversely, and similarly to men with children, working in different places and being self-employed lead to a lower number of out-of-work trips. As for the cross-spouse effects, these are more systematic for couples without children. Thus, all types of men's workplaces have effects on women's shopping and escort trips.

In particular, the effect of men's telework is significant here and leads to a reduction in women's shopping and escort trips. Conversely the effect of working in different places for men is linked to a lower number of out-of-work trips by women. There is therefore no trade-off here and, conversely, the lower off-work mobility of a man is accompanied by a lower off-work mobility of his wife. This lack of trade-off can also be observed in the case of a man with a part-time job whose spouse also makes more leisure/personal trips.

As a result, the effects of men's characteristics on the couple's CO2 emissions for nonwork trips are not very different between couples with and without children. However, these similar results are the outcome of slightly different trade-offs between men and women. In fact, beyond the partial balancing effects, we can observe here the effects of a shift towards fewer trips for couples in which the men work in different places and towards more trips for couples in which the men have a part-time job.

5-Discussion

This study developed a SEM approach using individual and household level data from the UK National Travel Survey (NTS) to shed light on the impacts of FWAs on travel behavior and CO2 emissions. While the impacts of FWAs on individual activities have been widely discussed in the academic literature, their effects on household joint decisions and overall travel behavior, such as potential trade-offs, have come in for little debate to date. Thus, with the increase in FWAs over the past decades, it is important to capture their impacts on travel arrangements not only at an individual level but also at household level.

The case is made for focusing exclusively on non-work trips, as previous investigations demonstrate that FWAs can entail substantial trade-offs between work and non-work trips and that the latter account for a significant share of CO2 emissions [START_REF] Cerqueira | Does working from home reduce CO2 emissions? An analysis of travel patterns as dictated by workplaces[END_REF]. In fact, nonwork trips are often disregarded in travel-behavior studies and current urban travel policies because most travel surveys use a one-day trip diary and so fail to capture the full scope of these trips.

Thus, the seven-day observation period provided in the United Kingdom NTS allows a better understanding of the full-array of non-work trips, including potential trade-offs within the household.

This study also contributes to the debates on the gendered allocation of activities within dual-income households. The main findings demonstrate that although the partner who benefits from FWAs tends to carry out domestic provision and maintenance activities, the allocation of tasks within the household is still asymmetrical. Thus, flexible work arrangements result in a greater allocation of domestic and maintenance activities to women than their male counterparts. Conversely, men with alternative work patterns tend to convert their free-time into leisure and personal trips, while still allocating a significant share of domestic-related trips to their partners. These results echo previous investigations that demonstrate that telework does not in itself transform the gendered allocation of domestic tasks and caregiving [START_REF] Sullivan | Home-based Telework, Gender, and the Synchronization of Work and Family: Perspectives of Teleworkers and their Co-residents[END_REF]. maintenance and domestic-related activities. For example, it is well known that personal and domestic provision shopping will entail completely different travel patterns and, consequently, household arrangements.

From a policy analysis perspective, the findings of this study imply that demandmanagement solutions directly impacting one adult could also result in changes to their partner's travel behavior and the household arrangements as a whole. Thus, travel demand management policies that influence individual work location and duration, such as telework, can also impact non-work activity participation choices of their partner, resulting in changes to the overall travel patterns of both adults in the household, and consequently, GHG emissions. The implications of the empirical results described in this paper are noteworthy because current travel and environmental policies often ignore non-work trips and potential trade-offs within household travel patterns. Such omissions could potentially reduce the effectiveness of such policies. In the event of an intense development of FWAs in a post-COVID period, this could lead to an increase in overall CO2 emissions by workers. In view of the results reported here and the potential increase in FWAs in the post COVID-19 era, the scope of travel-behavior analysis needs to be extended in order to come up with more effective policymaking on sustainability and travel.

  In the second stage of the model, in the structural model, the contribution of each travel motive for women and men to the CO2 emissions (of non-work travel) of the couple is studied. The choice of a CO2 indicator to represent the couple's total non-work mobility has the advantage of addressing the environmental issue of teleworking and working from home, which are intensively scrutinized. This second step makes it possible to measure the overall effect on emissions of the different employment characteristics of one or other of the spouses. But also, by considering the trade-offs between men and women we are able to measure CO2 emissions.

Figure 1 .

 1 Figure 1. Conceptual Framework

  Travel Survey 2002[START_REF] Motte-Baumvol | Who escort children: mum or dad? Exploring gender differences in escorting mobility among parisian dual-earner couples[END_REF], Tools: R with the Lavaan package Signif. codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 All values of the estimates are standardized and allow comparisons between reasons, between women and men, and between couples with and without children. The missing coefficients correspond to non-significant values, p < 0.1, and the categories/variables were removed from the final computation.

Available from https://www.gov.uk/government/publications/greenhouse-gas-reporting-conversion-factors-2020 (

accessed 20 March 2021).

Results also outline that the presence of children will intensely affect the allocation of activities and household arrangements, increasing gender inequalities. FWAs involve more escorting and caregiving-related trips, particularly for women. Although men also tend to increase the number of escorting trips when they benefit from alternative work patterns, the allocation of chauffeuring trips is significantly higher for their female counterparts, which increases the overall CO2 emissions for the household. These findings confirm results from previous investigations that outline marked gender inequalities in the allocation of escorting tasks within households [START_REF] Motte-Baumvol | Who escort children: mum or dad? Exploring gender differences in escorting mobility among parisian dual-earner couples[END_REF].

The trade-off effects observed are significantly different for dual-income couples without children. For women, FWAs have a less significant impact on non-work trips, particularly escorting trips, when compared to women with children. On the other hand, travel patterns of men without children tend to be similar to those of men with children, but escorting trips are less significant, while the number of personal and shopping trips is higher with FWAs. Trade-off effects between partners is significant for couples without children, as men with FWAs tend to reduce their partner's shopping and escorting trips.

6-Conclusion

While, over the past two decades, flexible work arrangements, such as teleworking, have received attention in the field of transport for their capacity to transform many aspects of commuting, the current COVID-19 outbreak has brought about dramatic changes in work arrangements, intensifying the debate about the increase of alternative work patterns in the postcrisis era. Thus, this paper presents an original exploration of the relationship between FWAs, CO2 emissions, and household travel allocation among dual-earner couples. We present a comprehensive analysis that extends traditional investigations by observing the impacts of alternative working patterns not only on individuals but also on households.

The interpretation of the findings should consider some key limitations, particularly regarding the database used. It is important to note that few variables in the National Travel Survey account for work flexibility. Therefore, the investigation could have benefited from a more detailed exploration if variables such as working hours were provided. Moreover, non-work trip purposes are not specific enough, limiting the analysis, particularly in terms of trade-off effects for


Alexandre Thorel
email: alexandre.thorel@univ-lehavre.fr

A biharmonic transmission problem in L p -spaces

Keywords: Analytic semigroups, biharmonic equations, functional calculus, interpolation spaces, maximal regularity. 2020 Mathematics Subject Classification: 35B65, 35J48, 35R20, 47A60, 47D06

In this work we study, by a semigroup approach, a transmission problem based on biharmonic equations with boundary and transmission conditions, in two juxtaposed habitats. We give a result of existence and uniqueness of the classical solution in L p -spaces, for p ∈ (1, +∞), using analytic semigroups and operators sum theory in Banach spaces. To this end, we invert explicitly the determinant operator of the transmission system in L p -spaces using the E ∞ -calculus and the Dore-Venni sums theory.

Introduction

In this work, we consider a system of linear biharmonic equations posed on two juxtaposed domains and coupled through transmission conditions at the interface. Throughout the paper we shall impose the continuity of the flux, of the dispersal and of its flux across the interface. Using an operator approach, we investigate the existence, uniqueness as well as maximal L p -regularity for such a problem.

Transmission problems arise in various applicative fields including engineering, physics and biology. Here, we refer the reader for instance to [START_REF] Barraza Martínez | Regularity and asymptotic behavior for a damped plate-membrane transmission problem[END_REF], [START_REF] Favini | Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces[END_REF] or [START_REF] Hassine | Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping[END_REF] for applications in plate theory, to [START_REF] Kotschote | Maximal L p -regularity for a linear three-phase problem of parabolic-elliptic type[END_REF], [START_REF] Perfekt | The transmission problem on a three-dimensional wedge[END_REF] or [START_REF] Yanga | Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues[END_REF] for applications in electromagnetism and to [START_REF] Favini | Analytic semigroups generated by the dispersal process in two habitats incorporating individual behavior at the interface[END_REF], [START_REF] Limam | On Some Problems Set in a Biological Cell, Analysis and Resolution[END_REF] or [START_REF] Wang | Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission[END_REF] for other applications in population dynamics. Let us also mention that mathematical models involving biharmonic operators also arise in various fields such as elasticity for instance see [START_REF] Costabel | On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[END_REF], [START_REF] Hrustalev | A boundary-value problem for the biharmonic equation in elasticity theory, (Russian) Izv. Vysš[END_REF] or [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF], electrostatic see [START_REF] Cakoni | On the boundary integral equation methodfor a mixed boundary value problem of the biharmonic equation[END_REF], [START_REF] Guo | Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions[END_REF] or [START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF], plate theory [START_REF] Barraza Martínez | Regularity and asymptotic behavior for a damped plate-membrane transmission problem[END_REF], [START_REF] Favini | Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces[END_REF] or [START_REF] Hassine | Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping[END_REF] or population dynamics [START_REF] Cohen | A generalized diffusion model for growth and dispersal in population[END_REF], [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF] or [START_REF] Ochoa | A generalized reaction-diffusion model for spatial structures formed by motile cells[END_REF].

In this work, we consider an n-dimensional (with n 2) straight cylinder of the form

Ω = (a, b) × ω,
where a < b are two given real numbers while the section ω ⊂ R n-1 denotes a smooth bounded domain. This cylinder Ω is split into two (open) sub-cylinders Ω ± and an interface Γ given for some γ ∈ (a, b) by Ω -= (a, γ) × ω, Ω + = (γ, b) × ω and Γ = {γ} × ω, so that Ω = Ω -∪ Γ ∪ Ω + . We consider the following biharmonic equations,

(EQ pde ) k -∆ 2 u -= g -in Ω - k + ∆ 2 u + = g + in Ω + ,
where g -∈ L p (Ω -), g + ∈ L p (Ω + ) are given and k + , k -> 0.

We denote by (x, y) the spatial variables with x ∈ (a, b) and y ∈ ω. Then, we consider the following conditions on ∂Ω \ Γ, the lateral boundary of Ω,

(BC pde )                (1) u -(x, ζ) = 0, x ∈ (a, γ), ζ ∈ ∂ω, u + (x, ζ) = 0, x ∈ (γ, b), ζ ∈ ∂ω ∆u -(x, ζ) = 0, x ∈ (a, γ), ζ ∈ ∂ω, ∆u + (x, ζ) = 0, x ∈ (γ, b), ζ ∈ ∂ω (2)    u -(a, y) = ϕ - 1 (y), u + (b, y) = ϕ + 1 (y), y ∈ ω ∂u - ∂x (a, y) = ϕ - 2 (y), ∂u + ∂x (b, y) = ϕ + 2 (y), y ∈ ω,
where ϕ ± 1 and ϕ ± 2 will be given in appropriated spaces. The system is coupled on the interface Γ where we impose the following continuity conditions,

(T C pde )                  u -= u + on Γ ∂u - ∂x = ∂u + ∂x on Γ k -∆u -= k + ∆u + on Γ k - ∂∆u - ∂x = k + ∂∆u + ∂x on Γ.
Let us now explain, for instance, in population dynamics framework, the boundary and transmission conditions.

The first line of (BC pde ) - [START_REF] Barraza Martínez | Regularity and asymptotic behavior for a damped plate-membrane transmission problem[END_REF], means that the individuals could not lie on the boundaries (a, b) × ∂ω, because, for instance, they die or the edge is impassable. The second line means that there is no dispersal in the normal direction. We deduce that the dispersal vanishes on (a, b)×∂ω.

In (BC pde ) -(2), the population density and the flux are given on {a, b} × ω. This means that the habitats are not isolated.

In (T C pde ), the two first transmission conditions mean the continuity of the density and its flux at the interface, while the two second express, in some sense, the continuity of the dispersal and its flux at Γ. This work is a natural continuation of that done in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF]. Moreover, it completes the study realized in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF] where the authors have considered equations

k ± ∆ 2 u ± -l ± ∆u ± = f ± in Ω ±
and on the interface Γ, the last condition of (T C pde ) is replaced by

∂ ∂x (k + ∆u + -l + u + ) on Γ.
In the present work, l ± = 0, which must be treated differently, since the proof in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF] uses the representation formula obained in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] which is not defined when l ± = 0. Thus, we use the representation formula obtained in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF] which is different and does not corresponds to the representation formula in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] when the limit of l ± tends to 0. Therefore, even if the proof follows the same steps than the one in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF], the calculus are different and cannot be deduce from those in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF], since they use a different representation formula. Note that, due to the transmission conditions, we can not obtain a solution u ∈ W 4,p (Ω) in all Ω. We only can obtain a solution u such that u

| Ω -∈ W 4,p (Ω -) and u | Ω + ∈ W 4,p (Ω + ).
The paper is organized as follows.

First, in section 2, we recall the PDE transmission problem (P pde ) and we rewrite it under operational form. Then, in section 3, we recall some definitions about BIP operator and interpolation spaces. We give our hypotheses and their consequences. We present our main result in Theorem 3.9 and Corollary 3.11 which states existence and uniqueness of the solution of problem (P pde ) that is (EQ pde ) -(BC pde ) -(T C pde ) quoted above. In section 4, we state technical results which allow us to prove our main result. Section 5, is devoted to the proof of Theorem 3.9.

Operational formulation

In this section, we first recall the PDE problem (P pde ) composed by (EQ pde ) -(BC pde ) -(T C pde ). Then, we define the Laplace operator A 0 and we use it to rewrite (P pde ). Note that this operational form is a vector values problem. Finally, we generalize this problem replacing A 0 by a more general operator A. We consider the following problem

(P pde )                                                      k -∆ 2 u - = g - in Ω - k + ∆ 2 u + = g + in Ω + u -(x, ζ) = 0, ∆u -(x, ζ) = 0, x ∈ (a, γ), ζ ∈ ∂ω u + (x, ζ) = 0, ∆u + (x, ζ) = 0, x ∈ (γ, b), ζ ∈ ∂ω u -(a, y) = ϕ - 1 (y), u + (b, y) = ϕ + 1 (y), y ∈ ω ∂u - ∂x (a, y) = ϕ - 2 (y), ∂u + ∂x (b, y) = ϕ + 2 (y), y ∈ ω u - = u + on Γ ∂u - ∂x = ∂u + ∂x on Γ k -∆u - = k + ∆u + on Γ k - ∂∆u - ∂x = k + ∂∆u + ∂x on Γ.
Let us define A 0 , the Dirichlet Laplace operator in R n-1 , n ∈ N \ {0, 1}, as follows

D(A 0 ) := {ψ ∈ W 2,p (ω) : ψ = 0 on ∂ω} ∀ψ ∈ D(A 0 ), A 0 ψ = ∆ y ψ.
(

Thus, using operator A 0 , problem (P pde ) becomes the following vector valued problem

                                           u (4) -(x) + 2A 0 u -(x) + A 2 0 u -(x) = f -(x), for a.e. x ∈ (a, γ) u (4) + (x) + 2A 0 u + (x) + A 2 0 u + (x) = f + (x), for a.e. x ∈ (γ, b) u -(a) = ϕ - 1 , u + (b) = ϕ + 1 u -(a) = ϕ - 2 , u + (b) = ϕ + 2 u -(γ) = u + (γ) u -(γ) = u + (γ) k -u -(γ) + k -A 0 u -(γ) = k + u + (γ) + k + A 0 u + (γ) k -u (3) -(γ) + k -A 0 u -(γ) = k + u (3) + (γ) + k + A 0 u + (γ),
where

f -∈ L p (a, γ; L p (ω)), f + ∈ L p (γ, b; L p (ω)) and p ∈ (1, +∞), with u ± (x) := u(x, •) and f ± (x) := g ± (x, •)/k ± .
Note that the boundary conditions on ∂ω in (P pde ) do not appear in the previous system since they are already included in the domain of A 0 . Thus, a classical solution of this problem satisfies the boundary conditions on ∂ω in (P pde ).

Then, using operator (A, D(A)) instead of (A 0 , D(A 0 )) and X instead of L p (ω), we can write that f -∈ L p (a, γ; X) and f + ∈ L p (γ, b; X).

Moreover, in all the sequel, we will study the following more general transmission problem :

(P)                                            (EQ)    u (4) -(x) + 2Au -(x) + A 2 u -(x) = f -(x), for a.e. x ∈ (a, γ) u (4) + (x) + 2Au + (x) + A 2 u + (x) = f + (x), for a.e. x ∈ (γ, b) (BC) u -(a) = ϕ - 1 , u + (b) = ϕ + 1 u -(a) = ϕ - 2 , u + (b) = ϕ + 2 (T C)                u -(γ) = u + (γ) u -(γ) = u + (γ) k -u -(γ) + k -Au -(γ) = k + u + (γ) + k + Au + (γ) k -u (3) -(γ) + k -Au -(γ) = k + u (3) + (γ) + k + Au + (γ).
The transmission conditions (T C) will be split into

(T C1) u -(γ) = u + (γ) u -(γ) = u + (γ),
and

(T C2)    k -u -(γ) + k -Au -(γ) = k + u + (γ) + k + Au + (γ) k -u (3)
-(γ) + k -Au -(γ) = k + u (3)
+ (γ) + k + Au + (γ). Note that (T C2) is well defined from Lemma 3.7, see Section 3.2 below.

We will search a classical solution of problem (P), that is a solution u such that

   u -:= u |(a,γ) ∈ W 4,p (a, γ; X) ∩ L p (a, γ; D(A 2 )), u -∈ L p (a, γ; D(A)), u + := u |(γ,b) ∈ W 4,p (γ, b; X) ∩ L p (γ, b; D(A 2 )), u + ∈ L p (γ, b; D(A)), (2)
and which satisfies (EQ) -(BC) -(T C).

3 Assumptions, consequences and statement of results

The class BIP(X, θ)

Throughout the article, (X, • ) is a complex Banach space. We give some definitions about UMD spaces, sectorial operators and BIP operators.

Definition 3.1.

A Banach space X is a UMD space if and only if for all p ∈ (1, +∞), the Hilbert transform is bounded from L p (R, X) into itself (see [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF] and [START_REF] Burkholder | A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional[END_REF]).

Definition 3.2. A closed linear operator

T 1 is called sectorial of angle θ ∈ [0, π) if i) σ(T 1 ) ⊂ S θ , ii) ∀ θ ∈ (θ, π), sup λ(λ I -T 1 ) -1 L(X) : λ ∈ C \ S ω < +∞,
where

S ω := {z ∈ C : z = 0 and | arg(z)| < ω} if ω ∈ (0, π), (0, +∞) if ω = 0, (3)
see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 19, such an operator is noted A ∈ Sect (ω).

Remark 3.3. From [START_REF] Komatsu | Fractional powers of operators[END_REF], p. 342, we know that any injective sectorial operator T 1 admits imaginary powers

T is 1 , s ∈ R, but, in general, T is 1 is not bounded. Definition 3.4. Let θ ∈ [0, π).
We denote by BIP(X, θ), the class of sectorial injective operators

T 1 such that i) D(T 1 ) = R(T 1 ) = X, ii) ∀ s ∈ R, T is 1 ∈ L(X), iii) ∃ C ≥ 1, ∀ s ∈ R, ||T is 1 || L(X) ≤ Ce |s|θ ,
see [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], p. 430.

Remark 3.5. From [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], proof of Proposition 3.2.1, c), p. 71, D(T 1 ) ∩ R(T 1 ) = X.

Interpolation spaces

Here we recall a definition given, for instance, in [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF], [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], [START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF] or in [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF] and a result from [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF] concerning real interpolation spaces.

Definition 3.6. Let T 2 : D(T 2 ) ⊂ X -→ X be a linear operator such that (0, +∞) ⊂ ρ(T 2 ) and ∃ C > 0 : ∀ t > 0, t(T 2 -tI) -1 L(X) C. (4)
Let k ∈ N \ {0}, θ ∈ (0, 1) and q ∈ [1, +∞]. We will use the real interpolation spaces

(D(T k 2 ), X) θ,q = (X, D(T k 2 )) 1-θ,q ,
defined, for instance, in [START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF], or in [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF].

In particular, for k = 1, we have the following characterization

(D(T 2 ), X) θ,q := ψ ∈ X : t -→ t 1-θ T 2 (T 2 -tI) -1 ψ X ∈ L q * (0, +∞) ,
where L q * (0, +∞) is given by

L q * (0, +∞) := f ∈ L q (0, +∞) : +∞ 0 f (t) q dt t 1/q < +∞ , for q ∈ [1, +∞),
and for q = +∞, by

L ∞ * (0, +∞; C) := f measurable on (0, +∞) : sup t∈(0,+∞) |f (t)| < +∞ ,
see [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] p. 325, or [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], p. 665, Teorema 3, or section 1.14 of [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], where this space is denoted by (X, D(T 2 )) 1-θ,q . Note that we can also characterize the space (D(T 2 ), X) θ,q taking into account the Osservazione, p. 666, in [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF]. We set also, for any k ∈ N \ {0}

(D(T 2 ), X) k+θ,q := ψ ∈ D(T k 2 ) : T k 2 ψ ∈ (D(T 2 ), X) θ,q , (X, D(T 2 )) k+θ,q := ψ ∈ D(T k 2 ) : T k 2 ψ ∈ (X, D(T 2 )) θ,q .
We recall the following lemma.

Lemma 3.7 ([12]

). Let T 2 be a linear operator satisfying [START_REF] Cakoni | On the boundary integral equation methodfor a mixed boundary value problem of the biharmonic equation[END_REF]. Let u such that

u ∈ W n,p (a 1 , b 1 ; X) ∩ L p (a 1 , b 1 ; D(T k 2 )),
where

a 1 , b 1 ∈ R with a 1 < b 1 , n, k ∈ N \ {0} and p ∈ (1, +∞).
Then for any j ∈ N satisfying the Poulsen condition 0 < 1 p + j < n and s ∈ {a 1 , b 1 }, we have

u (j) (s) ∈ (D(T k 2 ), X) j n + 1 np ,p .
This result is proved in [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], p. 678, Teorema 2'.

Hypotheses

Througout this article, k + , k -∈ R + \ {0} and A denotes a closed linear operator in X.

We assume the following hypotheses:

(H 1 ) X is a UMD space, (H 2 ) 0 ∈ ρ(A), (H 3 ) -A ∈ BIP(X, θ A ) for some θ A ∈ (0, π/2), (H 4 ) -A ∈ Sect(0).
Note that assumption (H 4 ) means that -A is a sectorial operator of any angle θ ∈ (0, π). This assumption is satisfied, for instance, by elliptic differential operators of second order. Now, we give some consequences on our hypotheses.

Consequences

1. Note that A 0 satisfies all the previous hypotheses with X = L q (ω), with q ∈ (1, +∞).

From [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF], Proposition 3, p. 207, X satisfies (H 1 ) and from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], Theorem 9.15, p. 241 and Lemma 9.17, p. 242, A 0 satisfies (H 2 ). Moreover, from [START_REF] Prüss | Imaginary powers of elliptic second order differential operators in L p -spaces[END_REF], Theorem C, p. 166-167, (H 3 ) is satisfied for every θ A ∈ (0, π), thus (H 4 ) is also satisfied.

2. In the scalar case, to solve each equation of (EQ), it is necessary to introduce the square roots ± √ -A of the characteristic equations

x 4 + 2Ax 2 + A 2 = 0,
this is why, in our operational case, we consider,

M := - √ -A. (5)
From (H 3 ), -A is a sectorial operator, thus the existence of M is ensured, see for instance [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], e), p. 25.

3. From (H 3 ), we have -A ∈ BIP(X, θ A ), then, from [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], Proposition 3.2.1, e), p. 71, we deduce -M ∈ BIP(X, θ A /2).

Since 0 < θ A /2 < π/2, we deduce that M generate a bounded analytic semigroup (e xM ) x 0 , see [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], Theorem 2, p. 437. Furthermore, due to [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], Theorem 4, p. 441, for n ∈ N \ {0}, we get -nM ∈ BIP(X, θ A /4 + ε), for any ε ∈ (0, π/2 -θ A /4). Then, due to [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], Theorem 2, p. 437, nM generate a bounded analytic semigroup (e nxM ) x 0 . The last results use also the works of [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] and [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF].

The main results

To solve our transmission problem (P), we introduce two auxiliary problems:

(P -)          u (4) -(x) + 2Au -(x) + A 2 u -(x) = f -(x), for a.e. x ∈ (a, γ) u -(a) = ϕ - 1 , u -(γ) = ψ 1 u -(a) = ϕ - 2 , u -(γ) = ψ 2 ,
and

(P + )          u (4) + (x) + 2Au + (x) + A 2 u + (x) = f + (x), for a.e. x ∈ (γ, b) u + (γ) = ψ 1 , u + (b) = ϕ + 1 u + (γ) = ψ 2 , u + (b) = ϕ + 2 .
Remark 3.8. Recall that a classical solution of (P ± ), in L p (J; X), with J = (a, γ) or (γ, b), is a solution to (P ± ) such that

u ± ∈ W 4,p (J; X) ∩ L p (J; D(A 2 )), u ∈ L p (J; D(A)).
We say that u is a classical solution of (P) if and only if there exist ψ 1 , ψ 2 ∈ X such that (i) u -is a classical solution of (P -), (ii) u + is a classical solution of (P + ), (iii) u -and u + satisfy (T C2).

Note that by construction, if there exist a classical solution u -of (P -) and u + of (P + ), then u - and u + satisfy (T C1).

Our goal is to prove that there exists a unique couple (ψ 1 , ψ 2 ) which satisfies (i), (ii) and (iii). This will lead us to obtain our main result. Theorem 3.9. Let f -∈ L p (a, γ; X) and f + ∈ L p (γ, b; X). Assume that (H 1 ), (H 2 ) and (H 3 ) be satisfied. Then, there exists a unique classical solution u of the transmission problem (P) if and only if

ϕ + 1 , ϕ - 1 ∈ (D(A), X) 1+ 1 2p ,p and ϕ + 2 , ϕ - 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p . (6)
Remark 3.10.

1. In the proof of Theorem 3.9, we use operator M and interpolation spaces (D(M ), X) 3-j+ 1 p ,p , j = 0, 1, 2, 3. But from the reiteration Theorem, we get

     (D(M ), X) 3+ 1 p ,p = (D(A), X) 1+ 1 2p ,p , (D(M ), X) 2+ 1 p ,p = (D(A), X) 1+ 1 2 + 1 2p ,p (D(M ), X) 1+ 1 p ,p = (D(A), X) 1 2p ,p , (D(M ), X) 1 p ,p = (D(A), X) 1 2 + 1 2p ,p . (7)
2. We can generalize the previous Theorem by considering a transmission problem between N juxtaposed habitats, with N ∈ N \ {0}. For instance, with N = 3, it suffices to use Theorem 3.9 on the two first habitats and then apply it on the transmission problem between the second and third habitat to solve the problem. By recurrence, we obtain the result for N habitats.

As consequence of Theorem 3.9, we deduce the following result for problem (P pde ). Consider the case A = A 0 (other cases can be treated).

Corollary 3.11. Assume that ω is a bounded open set of R n-1 where n ≥ 2 with C 2 -boundary. Let g + ∈ L p (Ω + ) and g -∈ L p (Ω -) with p ∈ (1, +∞) and p > n; let k + , k -∈ R + \ {0}. Then, there exists a unique solution u of (P pde ), such that

u -∈ W 4,p (Ω -), u + ∈ W 4,p (Ω + ), if and only if ϕ ± 1 , ϕ ± 2 ∈ W 2,p (ω) ∩ W 1,p 0 (ω), ∆ϕ ± 1 , ∈ W 2-1 p ,p (ω) ∩ W 1,p 0 (ω) and ∆ϕ ± 2 ∈ W 1-1 p ,p (ω) ∩ W 1,p 0 (ω).
Proof. The proof is quite similar to the one of Corollary 3.6 in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF], see also Corollary 2.7 in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF].

Taking into account the result of Theorem 3.9, we can also obtain anisotropic results by considering f -∈ L p (a, γ; L q (ω)) and f + ∈ L p (γ, b; L q (ω)) with p, q ∈ (1, +∞).

Preliminary results

Throughout this article, we set

c = γ -a > 0 and d = b -γ > 0.
From Remark 3.8, to solve problem (P), we first have to study problems (P -) and (P + ). To this end, we need the following invertibility result obtained in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF]. All these exponentials are analytic semigroups and they are well defined due to statement 3 of Section 3.4. For a detailed proof, see [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], Proposition 4.5.

Problem (P -)

Proposition 4.2. Let f -∈ L p (a, γ; X). Assume that (H 1 ), (H 2 ) and (H 3 ) hold. Then there exists a unique classical solution u -of problem (P -) if and only if

ϕ - 1 , ψ 1 ∈ (D(A), X) 1+ 1 2p ,p and ϕ - 2 , ψ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p . (9)
Moreover

u -(x) = e (x-a)M -e (γ-x)M α - 1 + (x -a)e (x-a)M -(γ -x)e (γ-x)M α - 2 + e (x-a)M + e (γ-x)M α - 3 + (x -a)e (x-a)M + (γ -x)e (γ-x)M α - 4 + F -(x), (10)
where

                         α - 1 := - 1 2 U -1 - I + (I + cM ) e cM ψ 1 -ce cM ψ 2 + φ- 1 α - 2 := 1 2 U -1 - I + e cM M ψ 1 + I -e cM ψ 2 + φ- 2 α - 3 := 1 2 V -1 - I -(I + cM ) e cM ψ 1 + ce cM ψ 2 + φ- 3 α - 4 := - 1 2 V -1 - I -e cM M ψ 1 + I + e cM ψ 2 + φ- 4 , (11)
with

                                                   φ1 -:= 1 2 U -1 - ϕ - 1 + e cM ϕ - 1 + c M ϕ - 1 + ϕ - 2 -F -(a) -F -(γ) φ2 -:= - 1 2 U -1 - M ϕ - 1 -ϕ - 2 + F -(a) + F -(γ) - 1 2 U -1 -e cM M ϕ - 1 + ϕ - 2 -F -(a) -F -(γ) φ3 -:= 1 2 V -1 - ϕ - 1 -e cM ϕ - 1 + c M ϕ - 1 + ϕ - 2 -F -(a) + F -(γ) φ4 -:= - 1 2 V -1 - M ϕ - 1 -ϕ - 2 + F -(a) -F -(γ) + 1 2 V -1 -e cM M ϕ - 1 + ϕ - 2 -F -(a) + F -(γ) , ( 12
)
and F -is the unique classical solution of problem

         u (4) -(x) + 2Au -(x) + A 2 u -(x) = f -(x), a.e. x ∈ (a, γ) u -(a) = u -(γ) = 0 u -(a) = u -(γ) = 0. ( 13
)
Proof. Due to [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], Theorem 2.8, statement 2., there exists a unique classical solution u -of problem (P -) if and only if (9) holds. Moreover, adapting the representation formula of u given by ( 31) in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], where u, f , b, F 0,f , ϕ 1 , ϕ 2 , ϕ 3 and ϕ 4 are respectively replaced by u -, f -, γ, F -, ϕ - 1 , ψ 1 , ϕ - 2 and ψ 2 , we obtain that the representation formula of u -is given by ( 10), [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] and [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF]. Remark 4.3. In the previous proposition, since [START_REF] Favini | Transmission Problem for an Abstract Fourth-order Differential Equation of Elliptic Type in UMD Spaces[END_REF], [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] and ( 12) hold, we have

α - 1 , α - 3 ∈ D(M 3 ) and α - 2 , α - 4 ∈ D(M 2 ).
Moreover, since F -is a classical solution of ( 13), due to Lemma 3.7, for j = 0, 1, 2, 3 and s = a or γ

F (j) -(s) ∈ (D(M ), X) 3-j+ 1 p ,p .

Problem (P + )

Proposition 4.4. Let f + ∈ L p (γ, b; X). Assume that (H 1 ), (H 2 ) and (H 3 ) hold. Then, there exists a unique classical solution u + of problem (P + ) if and only if

ϕ + 1 , ψ 1 ∈ (D(A), X) 1+ 1 2p ,p and ϕ + 2 , ψ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p . ( 14
)
Moreover

u + (x) = e (x-γ)M -e (b-x)M α + 1 + (x -γ)e (x-γ)M -(b -x)e (b-x)M α + 2 + e (x-γ)M + e (b-x)M α + 3 + (x -γ)e (x-γ)M + (b -x)e (b-x)M α + 4 +F + (x), (15)
where

                                 α + 1 := 1 2 U -1 + I + (I + dM ) e dM ψ 1 + de dM ψ 2 + φ+ 1 α + 2 := - 1 2 U -1 + I + e dM M ψ 1 -I -e dM ψ 2 + φ+ 2 α + 3 := 1 2 V -1 + I -(I + dM ) e dM ψ 1 -de dM ψ 2 + φ+ 3 α + 4 := - 1 2 V -1 + I -e dM M ψ 1 -I + e dM ψ 2 + φ+ 4 , (16)
with

                                                   φ1 + := - 1 2 U -1 + ϕ + 1 + e dM ϕ + 1 + d M ϕ + 1 -ϕ + 2 + F + (γ) + F + (b) φ2 + := 1 2 U -1 + M ϕ + 1 + ϕ + 2 -F + (γ) -F + (b) + 1 2 U -1 + e dM M ϕ + 1 -ϕ + 2 + F + (γ) + F + (b) φ3 + := 1 2 V -1 + ϕ + 1 -e dM ϕ + 1 + d M ϕ + 1 -ϕ + 2 -F + (γ) + F + (b) φ4 + := - 1 2 V -1 + M ϕ + 1 + ϕ + 2 + F + (γ) -F + (b) + 1 2 V -1 + e dM M ϕ + 1 -ϕ + 2 -F + (γ) + F + (b) , ( 17
)
and F + is the unique classical solution of problem

   u (4) + (x) + 2Au + (x) + A 2 u + (x) = f + (x), a.e. x ∈ (γ, b) u + (γ) = u + (b) = u + (γ) = u + (b) = 0. ( 18
)
Proof. Due to [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], Theorem 2.8, statement 2., there exists a unique classical solution u + of problem (P + ) if and only if [START_REF] Haase | The functional calculus for sectorial Operators[END_REF] holds. Moreover, adapting the representation formula of u given by ( 31) in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], where u, f , a, F 0,f , ϕ 1 , ϕ 2 , ϕ 3 and ϕ 4 are respectively replaced by u + , f + , γ, F + , ψ 1 , ϕ + 1 , ψ 2 and ϕ + 2 , we obtain that the representation formula of u + is given by ( 15), ( 16) and [START_REF] Komatsu | Fractional powers of operators[END_REF]. Remark 4.5. In the previous proposition, since [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], ( 16) and ( 17) hold, we have

α + 1 , α + 3 ∈ D(M 3 ) and α + 2 , α + 4 ∈ D(M 2 ).
Moreover, since F + is a classical solution of ( 18), due to Lemma 3.7, for j = 0, 1, 2, 3 and s = γ or b F

(j) + (s) ∈ (D(M ), X) 3-j+ 1 p ,p .

The transmission system

In this section we give the proof of Theorem 4.6 stated below. This theorem ensures the equivalence between the resolution of problem (P ) and the resolution of the following system

     P + 1 + P - 1 M ψ 1 -P + 2 -P - 2 ψ 2 = S 1 P + 2 -P - 2 M ψ 1 -P + 3 + P - 3 ψ 2 = S 2 . ( 19
)
The coefficients of the previous system are given by

                 P + 1 = k + U -1 + I + e dM 2 + V -1 + I -e dM 2 P + 2 = k + U -1 + + V -1 + I -e 2dM P + 3 = k + U -1 + I -e dM 2 + V -1 + I + e dM 2 (20)
and

                 P - 1 = k -U -1 - I + e cM 2 + V -1 - I -e cM 2 P - 2 = k -U -1 -+ V -1 - I -e 2cM P - 3 = k -U -1 - I -e cM 2 + V -1 - I + e cM 2 . ( 21
)
The seconds members are given by

S 1 = 2k + φ+ 2 + φ+ 4 + e dM φ+ 2 -φ+ 4 -2k - φ- 2 -φ- 4 + e cM φ- 2 + φ- 4 -M -2 Š ( 22
)
where Š = -k + F

(

+ (γ) + k + M 2 F + (γ) + k -F 3)
-(γ) -k -M 2 F -(γ) (3)
and

S 2 = 2k + φ+ 2 + φ+ 4 -e dM φ+ 2 -φ+ 4 + 2k - φ- 2 -φ- 4 -e cM φ- 2 + φ- 4 . ( 24
)
Theorem 4.6. Let f -∈ L p (a, γ; X) and f + ∈ L p (γ, b; X). Assume that (H 1 ), (H 2 ) and (H 3 ) hold. Then, problem (P) has a unique classical solution if and only if the data ϕ + 1 , ϕ - 1 , ϕ + 2 , ϕ - 2 satisfy (6) and system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] has a unique solution (ψ 1 , ψ 2 ) such that

(ψ 1 , ψ 2 ) ∈ (D(A), X) 1+ 1 2p ,p × (D(A), X) 1+ 1 2 + 1 2p ,p . (25)
Proof. Assume that problem (P ) has a unique classical solution u. We set

ψ 1 = u -(γ) = u + (γ) and ψ 2 = u -(γ) = u + (γ).
We get that u -(respectively u + ) is the classical solution of (P -) (respectively (P + )). Then, applying Proposition 4.2 (respectively Proposition 4.4), we obtain [START_REF] Costabel | On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[END_REF]. Moreover, from (7), we have [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. To this end, since u satisfies the transmission conditions (T C2). Then, we obtain the following system

ψ 1 ∈ (D(M ), X) 3+ 1 p ,p and ψ 2 ∈ (D(M ), X) 2+ 1 p ,p . It remains to prove that (ψ 1 , ψ 2 ) is solution of system
   k -u (3) -(γ) + k -Au -(γ) = k + u (3) + (γ) + k + Au + (γ) k -u -(γ) + k -Au -(γ) = k + u + (γ) + k + Au + (γ). Hence    k + u (3) + (γ) -M 2 u + (γ) -k -u (3) -(γ) -M 2 u -(γ) = 0 k + u + (γ) -M 2 u + (γ) -k -u -(γ) -M 2 u -(γ) = 0.
by ( 22), ( 23) and [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]. Then, we deduce that system [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF] writes

                                                                   - k + 2 U -1 + I + e dM I + e dM M ψ 1 -I -e dM ψ 2 - k + 2 V -1 + I -e dM I -e dM M ψ 1 -I + e dM ψ 2 - k - 2 U -1 - I + e cM I + e cM M ψ 1 + I -e cM ψ 2 - k - 2 V -1 - I -e cM I -e cM M ψ 1 + I + e cM ψ 2 = - S 1 2 - k + 2 U -1 + I -e dM I + e dM M ψ 1 -I -e dM ψ 2 - k + 2 V -1 + I + e dM I -e dM M ψ 1 -I + e dM ψ 2 + k - 2 U -1 - I -e cM I + e cM M ψ 1 + I -e cM ψ 2 + k - 2 V -1 - I + e cM I -e cM M ψ 1 + I + e cM ψ 2 = - S 2 2 .
Finally, we obtain

                                                               k + U -1 + I + e dM 2 + V -1 + I -e dM 2 M ψ 1 +k -U -1 - I + e cM 2 + V -1 - I -e cM 2 M ψ 1 -k + U -1 + + V -1 + I -e 2dM ψ 2 +k -U -1 -+ V -1 - I -e 2cM ψ 2 = S 1 k + U -1 + + V -1 + I -e 2dM M ψ 1 -k -U -1 -+ V -1 - I -e 2cM M ψ 1 -k + U -1 + I -e dM 2 + V -1 + I + e dM 2 ψ 2 -k -U -1 - I -e cM 2 + V -1 - I + e cM 2 ψ 2 = S 2 . ( 31
)
Then, using ( 20) and ( 21) system (31) become system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. So, (ψ 1 , ψ 2 ) is solution of system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. Conversely, if (6) holds and system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] has a unique solution (ψ 1 , ψ 2 ) such that

ψ 1 ∈ (D(A), X) 1+ 1 2p ,p and ψ 2 ∈ (D(A), X) 1+ 1 2 + 1 2p ,p ,
Then, considering u -(respectively u + ) the unique classical solution of problem (P -) (respectively problem (P + )), we obtain that

u = u -in Ω - u + in Ω + ,
is the unique classical solution of problem (P ).

Functional calculus

To prove Theorem 3.9, it remains, from Theorem 4.6, to solve system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. To this end, we have to show that the determinant operator of system ( 19) is an invertible operator using functional calculus.

We first recall some classical notations. Let θ ∈ (0, π), we denote by H(S θ ) the space of holomorphic functions on S θ (defined by ( 3)). Moreover, we consider the following subspace of H(S θ ):

E ∞ (S θ ) := f ∈ H(S θ ) : f = O(|z| -s ) (|z| → +∞) for some s > 0 .
Thus, E ∞ (S θ ) is the space of polynomially decreasing holomorphic functions at +∞.

Let T be an invertible sectorial operator of angle θ T ∈ (0, π) and let f ∈ E ∞ (S θ ), with θ ∈ (θ T , π), then, by functional calculus, we can define f (T ) ∈ L(X), see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 45.

We recall a useful invertibility result from [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], Lemma 5.3.

Lemma 4.7 ([19]

). Let P be an invertible sectorial operator in X with angle θ, for all θ ∈ (0, π).

Let G ∈ H(S θ ), for some θ ∈ (0, π), such that

(i) 1 -G ∈ E ∞ (S θ ), (ii) G(x) = 0 for any x ∈ R + \ {0}.
Then, G(P ) ∈ L(X), is invertible with bounded inverse. Now, to inverse the determinant of system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], we introduce some holomorphic functions which will play the role of G in the previous lemma. Then, we study them on the positive real axis.

Let δ > 0 and z ∈ C \ R -. We set

   u δ (z) = 1 -e -2δ √ z -2δ √ ze -δ √ z v δ (z) = 1 -e -2δ √ z + 2δ √ ze -δ √ z .
Then, we have

U + = u d (-A), U -= u c (-A), V + = v d (-A) and V -= v c (-A).
We set

C + = {z ∈ C : Re(z) 0}.
Moreover, from [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], Lemma 4.4, we have u δ (z) = 0 and v δ (z) = 0 for z ∈ C + \ {0}, thus we note

                     f δ,1 (z) = u -1 δ (z) 1 + e -δ √ z 2 + v -1 δ (z) 1 -e -δ √ z 2 f δ,2 (z) = u -1 δ (z) + v -1 δ (z) 1 -e -2δ √ z f δ,3 (z) = u -1 δ (z) 1 -e -δ √ z 2 + v -1 δ (z) 1 + e -δ √ z 2 g δ (z) = 16 u -1 δ (z)v -1 δ (z)e -2δ √ z .
Thus, we obtain that

P + 1 = k + f d,1 (-A), P + 2 = k + f d,2 (-A), P + 3 = k + f d,3 (-A) P - 1 = k -f c,1 (-A), P - 2 = k -f c,2 (-A), P - 3 = k -f c,3 (-A).
Moreover, for z ∈ C + \ {0}, we define

f (z) := k 2 + g d (z) + k 2 -g c (z) + k + k -(f d,1 (z)f c,3 (z) + f c,1 (z)f d,3 (z) + 2f d,2 (z)f c,2 (z)) .
Note that f ∈ H(S θ ), for all θ ∈ (0, π). It follows that

f (-A) = 16k 2 + U -1 + V -1 + e 2dM + 16k 2 -U -1 -V -1 -e 2cM + P + 1 P - 3 + P - 1 P + 3 + 2P + 2 P - 2 . ( 32
)
Lemma 4.8. For all x ∈ R + \ {0}, then f (x) does not vanish.

Proof. Let δ > 0, from Lemma 4.4, in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], for all z ∈ C + \ {0}, we have

0 < |u δ (z)| and 0 < |v δ (z)|.
Thus, for all x > 0, one has u δ (x), v δ (x) > 0. Moreover, for all x > 0, we deduce that

f δ,1 (x), f δ,2 (x), f δ,3 (x), g δ (x) > 0 and since k + k -> 0, it follows that f (x) = k 2 + g d (x) + k 2 -g c (x) + k + k -(f d,1 (x)f c,3 (x) + f c,1 (x)f d,3 (x) + 2f d,2 (x)f c,2 (x)) > 0.

Proof of the main result

Assume that (P) has a unique classical solution, then, (6) holds from Theorem 4.6. Conversely, Assume that (6) holds. From Theorem 4.6, it suffices to show that system (19) has a unique solution such that (25) is satisfies. The proof is divided in three parts. In the first part, we make explicit the determinant of system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. Then, in the second part, we show the uniqueness of the solution, to this end, we inverse the determinant with the help of functional calculus. Finally, in the last part, we prove that ψ 1 and ψ 2 have the expected regularity.

Calculus of the determinant

In this section, we make explicit the determinant. Recall system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]:

     P + 1 + P - 1 M ψ 1 -P + 2 -P - 2 ψ 2 = S 1 P + 2 -P - 2 M ψ 1 -P + 3 + P - 3 ψ 2 = S 2 .
Moreover, we writes the previous system as the matrix equation ΛΨ = S, where

Λ =   M P + 1 + P - 1 -P + 2 -P - 2 M P + 2 -P - 2 -P + 3 + P - 3   , Ψ = ψ 1 ψ 2 , S = S 1 S 2 .
To solve system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], we calculate the determinant of the associated matrix Λ: det(Λ) := -M P + 1 + P -

We begin to calculate each terms separately. Then, we have

P + 1 P + 3 = k 2 + U -1 + I + e dM 2 + V -1 + I -e dM 2 U -1 + I -e dM 2 + V -1 + I + e dM 2 = k 2 + U -2 + + V -2 + I -e 2dM 2 + U -1 + V -1 + I + e dM 4 + I -e dM 4
and I + e dM 4 + I -e dM 4 = 2 I + e 2dM 2 + 8e 2dM .

Thus

P + 1 P + 3 = k 2 + U -2 + + V -2 + I -e 2dM 2 + 2U -1 + V -1 + I + e 2dM 2 + 8U -1 + V -1 + e 2dM .
Moreover, we have

P + 2 2 = k 2 + U -1 + + V -1 + 2 I -e 2dM 2 = k 2 + U -2 + + V -2 + I -e 2dM 2 + 2k 2 + U -1 + V -1 + I -e 2dM 2 .
It follows

P + 1 P + 3 -P + 2 2 = k 2 + U -2 + + V -2 + I -e 2dM 2 + 2U -1 + V -1 + I + e 2dM 2
= 2k 2 + U -1 + V -1 + 4e 2dM + 4e 2dM = 16k 2 + U -1 + V -1 + e 2dM .
In the same way, replacing respectively k + , U -1 + , V -1 + and d by k -, U -1 -, V -1 -and c, we obtain

P - 1 P - 3 -P - 2 2 = 16k 2 -U -1 -V -1 -e 2cM .
Thus, the determinant of Λ writes det(Λ) = -M 16k 2 + U -1 + V -1 + e 2dM + 16k 2 -U -1 -V -1 -e 2cM + P + 1 P - 3 + P -

1 P + 3 + 2P + 2 P - 2 . ( 33
)
Finally, from [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF] and [START_REF] Wang | Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission[END_REF], we obtain det(Λ) = -M f (-A). [START_REF] Yanga | Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues[END_REF] Note that, since f (-A) ∈ L(X), it follows that D(det(Λ)) = D(M ).

Inversion of the determinant

Let C 1 , C 2 two linear operators in X. We note C 1 ∼ C 2 to means that C 1 = C 2 + Σ, where Σ is a finite sum of term of type kM n e αM , with k ∈ R, n ∈ N and α ∈ R + \ {0}. Note that Σ is a regular term in the sense: Moreover, we have

Σ ∈ L(X)
16k 2 + W 2 U -1 + V -1 + e 2dM ∼ 0 and 16k 2 -W 2 U -1 -V -1 -e 2cM ∼ 0.
We then deduce the following relation

-M -1 W 2 det(Λ) = 16k 2 + W 2 U -1 + V -1 + e 2dM + 16k 2 -W 2 U -1 -V -1 -e 2cM +W P + 1 W P - 3 + W P - 1 W P + 3 + 2W P + 2 W P - 2 ∼ 4k + k -I + 4k + k -I + 8k + k -I = 16k + k -I.
Thus, we get

det(Λ) = -W -2 M   16k + k -I + j∈J k j M n j e α j M   , ( 35
)
where J is a finite set and for all j ∈ J: k j ∈ R, n j ∈ N and α j ∈ R + \ {0}.

From (35), we have det(Λ) = -16k + k -W -2 M F, (

where

F = I + j∈J k j 16k + k - M n j e α j M . ( 37
) For z ∈ C \ R -, we set f (z) = 1 + j∈J k j 16k + k - - √ z n j e -α j √ z .
Then, F = f (-A) and from [START_REF] Yanga | Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues[END_REF] and (36), we have

f (-A) = -M -1 det(Λ) = 16k + k -W -2 f (-A).
Thus, by construction, for z ∈ C \ R -, the link between f and f is

f (z) = 16k + k -u -2 d (z)u -2 c (z)v -2 d (z)v -2 c (z) f (z). ( 38
)
Proposition 5.1. The operator F ∈ L(X) defined by (37), is invertible with bounded inverse.

Lemma 4 . 1 (

 41 [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF]). The operators U + , U -, V + , V -∈ L(X) defined by    U -:= I -e 2cM + 2cM e cM , U + := I -e 2dM + 2dM e dM V -:= I -e 2cM -2cM e cM , V + := I -e 2dM -2dM e dM , (8) are invertible with bounded inverse.

- 2 + + V - 2 +I -e 2dM 2 -2k 2 + U - 1 + V - 1 +I -e 2dM 2 = 2k 2 + U - 1 + V - 1 +I + e 2dM 2 -

 2221122112 I -e 2dM 2 + 4e 2dM

1 ∼ 1 ∼+ 2 ∼ 2 ∼+ 3 ∼ 3 ∼

 112233 with Σ(X) ⊂ D(M ∞ ) := k 0 D(M k ).Since U ± ∼ I and V ± ∼ I, then setting W = U + U -V + V -∼ I, we obtain 2k + I, W P - 2k -I, W P 2k + I, W P - 2k -I, W P 2k + I, W P - 2k -I.

Acknowledgments

This research is supported by CIFRE contract 2014/1307 with Qualiom Eco company and partially by the LMAH and the european funding ERDF through grant project Xterm. I would like to thank the referee for the valuable comments and corrections which help to improve this paper.

Moreover, for all x ∈ (a, γ), we have u -(x) = e (x-a)M -e (γ-x)M α - 1 + (x -a)e (x-a)M -(γ -x)e (γ-x)M α - 2 + e (x-a)M + e (γ-x)M α - 3 + (x -a)e (x-a)M + (γ -x)e (γ-x)M α -

4

+F -(x)

u -(x) = M e (x-a)M + e (γ-x)M α - 1 + M e (x-a)M -e (γ-x)M α - u -(x) = M 2 e (x-a)M -e (γ-x)M α - 1 + M 2 e (x-a)M + e (γ-x)M α - 3 + F -(x)

Thus, we obtain

Moreover, from (13), we have

Note that, from Remark 4.3, all terms of the previous equality are well defined.

In the same way, for all x ∈ (γ, b), we have

and

+ (x).

Hence, we obtain

Moreover, from ( 18), we have

As previously, from Remark 4.5, all terms of the previous equality are justified. Then, from ( 23), ( 26), ( 27), ( 28) and ( 29), we deduce that system [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] is equivalent to

Thus, we obtain the following system

Now, we regroup all source terms provided from ( 11), ( 12), ( 16) and ( 17) in S 1 and S 2 , defined

Proof. Note that f, f ∈ H(S θ ), for a given θ ∈ (0, π). Moreover, for z ∈ C \ R -and j ∈ J,

From Lemma 4.4 in [START_REF] Thorel | Operational approach for biharmonic equations in L p -spaces[END_REF], u d , u c , v d and v c do not vanish on C + \ {0}. Moreover, due to Lemma 4.8, f does not vanish on R + \ {0}, Thus, we deduce from (38) that f do not vanish on R + \ {0}.

Finally, we apply Lemma 4.7 with P = -A and G = f to obtain that F = f (-A) ∈ L(X) is invertible with bounded inverse.

We are now in position to prove the main result of this section.

Proposition 5.2. The operator det(Λ), defined by ( 36) is invertible with bounded inverse.

Proof. From Lemma 4.1, U + , U -, V + and V -are bounded invertible operators with bounded inverse. So we deduce that W -2 is invertible with bounded inverse. Moreover, from (H 2 ), (36) and Proposition 5.1, we obtain that det(Λ) is invertible with bounded inverse.

Regularity

To study the regularity, we need to recall the following technical result from [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], Lemma 5.1.

Lemma 5.3 ([19]

). Let V ∈ L(X) such that 0 ∈ ρ(I + V ). Then, there exists

and

From Proposition 5.2, system ( 19) has a unique solution (ψ 1 , ψ 2 ). From Theorem 4.6, it remains to prove that

To this end, we have to study the regularity of the inverse of the determinant det(Λ).

Lemma 5.4. There exists

Proof. From (36) and Proposition 5.2, we have

From Lemma 5.3, there exists R F ∈ D(M ∞ ), such that

Moreover, for δ > 0, we know that

We deduce that there exists

Now, we study the regularity of ψ 1 and ψ 2 . We recall that ΛΨ = S, where Λ is invertible from Proposition 5.2. From Lemma 5.3, there exist

From ( 39) and (40), there exist

where S 1 is given by ( 22) and S 2 is given by [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]. Since F + is a classical solution of problem (P + ) and F -is a classical solution of problem (P -), then from Remark 4.3 and Remark 4.5, we obtain that Š defined by ( 23) has the following regularity

Moreover, from (7), we have (43) So, from ( 22), ( 24), (39), ( 42) and (43), we have