Miloud Cherbi

Jérémy Joffre

Marion Beuzelin

Yves Roland

Louis Niquet

Raphael Favory

Hadi Khachab

Brahim Harbaoui

Gerald Vanzetto

Frédéric Bouisset

Eric Bonnefoy

Nicolas Lamblin

Edouard Gerbaud

Laurent Bonello

Bruno Levy

Pascal Lim

Nicolas Combaret

Benjamin Marchandot

Benoit Lattuca

Guillaume Leurent

Olivier Lairez

Etienne Puymirat

François Roubille

Clément Delmas
email: delmas.clement@chu-toulouse.fr

Characteristics, management, and mid-term prognosis of older adults with cardiogenic shock admitted to intensive care units: Insights from the FRENSHOCK registry

Keywords: ACE, angiotensin-converting enzyme, aHR, adjusted hazard ratio, AMI, acute myocardial infarction, aMCS, acute mechanical circulatory support, aOR, adjusted odds ratio, BNP, Brain natriuretic peptide, CI, confidence interval, CKD, chronic kidney disease, CS, cardiogenic shock, CVD, cardiovascular disease, DBP, diastolic blood pressure, ECLS, extracorporeal life support, GDMT, Guideline-directed medical therapy, GFR, glomerular filtration rate, HF, heart failure, HR, hazard ratio, HTx, heart transplantation, IABP, intraaortic balloon pump, ICU, intensive care unit, ICCU, intensive cardiac care unit, ICD, implantable cardioverter-defibrillator, IQR, interquartile range, LVEF, left ventricle ejection fraction, MBP, mean blood pressure, NYHA, New York Heart Association, OP, older patients, PCI, percutaneous coronary intervention, PSVtdi, peak systolic velocity tissue Doppler imaging, PT, prothrombin time, RRT, renal replacement therapy, SBP, systolic blood pressure, SD, standard deviation, TAPSE, tricuspid annular plane systolic excursion, VAD, ventricular assist device Cardiogenic shock Older adults Epidemiology Prognosis Mortality --

A B S T R A C T

Background:

The incidence of heart failure and cardiogenic shock (CS) in older adults is continually increasing due to population aging. To date, prospective data detailing the specific characteristics, management and outcomes of CS in this population are scarce.

Methods:

FRENSHOCK is a prospective registry including 772 CS patients from 49 centers. We studied 1-month and 1-year mortality among patients over 75-year-old, adjusted for independent predictors of 1-month and 1-year mortalities.

Results:

Out of 772 patients included 236 (30.6%) were 75 years old or more (mean age 81.9 ± 4.7 years, 63.6% male). Compared to patients <75 years old, older adults had a higher prevalence of comorbidities including hypertension, dyslipidemia, chronic kidney disease, and history of heart disease. Older adults were characterized by a lower blood pressure, as well as higher creatinine and lower haemoglobin levels at presentation. Yet, they were less likely to be treated with norepinephrine, epinephrine, invasive ventilation, and renal replacement therapy. They showed a higher 1-month (aHR: 2.5 [1.86-3.35], p < 0.01) and 1-year mortality (aHR: 2.01 [1.58-2.56], p < 0.01). Analysis of both 1-month and 1-year mortality stratified by age quartiles showed a gradual relationship between aging and mortality in CS patients.

Conclusion:

A third of patient with CS in critical care unit are older than 75 years and their risk of death at one month and one year is more than double compared to the younger ones. Further research is essential to identify best therapeutic strategy in this population.

--

Introduction

Heart failure (HF) and cardiogenic shock (CS) have an increasing incidence and now constitute one of the leading global causes of morbidity and mortality worldwide [START_REF] Conrad | Temporal trends and patterns in heart failure incidence: a population-based study of 4 million individuals[END_REF]. This is reinforced by the significant aging of the global population [START_REF] Triposkiadis | Cardiovascular aging and heart failure: JACC review topic of the week[END_REF], which has transformed agerelated healthcare into one of the most challenging domains of the 21st century [START_REF] Christensen | Ageing populations: the challenges ahead[END_REF]. As longevity extends and medical advances improves, the proportion of older patients within our population continues to grow, while age constitutes an unwavering risk factor for cardiovascular diseases (CVD) [START_REF] North | The intersection between aging and cardiovascular disease[END_REF], inevitably leading to a proportional increase in the number of patients suffering from HF [START_REF] Triposkiadis | Cardiovascular aging and heart failure: JACC review topic of the week[END_REF][START_REF] Curtis | Incidence and prevalence of heart failure in elderly persons, 1994-2003[END_REF]. As healthcare systems continue to improve, these conditions have become an all-too-common endpoint for many cardiac diseases, requiring further research and understanding [START_REF] Hongisto | CardShock Study Investigators and the GREAT Network, Mortality risk prediction in elderly patients with cardiogenic shock: results from the CardShock study[END_REF][START_REF] Jentzer | Influence of age and shock severity on short-term survival in patients with cardiogenic shock[END_REF].

Managing CS in older patients (OP) remains challenging: on the one hand, they more often present CS and severe forms, on the other their treatment options are restricted [START_REF] Jentzer | Comparison of mortality risk prediction among Patients ≥70 versus <70 years of age in a cardiac intensive care unit[END_REF][START_REF] Brunker | Elderly patients and management in Intensive Care Units (ICU): clinical challenges[END_REF]. Indeed, the aggressive therapeutic interventions typically utilized in CS (mechanical circulatory support, renal replacement therapy) remain limited in the OP probably due to fear of harm ("primum non nocere") and fear of futile treatment [START_REF] Martillo | Survival of elderly patients using temporary mechanical circulatory support[END_REF][START_REF] Salath´e | Epidemiology and outcomes of elderly patients requiring renal replacement therapy in the intensive care unit: an observational study[END_REF].. Ethical considerations often arise, questioning the balance between extending life and maintaining its quality [START_REF] Kaarlola | Long-term survival, quality of life, and qualityadjusted life-years among critically ill elderly patients[END_REF]. Further, this population's vulnerability to complications, resulting from multiple comorbidities and physiological frailty, often limits the application of standard therapies [START_REF] Inaba | the LIFE Study Investigators, Impact of frailty on long-term mortality in older patients receiving intensive care via the emergency department[END_REF].

Additionally, even though older individuals currently constitute a significant proportion of patients hospitalized for CS, they are nonetheless often underrepresented in clinical trials on this topic that often exclude patients above 75, 80 or 90 years [START_REF] Heiat | Representation of the elderly, women, and minorities in heart failure clinical trials[END_REF][START_REF] Lazzarini | Heart failure in elderly patients: distinctive features and unresolved issues[END_REF]. This lack of representation impedes the development of appropriate personalized care for these patients.

This study aims to compare characteristics and outcomes of CS between patients 75 years or older and the younger patients, based on database from a large prospective multicentric registry of unselected CS. 2.

Materials and methods

Patient population

As previously reported [START_REF] Delmas | FRENSHOCK investigators, Design and preliminary results of FRENSHOCK 2016: a prospective nationwide multicentre registry on cardiogenic shock[END_REF][START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF], FRENSHOCK is a prospective, observational, and multicentre survey, conducted between April and October 2016, including 772 patients admitted for CS in ICU/ICCU in France, coming from various types of institutions (primary to tertiary centres, university and non-university, public and private hospitals) All adult patients (≥18 years old) with CS were prospectively included in this registry if they met at least one criterion of each of the following three components: (1) Low cardiac output: low SBP <90 mmHg and/or the need for maintenance with vasopressors/inotropes and/or a low cardiac index <2.2 L/min/m 2 ; (2) Left and/or right heart filling pressure elevation, defined by clinical signs, radiology, blood tests, echocardiography, or signs of invasive hemodynamic over lOPd and (3) Signs of organ malperfusion, which could be clinical (oliguria, confusion, pale and/or cold extremities, mottled skin) and/or biological (lactate >2 mmol/L, metabolic acidosis, renal failure, liver insufficiency).

OP subjects were defined as those aged 75 years or more based on previous published series [START_REF] Hongisto | CardShock Study Investigators and the GREAT Network, Mortality risk prediction in elderly patients with cardiogenic shock: results from the CardShock study[END_REF][START_REF] Jentzer | Influence of age and shock severity on short-term survival in patients with cardiogenic shock[END_REF]. Ischemic CS was defined by the presence of at least one hemodynamically significant culprit lesion on coronary angiography (stenosis, thrombosis).

Data collection

Data collecting protocol has already been presented elsewhere [START_REF] Delmas | FRENSHOCK investigators, Design and preliminary results of FRENSHOCK 2016: a prospective nationwide multicentre registry on cardiogenic shock[END_REF][START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF], including medical history, cardiovascular risk factors, previous treatments, clinical, biological, and echocardiographic parameters (at admission and at 24 h), and in-hospital CS management [inotropes/ vasopressors, mechanical ventilation, renal replacement therapy (RRT) and acute mechanical circulatory support (aMCS)].

For each patient, investigators had to specify one to three triggers among the following: ischemic (type 1 or 2 AMI), mechanical complications (valvular injury, ventricular septal defect), ventricular and supraventricular arrhythmia, severe bradycardia, iatrogenesis (medication induced), infections, non-observance of previous medication.

Outcomes

All-cause mortality was assessed at 1-month by local investigator and at 1-year by dedicated research technicians based at the French Society of Cardiology as previously described [START_REF] Delmas | FRENSHOCK investigators, Design and preliminary results of FRENSHOCK 2016: a prospective nationwide multicentre registry on cardiogenic shock[END_REF]. It constitutes respectively, our primary and secondary endpoints. Moreover, heart transplantation (HTx) and ventricular assist devices (VAD) were assessed at one year.

Ethics

The study was conducted in accordance with the Helsinki declaration and French law. Written consent was obtained for all patients. Recorded data and their storage were approved by the CCTIRS (French Health. Research Data Processing Advisory Committee) (n• 15.897) and the CNIL (French Data Protection Agency) (n• DR-2016-109).

Statistical analysis

Continuous variables are reported as means and standard deviation (SD) or medians and interquartile ranges (IQR) when appropriate. Categorical variables are described as frequencies and percentages. Comparisons were made using Mann Whitney non-parametric test for continuous variables and chi-square test or Fisher's exact test for categorical variables. Paired data were analysed with Wilcoxon signed-rank test.

The main analysis was a comparison of characteristics, management and outcomes between OP, defined as being 75 years or older, and the remaining younger population. Two sensitivity analyses were made, examining the above primary and secondary outcomes including only patients older than 80 and 85 years.

To find independent predictors for each outcome of interest, multivariate stepwise logistic regression analysis was performed. Based on previous publications [START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF], the covariates included in the model consisted of all baseline characteristics (age, sex ratio, body mass index (BMI), cardiovascular risk factors, comorbidities, previous NYHA, history of previous heart disease, medications, initial cardiac arrest, sinus rhythm), as well as treatment modalities (diuretics, norepinephrine, epinephrine, dobutamine, renal replacement therapy, aMCS, invasive and non-invasive ventilation), CS' triggers (ischemic, mechanical complication, ventricular and supraventricular arrhythmias, infections, non-observance, iatrogenesis), and markers of CS severity (left ventricular ejection fraction (LVEF) ≤ 30%, lactates ≥4 mmol/L, eGFR ≤30 mL/min). First, the association of all these characteristics and each primary and secondary outcome was assessed using univariable logistic regression analyses. Then, all significant independent predictors (defined at the level of 0.1) were integrated in multivariable analyses for each outcome and backwards reduced to only significant characteristics (p ≤ 0.05). Finally, these significant characteristics were incorporated in multivariable models as fixed covariates for each adjusted outcome analysis. The variance inflation factor was used to rule out multicollinearity among the variables. Primary and secondary outcome of all-cause mortality was assessed using Kaplan-Meier time-to-event analysis, and Cox proportional hazards models were used to determine the adjusted hazard ratio (HR), 95% CI and p values. HTx and VAD at 1-year are reported as their rate and their adjusted OR and 95% CI.

To discern the potential incremental association between age and mortality, we further stratified the entire cohort into quartiles based on age and evaluated mortality across these strata. Lastly, we undertook an additional comparison between individuals suffering from ischemic and non-ischemic CS within the OP cohort.

All tests were two-tailed. A value of p ≤ 0.05 was accepted as statistically significant. Analyses were performed using R software (version 4.1.2 (2021/ 11/ 01)).

Results

Baseline characteristics

Fig. 1 presents an overview of our study cohort: out of 772 CS patients incorporated from 49 centers, 236 (30.6%) were aged 75 or older, whose baseline characteristics are depicted in Table 1. The distribution of age groups within the cohort is provided in Supplemental Fig. 1. OP were more often females (36.4 vs 25%, p < 0.01), and exhibited a lower mean BMI (25.0 vs 26.2, p = 0.02), with higher rates of hypertension (68.2 vs 37.9%, p < 0.01), dyslipidaemia (42.4, 33.1%, p = 0.02) and chronic kidney disease (34.3 vs 15.5%, p < 0.01). In contrast, their smoking's prevalence was reduced (9.4 vs 35.9%, p < 0.01).

OP demonstrated a higher frequency of history of cardiac disease (62.7 vs 53.3%, p = 0.02), with an emphasis on ischemic (41.1 vs 24.8%, p < 0.01) and valvular (13.1 vs 6.3%, p < 0.01) cardiomyopathies, even though toxic heart disease was more prevalent in the younger group (1.7 vs 5.6%, p = 0.02). While the incidence of implantable cardioverter-defibrillator (ICD) was lower in the OP (10.2 vs 19.3%, p < 0.01), they were more commonly treated with ACE inhibitors (43.6 vs 35.4%, p = 0.04) and loop diuretics (55.5 vs 45.9%, p = 0.02).

No difference was found between the groups regarding CS triggers (Supplementary Table 1), with a predominance of ischemic (39.8 vs 34.7%, p = 0.2), supra-ventricular arrhythmias (11 vs 14.4%, p = 0.25), ventricular arrhythmias (9.7 vs 13.8%, p = 0.15), and infections (11 vs 12.3%, p = 0.7).

Significant associations between baseline characteristics and each outcome of interest can be found in Supplementary table 2.

CS presentation and evolution at 24 h according to age

Table 2 outlines the differences between both groups regarding clinical, biological, and echocardiographic parameters at admission and at 24 h. OP presented with lower blood pressure but without difference in terms of rate of HF signs and skin mottling. Biologically, OP presented with more elevated levels of creatinine (149.5 vs 124.0 μmol/L, p < 0.01) and Nt-proBNP (16,000 vs 6787 pg/mL, p < 0.01), whereas lactate levels remained consistent between groups (3.0 vs 3.0 mmol/L, p = 0.49). The echocardiographic evaluations showed a relatively higher LVEF (29.3 vs 25%, p < 0.01) in the OP group, notably including a greater proportion of patients with preserved LVEF (≥ 50%) (11.9% vs 6.5%, p = 0.02), alongside an increased prevalence of aortic stenosis (9.6 vs 2.6%, p < 0.01).

In hospital management according to age

Table 3 underscores the variances in therapeutic approaches between groups. Dobutamine was the main inotrope used whatever the age group (80.9 vs 82.9%, p = 0.58) even though OP received less norepinephrine (47.9 vs 55.8%, p = 0.05) and epinephrine (7.2 vs 14.7%, p < 0.01). Loop diuretics (74.1 vs 64.4%, p = 0.01) were more routinely administered in OP. While no significant difference was observed between groups regarding the use of noninvasive ventilation, OP were less frequently subjected to various forms of organ support. This includes invasive ventilation (24.6% vs 43.8%, p < 0.01), renal replacement therapy (9.3% vs 18.7%, p < 0.01), and acute mechanical circulatory support (aMCS) (6.4% vs 24%, p < 0.01). However, OP benefited more from percutaneous coronary intervention (PCI) (34.7% vs 27%, p = 0.04).

Short and long-term outcomes

An age of 75 years or older was associated with a significantly higher 1-month (38.6 vs 20.5%, aHR: 2.5 [1.86-3.35], p < 0.01) and 1-year all-causes mortality (60.2 vs 38.6%, aHR: 2.01 [1.58-2.56], p < 0.01), as shown in Fig. 2. Conversely, the 1-year recourse to VAD and/or HTx was lower for OP (3 vs 13.2%, aOR 0.36 [0.15-0.91], p = 0.03). No significant difference was found between ischemic and non-ischemic CS in the OP group (Supplementary Fig. 2). Additional analyses with age thresholds >80 and 85 years showed similar results, with higher 1-month and 1-year mortality, as well as a trend towards a lower number of VAD or HTx (Supplementary Fig. 3 &4, Supplementary Table 3). Fig. 3 shows that, based on quartiles subdivision, it exists a graded relationship between age and mortality, demonstrating a proportional increase in mortality as age escalates, with a significant cut-off starting from the age of 66 for 1-month mortality, and 58 for 1-year mortality.

ICU-related complications

No difference was reported regarding the most common complications, whether for ischemic (3.2 vs 4.1%, p = 0.69) or hemorrhagic (0.9 vs 1.4%, p = 0.73) strokes, gastrointestinal bleeding (4.1 vs 4.1%, p = 1), pneumonia (16.1 vs 21.4%, p = 0.12) and urinary (4.9 vs 3.7%, p = 0.59), or bloodstream (11.2 vs 14.3%, p = 0.29) infections. However, it is worth noting a higher incidence of pericardial effusion in the cohort of younger patients (0 vs 2.2%, p = 0.04) (Supplementary Table 4).

Discussion

Within this prospective, multicentre study including 772 CS patients with unselected aetiology managed in routine practice, OP patients (≥ 75 years old) constituted almost onethird of the population, affected by more comorbidities (heart failure, chronic kidney disease), and treated less invasively (lower rate of norepinephrine, epinephrine, RRT and aMCS use). We observed a strong, independent, and graded relationship between older age and both lower 1-month and 1-year survival.

The OP constituted one third of the patients in this study, which seems consistent with prior recent CS studies [START_REF] Aissaoui | Fifteen-year trends in the management of cardiogenic shock and associated 1-year mortality in elderly patients with acute myocardial infarction: the FAST-MI programme[END_REF][START_REF] Gasior | Comparison of invasive and non-invasive treatment strategies in older patients with acute myocardial infarction complicated by cardiogenic shock (from the polish registry of Acute coronary syndromes -PL-ACS)[END_REF]. There was no difference in the distribution of CS triggers among age groups, with ischemic aetiology being the most prevalent at about 35-40%. This proportion is lower than reported in older studies, where ischemic aetiology typically accounts for around 70-80% [START_REF] Hongisto | CardShock Study Investigators and the GREAT Network, Mortality risk prediction in elderly patients with cardiogenic shock: results from the CardShock study[END_REF][START_REF] Jentzer | Influence of age and shock severity on short-term survival in patients with cardiogenic shock[END_REF], likely related to the varied care settings included in our study, spanning from primary to tertiary centers, as well as both university and non-university hospitals, and private institutions. So, although ischemic cause remains the most common individual trigger of CS, non-ischemic causes collectively outweighed it in terms of frequency, underlying the growing awareness regarding non-ischemic CS, which, while predominant in everyday practice, remain underrepresented in previous publications [START_REF] Schrage | Eligibility for mechanical circulatory support devices based on current and past randomised cardiogenic shock trials[END_REF]. Our findings did not demonstrate any difference in prognosis between ischemic and non-ischemic CS in the specific OP as we previously found in the general FRENSHOCK population analysis [START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF]. Possible explanation could be the absence of collection of other geriatric factors that may override this difference such as the degree of dependence or frailty before hospitalization. Nevertheless, data on general population comparing prognosis in ischemic and non-ischemic CS remains conflicting [START_REF] Schrage | Patient characteristics, treatment and outcome in non-ischemic vs ischemic cardiogenic shock[END_REF][START_REF] Yandrapalli | Cardiogenic shock during heart failure hospitalizations: age-, sex-, and race-stratified trends in incidence and outcomes[END_REF][START_REF] Harjola | CardShock study investigators; GREAT network. Clinical picture and risk prediction of short-term mortality in cardiogenic shock[END_REF]. A reassuring point is the PCI rate in the OP group, approaching 35% as in the younger cohort, which is coherent with previous studies indicating the benefit of myocardial revascularization even for OP [START_REF] Gasior | Comparison of invasive and non-invasive treatment strategies in older patients with acute myocardial infarction complicated by cardiogenic shock (from the polish registry of Acute coronary syndromes -PL-ACS)[END_REF][START_REF] Dauerman | Outcomes and early revascularization for patients > or = 65 years of age with cardiogenic shock[END_REF].

Older age is already known as a strong risk factor for mortality whatever the pathology or condition studied including among patients with CS [START_REF] Hongisto | CardShock Study Investigators and the GREAT Network, Mortality risk prediction in elderly patients with cardiogenic shock: results from the CardShock study[END_REF][START_REF] Jentzer | Influence of age and shock severity on short-term survival in patients with cardiogenic shock[END_REF][START_REF] Thiele | CULPRIT-SHOCK Investigators, PCI strategies in patients with acute myocardial infarction and cardiogenic shock[END_REF], illustrated by an odds ratio of 1.06 per additional year of age for 1-month mortality in our previous analysis [START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF]. This study confirms a gradual association between age and CS mortality with an emphasis for patients aged 75 years and older and extends this prognostic decline to non-ischemic CS, which constitutes most cases in our cohort. In addition, our quartile analysis revealed that mortality began to significantly increase from the age of 66 years for 1-month mortality and 58 years for 1-year mortality, consistent with prior studies that have identified age cut-offs ranging from ≥65 to >75years for prediction of higher mortality among patients with CS [START_REF] Hongisto | CardShock Study Investigators and the GREAT Network, Mortality risk prediction in elderly patients with cardiogenic shock: results from the CardShock study[END_REF][START_REF] Jentzer | Influence of age and shock severity on short-term survival in patients with cardiogenic shock[END_REF]. This observed excess mortality in older subjects is undoubtedly explained by various agerelated factors hindering their response capabilities to the hemodynamic stresses of CS, such as frailty, lower access to admissions or reluctance, cellular aging, atypical clinical presentation, multimorbidity, altered drug metabolism or reduced functional reserve [START_REF] Christensen | Ageing populations: the challenges ahead[END_REF][START_REF] Damluji | Older adults in the cardiac intensive care unit: factoring geriatric syndromes in the management, prognosis, and process of care: a scientific statement from the American Heart Association[END_REF]. Otherwise, in our study, the initial clinical presentation of the OP cohort seemed slightly more severe than younger patients, with more comorbidities, lower blood pressure and more pronounced renal impairment, although the higher prevalence of CKD at baseline must be considered.

Despite their more severe presentation, older subjects were less frequently treated with norepinephrine, renal replacement therapy, invasive ventilation and aMCS suggesting potential limitations on the aggressiveness of care in the setting of critical illness and a poor anticipated prognosis in this population. This remark underlines the interest of our real-life approach as this population remains underrepresented in classical trials.

Meanwhile, even if older patients had more comorbidities, worse kidney function, and decreased use of MCS devices, they had a significantly higher risk of mortality despite adjusting for these factors in addition to shock severity. Moreover, while advanced therapies for CS provide encouraging results, they often come with a high risk of complications that increase with age and is negatively associated with outcomes [START_REF] Bruoha | Mechanical circulatory support devices for the treatment of cardiogenic shock complicating acute myocardial infarction-a review[END_REF], which may discourage physicians from using them. However, even though age is not a contraindication per se to the use of such treatments, patients who might benefit from them should be carefully selected, considering their general condition before CS, frailty criteria, and expected chances of recovery [START_REF] Alonso-Fernandez-Gatta | Short-term mechanical circulatory support in elderly patients[END_REF].

Nevertheless, it is uncertain whether the higher mortality observed in the OP group is only age-related or secondary to others unconsidered factors whether related to the patient (other comorbidities, dependence, or frailty for example), or to the physician (age-related differences in history taking, shock recognition, diagnostic testing, or treatment implementation). Since they benefited from less invasive management and they die twice, inequalities in the quality of care should be questioned. Besides, unlike younger patients, some advanced heart failure treatments such as chronic VAD and HTx may seem unreasonable especially in this CS context. Instead, the management plan should focus on optimizing medical treatment and implementing GDMT. Remote monitoring also presents a compelling approach for early detection of HF recurrences and the corresponding adjustment of medication treatments [29], aiming to enhance quality of life, which remains the primary objective to achieve.

All these findings underline the frequency, severity, and complexity of managing CS in OP, with both short-and long-term prognoses appearing to be encumbered, highlighting the importance of global evaluation of the patient considering not only age but dependence and frailty, to avoid potential ethical dilemmas. As a result, to provide the most thorough and individualized care, the management of these patients should be more extensively discussed within multidisciplinary teams involving geriatrician, intensivist and cardiologist, ideally within the framework of geriatric intensive care units, representing an interesting future perspective given the expectation of facing increasingly critical situations with OP patients.

Limitations

The main limitation is the lack of data regarding the geriatric assessment of the general and functional status, including frailty criteria or comorbidity score (such as Charlson Comorbidity Index) which would have allowed for a study stratified by frailty group to assess the potential benefit of more aggressive treatment in the most robust older subjects. Future studies should endeavor to collect this data to improve the management of elderly and very elderly patients in the future. Similarly, we also did not have information regarding established care limitations, such as "not to be resuscitated" orders, or transitions to palliative care, which would have likewise allowed a more nuanced study of prognosis according to the degree of robustness. Besides mortality, it would also be pertinent to examine the changes in functional capacities before and after CS, and the prospects for medium-term recovery It is noteworthy that there was a higher prevalence of preserved LVEF among those aged 75 and over, possible owing to a higher occurrence of specific cardiac conditions like cardiac amyloidosis, relatively frequent among the elderly, but also frequently underdiagnosed [30]. As our cohort's data traces back to 2016, a time when the understanding and detection of amyloidosis were less advanced, we lack sufficient data to properly address this issue.

Another notable limitation is the year of our data collection, which dates back to 2016. The concern arising from this temporal gap is understood, especially in rapidly evolving fields. However, this dataset, despite its age, captures a distinctive and realistic snapshot of the clinical practices and patient demographic characteristics prevalent at that time. Subsequent studies, when juxtaposed against this dataset, stand to gain a more nuanced understanding, allowing researchers to discern intricate trends, evolutions in treatment modalities, and demographic shifts over the ensuing years.

As previously described [START_REF] Delmas | FRENSHOCK investigators, Design and preliminary results of FRENSHOCK 2016: a prospective nationwide multicentre registry on cardiogenic shock[END_REF][START_REF] Delmas | Baseline characteristics, management, and predictors of early mortality in cardiogenic shock : insights from the FRENSHOCK registry[END_REF], the FRENSHOCK registry targeted patients from ICU and ICCU, either admitted directly or post-transfer from another centre. Consequently, we cannot rule out the possibility that individuals with severe comorbidities, advanced age, or extensive multi-organ failure might not have been transferred due to perceived futility. Additionally, our data collection did not include those who succumbed early, specifically patients who died before informed consent could be obtained, leading to a potential underestimation of mortality within our cohort. Nevertheless, the representativeness of our dataset is still reasonably acceptable, as underscored by the fact that over a third of our studied population was aged 75 years or above, >8% exceeded 85 years of age, and the eldest participant was 98 years old. Finally, we were not able to use the SCAI SHOCK Stage Classification given that it was not yet available at the time of our study.

Conclusion

A third of CS patients in the intensive care unit are over 75 years old. Although their risk of death at one month and one year is more than doubled compared to the youngest, they benefited less from invasive management, raising the question of the therapeutic choices made in this population. Further research is essential to identify best therapeutic strategy in this population.

Acknowledgments

FRENSHOCK is a registry of the French Society of Cardiology, managed by its Emergency and Acute Cardiovascular Care Working Group. Our thanks go out to all the devoted personnel of Société Française de Cardiologie who participate in the upkeep of the registry. The authors are deeply indebted to all the physicians who took care of the patients at the participating institutions.

Funding

The study was sponsored by the Fédération Française de Cardiologie and was funded by unrestricted grants from Daiichi-Sankyo and Maquet SAS.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.


Thomas Frapard

Giuliana Amaddeo

Maxens Decavele

Paer-Selim Abback

Antoine Gaillet

Charlotte Bouzbib

Claire Vanlemmens

Romy Younan

Emmanuel Canet

Anne Sophie Moreau

Mathilde Neuville

Elie Azoulay

Alexandre Sitbon

Djamel Mokart

Sylvie Radenne

Armand Abergel

Céline Guichon

Olivier Roux

Agnes Bonadona

Armand Mekontso Dessap

Audrey De Jong

Jerome Dumortier

Nicolas De Prost1

MD Nicolas De Prost
email: nicolas.de-prost@aphp.fr

Herpes Simplex Virus Hepatitis in Patients Requiring Intensive Care Unit Admission: A Retrospective, Multicenter, Observational Study

Keywords: ICU, acyclovir, acute liver failure, hemophagocytic syndrome, herpes simplex virus

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Acute liver failure is a rare but life-threatening critical illness [START_REF] Stravitz | Acute liver failure[END_REF][START_REF] Bernal | Acute liver failure[END_REF]. One of the least reported causes is severe herpes simplex virus (HSV) hepatitis (HH). Its pathophysiology is not fully understood. Although primary infection is a major cause of herpes hepatitis [START_REF] Ichai | Herpes simplex virus-associated acute liver failure: a difficult diagnosis with a poor prognosis[END_REF][START_REF] Pinna | Five cases of fulminant hepatitis due to herpes simplex virus in adults[END_REF][START_REF] Norvell | Herpes simplex virus hepatitis: an analysis of the published literature and institutional cases[END_REF][START_REF] Yokoi | Fatal fulminant herpes simplex hepatitis following surgery in an adult[END_REF], a few cases of HH associated with HSV reactivation, defined by immunoglobulin G (IgG) serology positivity, have been reported in immunocompromised patients [START_REF] Chase | Herpes simplex viral hepatitis in adults: two case reports and review of the literature[END_REF]. The prevalence of HH appears to be around 0.8% of cases of severe hepatitis [START_REF] Ganger | Acute liver failure of indeterminate etiology: a comprehensive systematic approach by an expert committee to establish causality[END_REF][START_REF] Little | Acute liver failure study group. Herpes simplex virus-associated acute liver failure often goes unrecognized[END_REF].

Most of the available data consist of case reports with literature reviews [START_REF] Chase | Herpes simplex viral hepatitis in adults: two case reports and review of the literature[END_REF][START_REF] Kaufman | Herpes simplex virus hepatitis: case report and review[END_REF]. The clinical picture of HH is a febrile hepatic insufficiency with cutaneous signs and marked serum transaminase and lactate dehydrogenase elevation occurring in immunocompromised or pregnant patients, and to a lesser extent in the postoperative setting [START_REF] Ichai | Herpes simplex virus-associated acute liver failure: a difficult diagnosis with a poor prognosis[END_REF][START_REF] Pinna | Five cases of fulminant hepatitis due to herpes simplex virus in adults[END_REF][START_REF] Norvell | Herpes simplex virus hepatitis: an analysis of the published literature and institutional cases[END_REF][START_REF] Yokoi | Fatal fulminant herpes simplex hepatitis following surgery in an adult[END_REF][START_REF] Chase | Herpes simplex viral hepatitis in adults: two case reports and review of the literature[END_REF][START_REF] Little | Acute liver failure study group. Herpes simplex virus-associated acute liver failure often goes unrecognized[END_REF]. The prognosis of HH has been reported to be poor, with a mortality rate of up to 60%, compared with 19%-36% for other causes of severe acute hepatitis. A diagnostic delay has been frequently reported, with only 30% of patients receiving early antiviral treatment with acyclovir [START_REF] Little | Acute liver failure study group. Herpes simplex virus-associated acute liver failure often goes unrecognized[END_REF]. There are to our knowledge no previous studies describing the clinical characteristics, management, and shortterm outcomes of patients admitted to the intensive care unit (ICU) for HH. We conducted a large multicenter retrospective cohort study in adult patients admitted to French ICUs for acute hepatitis and proven HSV infection and aimed at describing their clinical and biological features, management, and ICU mortality.

METHODS

Patients

We performed a 16-year multicenter retrospective observational cohort study in 27 adult ICUs located throughout metropolitan France. All consecutive patients hospitalized in participating ICUs between 2006 and 2022 with a compatible history and a proven HSV infection (positive polymerase chain reaction [PCR] in blood or liver biopsy) were included (Supplementary Method 1).

Ethical Considerations

This observational, noninterventional analysis of medical records was approved by the Institutional Review Board of the French Intensive Care Society in December 2021 (CE SRLF 21-108).

Collection of Data

Data pertaining to demographics, comorbidities, clinical examinations, laboratory findings, microbiological investigations, and therapeutic management at ICU admission and during ICU stay were collected. Duplex HSV PCR methods were used in most centers and were qualitative for half the patients. Quantitative assessment of circulating HSV DNA was provided when available and expressed as log DNA copy numbers/mL of serum. Simplified Acute Physiology Score II (SAPS II) and Sequential Organ Failure Assessment (SOFA) scores were computed using the worst values recorded within the first 24 hours of admission. Missing data were retrieved by queries to the investigators.

Statistical Analysis

Quantitative data were expressed as mean (SD) or median (interquartile range [IQR]) and compared using a Student t test or nonparametric Mann-Whitney test, as appropriate. Qualitative data were compared by χ2 test or Fisher exact test. Factors associated with ICU mortality were analyzed by univariable logistic regression. The tests were 2-tailed, and a value of P < .05 was considered statistically significant. Statistical analysis was performed using the software R 4.2.2 (The R Foundation for Statistical Computing, Vienna, Austria).

Control Group

A bicentric control group cohort of ICU acetaminophen hepatitis patients was assembled to explore the specificity of the features of hemophagocytic syndrome observed in patients with HH (Supplementary Method 2).

RESULTS

Among the 27 participating ICUs, 15 identified a total of 33 patients during the study period.

Clinical and Biological Features

The median age (IQR) was 55 (37-65) years, with 15 patients (45.4%) of male gender. Risk factors for HH commonly reported in the literature included 22 (66.7%) immunosuppressed patients and 4 (12.1%) pregnant patients (3 were in the third trimester and 1 in the second trimester) (Table 1). Upon ICU admission, 30 (90.9%) patients were febrile, with a median temperature (IQR) of 39.2°C (39.1°C-40.2°C). Skin signs were present in 13 (39.4%) patients, and 27 (81.8%) patients had an altered mental status. All patients had at least 1 organ failure on admission, with a median SOFA score (IQR) of 14 [START_REF] Kaufman | Herpes simplex virus hepatitis: case report and review[END_REF][START_REF] Mihalcea-Danciu | Herpes simplex hepatitis with macrophage activation syndrome in an immunocompetent patient [in French][END_REF][START_REF] Fardet | Development and validation of the HScore, a score for the diagnosis of reactive hemophagocytic syndrome[END_REF][START_REF] Henter | HLH-2004: diagnostic and therapeutic guidelines for hemophagocytic lymphohistiocytosis[END_REF][START_REF] Luyt | Acyclovir For mechanically ventilated patients with herpes simplex virus oropharyngeal reactivation: a randomized clinical trial[END_REF][START_REF] Lasserre | Acute severe herpes simplex hepatitis with virusassociated hemophagocytic syndrome in an immunocompetent adult[END_REF](16)(17). Thrombocytopenia was almost constant, with a median platelet count on admission (IQR) of 50 (22-82) G/L and a median (IQR) prothrombin time (PT) of 38% (28%-50%). There was a marked transaminase elevation, which was more pronounced on aspartate transaminase (AST); median serum value [IQR], 2694 [932-6217] IU/L) than on alanine transaminase (ALT) (Table 1). HH diagnosis was confirmed in all cases by a blood PCR, and a liver biopsy was performed in 16 (48.5%) patients, confirming diagnosis in all cases when performed (Table 2). When available, HSV IgG was negative for 94.1% of patients (n = 16/17; data missing for 16 patients) (Table 2).

Treatment

The vast majority of patients (n = 32/33, 97%) received acyclovir during their hospital stay (Table 2). The median time from symptom onset to initiation of acyclovir (IQR) was 8 (5-10) days, and the median time from ICU admission to initiation of acyclovir (IQR) was 1 (0-3) day.

Regarding organ support, 21 (63.6%) patients required vasopressors, 12 (36.4%) required renal replacement therapy, and 17 (51.5%) required invasive mechanical ventilation. Of the pregnant patients, 2 required emergency fetal extraction before ICU admission.

Outcomes and Prevalence of Hemophagocytic Syndrome

During their ICU stay, 27 (82%) patients developed hepatic encephalopathy, 6 (18%) had proven HSV meningo-encephalitis, 11 (33.3%) developed hemorrhagic shock, and 14 (42.4%) developed disseminated intravascular coagulation (DIC). The serum AST level peaked (IQR) at 7000 (2714-10 000) IU/L, and the prothrombin time (PT) nadir (IQR) was 20% (8%-35%).

Unexpectedly, 16 (48.5%) patients developed hemophagocytic syndrome (HS) [START_REF] Henter | HLH-2004: diagnostic and therapeutic guidelines for hemophagocytic lymphohistiocytosis[END_REF] during the course of their ICU stay. Hemophagocytosis was present on myelogram in 12 (41%) patients (Table 2). In order to establish whether HS was related to the severity of liver failure or a specific feature of severe HSV hepatitis, we compared patients from the current cohort with a control cohort of patients with severe acetaminophen hepatitis with regard to HS features (Supplementary Table 1). There were significantly fewer immunosuppressed patients and pregnant women in the acetaminophen hepatitis group than in the HH group (P < .001), with no fever (P < .001) and no cytopenia (P < .001). The HScore [START_REF] Fardet | Development and validation of the HScore, a score for the diagnosis of reactive hemophagocytic syndrome[END_REF] was significantly lower, with no hemophagocytic syndrome diagnosed in the acetaminophen group, suggesting that HS was not related to the severity of liver failure itself.

Finally, 23 patients (69.7%) died in the intensive care unit. Four (12.1%) patients underwent unsuccessful liver transplantation and eventually died (Table 2).

Factors Associated With ICU Death

Factors associated with mortality in univariable analysis included severity of illness scores (SAPS 2 and SOFA scores), an altered mental status and serum creatinine at ICU admission, and need for vasopressor support, occurrence of hepatic encephalopathy or DIC, and PT nadir during ICU stay (Supplementary Table 2).

DISCUSSION

We describe the largest multicenter cohort of patients admitted to the ICU for herpetic hepatitis. The main findings of our study are as follows: (1) the 2 main causes of ICU admission were multivisceral and acute liver failure; (2) 67% of patients were immunocompromised and 12% were pregnant women; (3) clinical presentation was associated with frank fever in 91% of patients, cutaneous signs in 39% of patients, and marked transaminase elevation predominating on AST; and (4) ICU mortality was 70%. As expected, the factors associated with ICU mortality included severity of illness scores and hepatic encephalopathy, which are indicative of liver and other organ failures. We found no statistically significant association between time to initiation of acyclovir and mortality. Nevertheless, the high mortality observed in our series suggests that clinicians should consider initiating empiric treatment with acyclovir in patients admitted for acute febrile hepatitis with a compatible clinical history. Importantly, studies comparing acyclovir with placebo in mechanically ventilated, critically ill patients have reported no statistically significant excess risk of renal failure [START_REF] Luyt | Acyclovir For mechanically ventilated patients with herpes simplex virus oropharyngeal reactivation: a randomized clinical trial[END_REF]. For the most severe patients, liver transplantation may be considered [START_REF] Little | Acute liver failure study group. Herpes simplex virus-associated acute liver failure often goes unrecognized[END_REF]. However, in our series, only 4 patients required transplantation, and, regrettably, they eventually died.

As previously reported [START_REF] Mihalcea-Danciu | Herpes simplex hepatitis with macrophage activation syndrome in an immunocompetent patient [in French][END_REF][START_REF] Lasserre | Acute severe herpes simplex hepatitis with virusassociated hemophagocytic syndrome in an immunocompetent adult[END_REF], there was a significant proportion (48.5%) of patients developing hemophagocytic syndrome, confirmed on myelogram in most cases, during the course of ICU stay. In our series, the diagnosis was made on the basis of severe fever, almost constant immunosuppression, profound thrombocytopenia, hepatic abnormalities, and coagulation disorders (eg, DIC; 42.4%). Hemophagocytic syndrome in a patient at risk is a supplemental feature that should lead clinicians to suspect herpetic hepatitis.

Our study certainly has limitations, related to its retrospective and observational design and the long duration of the inclusion period, required by the rareness of HH. The strengths of this study lie in its national multicenter design, involving liver and general ICUs throughout France. The relatively high number of cases, considering the rarity of the disease, makes it the largest worldwide series to our knowledge. The number of cases with histologic evidence (n = 16/33, 48.5%) also limits the subjectivity of the diagnosis.

Among the perspectives of this work, the delay in diagnosis leads to a reflection on diagnostic criteria. Whether patients with negative HSV PCR may have been missed in the current series cannot be excluded and raises the question of the best diagnostic and therapeutic strategy in this setting (eg, repeated PCR testing and empiric acyclovir initiation). Given the extreme rarity of this disease, national prospective registries would probably be needed to validate diagnostic criteria, reassess the prevalence of the disease, and evaluate more aggressive therapeutic strategies.

Acknowledgments

The French GrrrOH study group includes the following investigators: Guillaume VOIRIOT, MD, PhD, Julien SCHMIDT, MD, Jean Pierre QUENOT, MD, PhD, Maxens DECAVELE, MD, Juliette AUDIBERT, MD, Emmanuel CANET, MD, PhD, Fabrice UHEL, MD, PhD, Olivier LESIEUR, MD, Romy YOUNAN, MD, Anne Sophie MOREAU, MD, Audrey DE JOND, MD, PhD, Djamel MOKART, MD, PhD, Mathilde NEUVILLE, MD, Elie AZOULAY, MD, PhD. The French Gref2 study group includes the following investigators: Giuliana AMADDEO, MD, PhD, Paer-Selim ABBACK, MD, Charlotte BOUZBIB, MD, Claire VANLEMMENS, MD, Jerome DUMORTIER, MD, PhD, Sylvie RADENNE, MD, Armand ABERGEL, MD, PhD, Céline GUICHON, MD, Olivier ROUX, MD, Agnes BONADONA, MD, Nicolas CARBONELL, MD, PhD, Christine SILVAIN, MD, Helene MONTIALOUX, MD, Jose URSIC, MD.

CONCLUSIONS

HH in patients admitted to the ICU is an extremely rare entity associated with very high mortality. Despite known risk factors (immunodepression and pregnancy) and a clinico-biologic picture of febrile hepatitis, characterized by a marked transaminase elevation predominating on AST, with cutaneous signs and cytopenias, diagnosis remains difficult. Early acyclovir treatment is recommended.

Author contributions.

T.F., G.A., J.D., and N.D.P. designed the study; T.F. and N.D.P. submitted the study to the ethical committee; T.F., G.A., and N.D.P. designed the first draft of the study. All authors collected the data and corrected and approved manuscript.

Patient consent.

This observational, noninterventional analysis of medical records was approved by the Institutional Review Board of the French Intensive Care Society in December 2021 (CE SRLF 21-108). According to French law, written informed consent was not required due to the retrospective design of the study.

Financial support.

There was no specific funding for this study.

Potential conflicts of interest.

All authors: no reported conflicts.


Guillaume Leloup

Coupled system involving eddy coefficients and a right hand side in L 1 : Review chapter

Introduction

This chapter is focused on the following system: 2 in Ω,

         -div(A(k)∇u) = f in Ω, -div(B(k)∇k) = A(k)|∇u|
-A(k) ∂u ∂n = αu on Γ, k = 0 on Γ. (1.1)
Where Ω is an open bounded set in R N (N ≥ 2), and Γ = ∂Ω its boundary. The viscosity functions are bounded and satisfy: ∃ ν, µ > 0, ∀x ≥ 0, ν ≤ A(x), B(x) ≤ µ.

(1.2)

We also assume that f ∈ L 2 (Ω) and α > 0.

The aim of this chapter is proving the existence of a couple (u, k) of distributional solutions of (1.1). This is not the only kind of solutions commonly searched. Indeed, some articles deal with renormalized solutions, which are defined and studied in [START_REF] Dal Maso | Renormalized solutions of elliptic equations with general measure data[END_REF]. This notion is interesting for the study of a system having one source term in L 1 (Ω), as shown in [START_REF] Lewandowski | Analyse mathématique et océanographie[END_REF].

The source term of the equation in k in (1.1) is in L 1 (Ω). Without any better hypothesis,it remains possible to find distributional solutions of (1.1).

This figures in the works of L. Boccardo and T. Gallouët in [START_REF] Boccardo | Non-linear elliptic and parabolic equations involving measure data[END_REF], on which is based this chapter. The associated result, proved in section 4, is a consequence of the washer Lemma, stated and proved in section 2. This Lemma brings a bound for a solution in W 1,q 0 (Ω), where q < N N -1 . Consequently, some particular function sets are needed. Then, in section 4, we search weak solutions in V = 1<q<N ′ W 1,q 0 (Ω), where N ′ refers to N N -1 ; with test functions in W = q>N W 1,q 0 (Ω). This method was also used by T. Gallouët and R. Herbin in [START_REF] Gallouët | Existence of a solution to a coupled elliptic system[END_REF].

Along this chapter, the viscosities A and B are bounded (assumption (1.2)). This does not really correspond to the physical situation, but is a right way to establish some interesting mathematical results, as a priori estimates, such as in section 3 and 4. The assumption of bounded viscosities appears in [START_REF] Gallouët | Existence of a solution to a coupled elliptic system[END_REF], but also in the works of S. Clain and R. Touzani (see [START_REF] Clain | A two-dimensional stationary induction heating problem[END_REF]), in which an existence result is proved in the two-dimensional case. In the present work, the existence results are proved when N = 3.

Of course it is possible to deal with unbounded viscosities, such as in [START_REF] Lewandowski | Analyse mathématique et océanographie[END_REF], where they are only bounded from below. Otherwise, if they are not, other assumptions are required, concerning the decreasing. In addition, other function spaces need to be developped, like in [START_REF] Gallouët | On a turbulent system with unbounded eddy viscosities[END_REF].

In order to find the existence of a solution of (1.1), this work will be divided into two steps, that correspond to sections 3 and 4. Section 3 is devoted to the equation in u, which amounts to a linear problem. But when the problem is obviously non-linear, like the equation in k considered in section 4, another method is required. This consists in regularizing the system, then solving it by a linearization and a fixed-point theorem. This method is already used in [START_REF] Clain | A two-dimensional stationary induction heating problem[END_REF].

Given m ∈ N and p ∈ [1, +∞], the norm of W m,p (Ω) will be simply written: ∥ • ∥ m,p,Ω . In particular, ∥ • ∥ 0,p,Ω is the L p (Ω) norm.

Preliminary results

The washer lemma

This section aims at proving the following result (already proved by Boccardo-Gallouët in [START_REF] Boccardo | Non-linear elliptic and parabolic equations involving measure data[END_REF]).

Lemma 1 (Washer lemma). Let q ∈]1, N ′ [ where N ′ = N N -1

, and u ∈ W 1,q 0 (Ω). Let us write:

B n = {x ∈ Ω / n ≤ |u(x)| < n + 1}. (2.1)
We assume that there is a constant C > 0 such that:

∀n ∈ N, Bn |∇u(x)| 2 dx ≤ C. (2.2)
Then we have the inequality: ∥u∥ 1,q,Ω ≤ A.

(2.3)

Where A depends on Ω, q and C that is introduced by (2.2).

Remark 1. The most important in this result is the fact that the two constants C and A are independent of u. Actually, the result that will be used is: let (u j ) j∈J a family of functions in

W 1,q 0 (Ω) with q ∈]1, N ′ [ satisfying: ∀j ∈ J, ∀n ∈ N, B j n |∇u j (x)| 2 dx ≤ C, (2.4)
where B j n refers to:

B j n = {x ∈ Ω / n ≤ |u j (x)| < n + 1}.
Then there exists a constant A > 0 only depending on Ω, q and C such that:

∀j ∈ J, ∥u j ∥ 1,q,Ω ≤ A. (2.5)
The proof of lemma 1 is based on a decomposition given by the following lemma.

Lemma 2. Let u satisfying the assumptions of lemma 1. Let q * be the critical exponent in Sobolev embedding W 1,q (Ω) → L q * (Ω), and any index n 0 ∈ N. There exist two constants λ 1 and λ 2 depending on n 0 , q and Ω satisfying:

∥∇u∥ q 0,q,Ω ≤ C q/2 λ 1 + λ 2 ∥u∥ q * (2-q) 2 0,q * ,Ω . (2.6)
Moreover, we have: lim

n 0 →+∞ λ 2 (n 0 ) = 0.
Proof of lemma 2. At first, we decompose Ω |∇u(x)| q dx as a sum:

Ω |∇u(x)| q dx = +∞ n=0 Bn |∇u(x)| q dx. (2.7)
Let us apply Hölder inequality, recalling that Ω is bounded, thus for n ∈ N, B n is bounded.

∀n ∈ N, Bn |∇u(x)| q dx ≤ Bn |∇u(x)| 2 dx q/2 Bn dx 1-q 2 = Meas(B n ) 2-q 2 Bn |∇u(x)| 2 dx q/2 ≤ Meas(B n ) 2-q 2 C q/2 .
(2.8)

Where "Meas" refers to the Lebesgue measure of any borelian subset of R N . The last inequality comes from (2.2).

We set n 0 ∈ N, and we focus on the case n ≤ n 0 , then n > n 0 .

On one hand, let n ≤ n 0 , we have

Bn |∇u(x)| q dx ≤ Meas(B n ) 2-q 2 C q/2 ≤ Meas(Ω) 2-q 2 C q/2 , (2.9)
then by addition

n 0 n=0 Bn |∇u(x)| q dx ≤ (n 0 + 1)Meas(Ω) 2-q 2 C q/2 .
(2.10)

On the other hand, let n > n 0 , we have

Bn |u(x)| q * dx ≥ Bn n q * dx = n q * Meas(B n ), then Meas(B n ) ≤ 1 n q * Bn |u(x)| q * dx. (2.11) Inequalities (2.8) and (2.11) yield ∀n > n 0 , Bn |∇u(x)| q dx ≤ C q/2 1 n q * 2-q 2 Bn |u(x)| q * dx 2-q 2 , (2.12)
which yields by addition

+∞ n=n 0 +1 Bn |∇u(x)| q dx ≤ C q/2 +∞ n=n 0 +1 n (q-2)q * 2 Bn |u(x)| q * dx 2-q 2 .
(2.13)

We then apply the discrete Hölder inequality on the right hand side of (2.13) to obtain

+∞ n=n 0 +1 n (q-2)q * 2 Bn |u(x)| q * dx 2-q 2 ≤ +∞ n=n 0 +1 Bn |u(x)| q * dx 2-q 2 +∞ n=n 0 +1 n N (q-2) N -q q/2
.

(2.14)

We need to prove that the series n≥n 0 +1 n

N (q-2) N -q converges, which is equivalent to N (2 -q) N -q > 1.
By simple calculations, this inequality is equivalent to q < N ′ .

Then we set λ 2 (n 0 , q)

= +∞ n=n 0 +1 n N (q-2) N -q q/2
. And since the series converges, we directly have:

λ 2 (n 0 , q) -→ n 0 →+∞ 0. With this notation, (2.14) becomes +∞ n=n 0 +1 n (q-2)q * 2 Bn |u(x)| q * dx 2-q 2 ≤ λ 2 (n 0 , q) +∞ n=n 0 +1 Bn |u(x)| q * dx 2-q 2 ≤ λ 2 (n 0 , q) Ω |u(x)| q * dx 2-q 2 .
(2.15)

Hence, (2.15) combined with (2.13) yield

+∞ n=n 0 +1 Bn |∇u(x)| q dx ≤ C q/2 λ 2 (n 0 , q) Ω |u(x)| q * dx 2-q 2
= C q/2 λ 2 (n 0 , q)∥u∥ q * (2-q) 2 0,q * ,Ω .

(2.16) We set λ 1 (n 0 , q) = (n 0 + 1)Meas(Ω) 2-q 2 , and we add (2.10) to (2.16) to obtain, by (2.7)

Ω |∇u(x)| q dx ≤ C q/2 λ 1 (n 0 , q) + λ 2 (n 0 , q)∥u∥ q * (2-q) 2 0,q * ,Ω .
(2.17)

That corresponds to (2.6), which concludes the proof.

Then we use this result to prove the washer lemma.

Proof of lemma 1. We start with the case N = 2, which is easier. In that case, we have q * (2 -q) 2 = N q(2 -q) 2(N -q) = q. Then (2.6) becomes ∥∇u∥ q 0,q,Ω ≤ C q/2 λ 1 + λ 2 ∥u∥ q 0,q * ,Ω .

(2.18)

As q < 2, we can apply Sobolev embedding W 1,q (Ω) → L q * (Ω) on the right hand side of (2.18), and then the Poincaré-Friedrichs inequality ∥u∥ q 0,q * ,Ω ≤ K∥∇u∥ q 0,q,Ω ≤ KC q/2 λ 1 + λ 2 ∥u∥ q 0,q * ,Ω ,

where K > 0 depends on Ω and q. We recall that if b < 1, the following equivalence stands

x ≤ a + bx ⇔ x ≤ a 1 -b . (2.20)
Since λ 2 (n 0 , q) -→ n 0 →+∞ 0, it is possible to choose n 0 ∈ N large enough to have

KC q/2 λ 2 < 1.
Then by (2.20), we obtain

∥u∥ q 0,q * ,Ω ≤ KC q/2 λ 1 1 -KC q/2 λ 2 .
(2.21) Thus, from (2.18):

∥∇u∥ q 0,q,Ω ≤ C q/2 λ 1 + KC q/2 λ 1 λ 2 1 -KC q/2 λ 2 . ( 2

.22)

And the Poincaré-Friedrichs inequality yields

∥u∥ q 1,q,Ω ≤ LC q/2 λ 1 + KC q/2 λ 1 λ 2 1 -KC q/2 λ 2 , (2.23)
where L depends on q and Ω. This yields the result with

A = L 1/q C 1/2 λ 1 + KC q/2 λ 1 λ 2 1 -KC q/2 λ 2 1/q . ( 2

.24)

Let us focus now on the case N > 2. Since q < N ′ < N , the Sobolev embedding used before yields ∥u∥ q 0,q * ,Ω ≤ K∥∇u∥ q 0,q,Ω ≤ KC q/2 λ 1 + λ 2 ∥u∥ q * (2-q) 2 0,q * ,Ω .

(2.25)

Inequality (2.25) is of the form x ≤ a + bx γ , with γ = q * (2 -q) 2q = N (2 -q) 2(N -q) .
(2.26)

We have already shown that if q ∈]1, N ′ [, then:

1 < N (2 -q) N -q < N ′ hence 1 2 < γ < N ′ 2 < 1.
(2.27)

Thus, the inequation we need to solve is of the form: x ≤ a + bx γ , with γ ∈]0, 1[ and a, b > 0.

The graph on the side shows the existence of τ ∈]0, +∞[ depending on a, b and γ such that x ≤ a + bx γ ⇔ x ≤ τ .

F ′ (x) = bγx γ-1 -1 F ′′ (x) = bγ(γ -1)x γ-2
, therefore F ′′ (x) < 0. We obtain the following variations x F ′′ (x)

F ′ (x) F (x) 0 α +∞ - - +∞ -1 a F (α) -∞ 0 τ (a, b, γ) F (x)
x 0 α Inequality (2.25) finally ensures the existence of τ = τ (λ 1 , λ 2 , C) > 0 such that ∥u∥ q 0,q * ,Ω ≤ τ.

(2.28) Thus, (2.6) and Poincaré-Friedrichs inequality yield

∥∇u∥ q 0,q,Ω ≤ C q/2 (λ 1 + λ 2 τ ) hence ∥u∥ q 1,q,Ω ≤ LC q/2 (λ 1 + λ 2 τ ) , (2.29)
where L only depends on Ω and q. With A = L 1/q C 1/2 (λ 1 + λ 2 τ ) 1/q , lemma 1 is proved in case N > 2.

The first hypothesis of lemma 1 is q ∈]1, N ′ [, where N ′ = N N -1 . It has been used many times, and we are able to show that is impossible to obtain similar results with higher values of q.

Indeed, considering the case N > 2, we recall that α is the only point where F ′ vanishes, which means bγα γ-1 = 1 thus α = (bγ)

1 1-γ . (2.30)
Replacing b by the constants of lemma 2 yields b(q) = M (q)λ 2 (q), where M (q) is bounded if q ∈]1, N ′ [. Consequently, b(q) -→ q→N ′ +∞ and then There is a similar result in the case N = 2. The proof of lemma 1 requires to choose an index n 0 great enough to have b < 1. But as λ 2 (n 0 , q) -→ q→N ′ +∞, an increase of q implies an increase of n 0 . Thus, as

γ(q) = N (2 -q) 2(N -q) -→ q→N ′ 2 hence 1 1 -γ -→ q→N ′ 1 2 . ( 2
2-q 2 = 1 -q 2 -→ q→N ′ 1 -N ′ 2 = 0, it follows λ 1 (n 0 , q) = (n 0 + 1)Meas(Ω) 2-q 2 -→ q→N ′ n 0 →+∞ +∞ hence A -→ q→N ′ +∞.
(2.32)

We have shown that q cannot be larger than the assumptions of lemma 1.

The result shown in this first section will be useful in the following. Generally, solving a PDE system is divided into two steps:

• simplifying the problem (for example by moving to a finite-dimension space, or by using a truncation function),

• coming back to the initial problem with the help of compacity results.

A general result on norms

Prop./Def. 3 (Trace norm on H 1 ). Let us define the application:

N 0 (u) = ∥∇u∥ 2 0,2,Ω + ∥u∥ 2 0,2,Γ 1/2
, "u |Γ " referring to the trace of u on Γ.

Then N 0 is a norm on H 1 (Ω).

Proof. N 0 is non negative, homogeneous and satisfies triangular inequality (direct consequence of the fact that the norms L 2 also do). Let us consider u ∈ H 1 (Ω) such that N 0 (u) = 0. It comes ∥∇u∥ 0,2,Ω = 0 and ∥u∥ 0,2,Γ = 0.

First, this yields that ∇u = 0 almost everywhere on Ω. Thus, we write: u ae.

= λ, where λ is a real constant. And if u is constant almost everywhere, its trace has the same value almost everywhere on Γ. Yet, tr(u) = 0, therefore u ae.

= 0. Consequently, N 0 is a norm on H 1 (Ω).

Theorem 4 (Two equivalent norms). Let us recall the definition of the classical Sobolev norm on H 1 (Ω):

∥u∥ 1,2,Ω = ∥u∥ 2 0,2,Ω + ∥∇u∥ 2 0,2,Ω 1/2 . (2.33)
Then this norm and N 0 previously defined are equivalent.

Proof. At first, the trace operator tr :

H 1 (Ω) → H 1/2 (Γ) is surjective (see [7]
), and by Sobolev embedding,

H 1/2 (Γ) → L q (Γ), with 1 q = 1 2 -1 2N , thus q = 2N N -1 ≥ 2. Then, since Ω is bounded, we have: H 1/2 (Γ) → L 2 (Γ). Hence ∃C > 0, ∀u ∈ H 1 (Ω), ∥u∥ 2 0,2,Γ ≤ C∥u∥ 2 1,2,Ω . (2.34) We deduce ∀u ∈ H 1 (Ω), N 0 (u) 2 ≤ (1 + C)∥u∥ 2 1,2,Ω . (2.35) Therefore N 0 is dominated by ∥ • ∥ 1,2,Ω .
Conversely, we intend to show

∃C > 0, ∀u ∈ H 1 (Ω), ∥u∥ 2 1,2,Ω ≤ CN 0 (u) 2 .
(2.36)

By contradiction, let us suppose the negation of (2.36), which means

∀C > 0, ∃u ∈ H 1 (Ω), ∥u∥ 2 1,2,Ω > CN 0 (u) 2 . (2.37) Then in particular ∀n ∈ N, ∃u n ∈ H 1 (Ω), ∥u n ∥ 2 1,2,Ω > nN 0 (u n ) 2 .
(2.38)

Inequality (2.38) yields that ∀n ∈ N * , ∥u n ∥ 0,2,Ω > 0.
Then, both sides of (2.38) van be divided by ∥u n ∥ 0,2,Ω . Which means, keeping the same notation

∀n ∈ N, ∃u n ∈ S, ∥u n ∥ 2 1,2,Ω > nN 0 (u n ) 2 , (2.39)
where S refers to the "L 2 -sphere" defined by: S = {v ∈ H 1 (Ω) / ∥v∥ 0,2,Ω = 1}. Then (2.39) can be rewritten as

∀n ≥ 2, 1 n + 1 n ∥∇u n ∥ 2 0,2,Ω > ∥u n ∥ 0,2,Γ + ∥∇u n ∥ 2 0,2,Ω . (2.40) Since ∥u n ∥ 2 0,2,Γ ≥ 0, (2.40) yields in particular 1 - 1 n ∥∇u n ∥ 2 0,2,Ω ≤ 1 n , then ∥∇u n ∥ 2 0,2,Ω ≤ 1 n -1 . (2.41) Therefore, the sequence (∇u n ) n∈N * is bounded in L 2 (Ω) ; and as (u n ) n∈N * is a sequence of points of S, (u n ) n∈N is also bounded in L 2 (Ω). We deduce (u n ) is bounded in H 1 (Ω).
Thus, Banach-Alaoglu theorem ensures the existence of a subsequence still denoted

(u n ) n∈N * satisfying u n H 1 ⇀ n→+∞ u. But (2.40) yields ∀n ≥ 2, 1 n > ∥u n ∥ 2 0,2,Γ + 1 - 1 n ∥∇u n ∥ 2 0,2,Ω ≥ 1 2 N 0 (u n ) 2 , (2.42) hence N 0 (u n ) -→ n→+∞ 0, then u = 0.
Last, by Rellich theorem:

u n L 2 -→ n→+∞ 0, which implies: ∥u n ∥ 0,2,Ω -→ n→+∞ 0. But we have assumed that ∀n ∈ N, ∥u n ∥ 0,2,Ω = 1, which is a contradiction.

Analysis of the first equation

In this subsection, given p ∈]1, +∞[ and k ∈ W 1,p 0 (Ω), we consider the system:

-div(A(k)∇u) = f in Ω, -A(k) ∂u ∂n = αu on Γ. (3.1)
Where f ∈ L 2 (Ω), and α > 0. We recall that A : R → R + is bounded from above and below:

∀k ∈ R, 0 < ν ≤ A(k) ≤ µ < +∞. (3.2)

Variational formulation

Let us write the variational form of this equation. To do so, we need the Green formula.

Theorem 5 (Green formula). Let f ∈ H 1 (Ω) and g ∈ H 1 (Ω) N . Assume that n is the normal vector of Ω, then the following formula stands

Ω div(g)f dx = - Ω g • ∇f dx + Γ (g • n)f dS(x). (3.3)
Let us consider w ∈ H 1 (Ω) and multiply the first equation of (3.1) by w, then integrate.

-

Ω div(A(k)∇u)w dx = Ω f w dx. (3.4)
The Green formula (3.3) yields

- Ω div(A(k)∇u)w dx = Ω A(k)∇u • ∇w dx - Γ A(k) ∂u ∂n w dS(x) = Ω A(k)∇u • ∇w dx + α Γ uw dS(x). (3.5)
That leads to the variational formulation of (3.1):

   u ∈ H 1 (Ω) ∀w ∈ H 1 (Ω), Ω A(k)∇u • ∇w dx + α Γ uw dS(x) = Ω f w dx. (3.6)
All the terms of (3.6) are well defined, since u, w ∈ H 1 (Ω), hence ∇u, ∇w ∈ L 2 (Ω). And A is bounded. Moreover, by the trace theorem (see [START_REF] Louis | Non-homogeneous boundary value problems and applications[END_REF]), u, w

∈ H 1/2 (Γ) → L 2 (Γ).
The main result of this section is the following. Theorem 6. There exists a unique function u ∈ H 1 (Ω) satisfying (3.6).

Proof. This proof uses the Lax-Milgram theorem on the Hilbert space H 1 (Ω). Let a(k, •, •) be the bilinear form given by

∀u, w ∈ H 1 (Ω), a(k, u, w) = Ω A(k)∇u • ∇w dx + α Γ uw dS(x). (3.7) Let u, w ∈ H 1 (Ω) |a(k, u, w)| ≤ Ω A(k)|∇u • ∇w| dx + α Γ |uw| dS(x) ≤ µ∥∇u∥ 0,2,Ω ∥∇w∥ 0,2,Ω + α∥u∥ 0,2,Γ ∥w∥ 0,2,Γ ≤ (µ + αM )∥u∥ 1,2,Ω ∥w∥ 1,2,Ω , (3.8)
where M does not depend on u. Therefore, a(k,

•, •) is continuous on H 1 (Ω) × H 1 (Ω).
Then we show the coercivity. Let w ∈ H 1 (Ω), we have

a(k, w, w) = Ω A(k)|∇w| 2 dx + α Γ |w| 2 dS(x) ≥ ν∥∇w∥ 2 0,2,Ω + α∥w∥ 2 0,2,Γ ≥ inf(ν, α)N 0 (w) 2 , (3.9)
hence the coercivity of a(k, •, •) by theorem 4.

Finally, it is clear that the linear form [w → Ω f w dx] is continuous. Problem (3.6) then admits a unique solution u ∈ H 1 (Ω), by the Lax-Milgram theorem.

A priori estimates for u

Previously, we have shown that for k ∈ W 1,p 0 (Ω), (3.6) admits a unique solution u = u(k). In this subsection, we give some estimates for this function u.

Proposition 7. The solution u of (3.6) satisfies the inequality:

∥u∥ 1,2,Ω ≤ C∥f ∥ 0,2,Ω inf(α, ν) . ( 3

.10)

Where C > 0 only depends on Ω.

Proof. Taking w = u as a test in (3.6) leads to

Ω A(k)|∇u| 2 dx + α Γ |u| 2 dS(x) = Ω f u dx. (3.11)
We get by Cauchy-Schwarz inequality

ν∥∇u∥ 2 0,2,Ω + α∥u∥ 2 0,2,Γ ≤ ∥f ∥ 0,2,Ω ∥u∥ 0,2,Ω . (3.12)
Then by Theorem 4

ν∥∇u∥ 2 0,2,Ω + α∥u∥ 2 0,2,Γ ≥ inf(ν, α)N 0 (u) 2 ≥ inf(ν, α) C ∥u∥ 2 1,2,Ω . (3.13) Then (3.12) becomes inf(ν, α) C ∥u∥ 2 1,2,Ω ≤ ∥f ∥ 0,2,Ω × ∥u∥ 0,2,Ω ≤ ∥f ∥ 0,2,Ω ∥u∥ 1,2,Ω , (3.14)
which yields (3.10).

Proposition 8. Let k ∈ W 1,p 0 (Ω), and u = u(k) the solution of (3.6). Then u and k satisfy the inequality

∥ A(k)∇u(k)∥ 0,2,Ω ≤ C∥f ∥ 0,2,Ω inf(α, ν) , (3.15)
where C > 0 only depends on Ω.

Proof. Equality (3.11) can be rewritten with norms

∥ A(k)u∥ 2 0,2,Ω + α∥u∥ 2 0,2,Γ = Ω f u dx. (3.16)
Let us bound from below the left hand side of (3.16):

∥ A(k)u∥ 2 0,2,Ω + α∥u∥ 2 0,2,Γ = 1 2 ∥ A(k)u∥ 2 0,2,Ω + 1 2 Ω A(k)|∇u| 2 dx + α∥u∥ 2 0,2,Γ ≥ 1 2 ∥ A(k)u∥ 2 0,2,Ω + ν 2 ∥∇u∥ 2 0,2,Ω + α∥u∥ 2 0,2,Γ ≥ 1 2 ∥ A(k)u∥ 2 0,2,Ω + inf(α, ν) 2 N 0 (u) 2 ≥ 1 2 ∥ A(k)u∥ 2 0,2,Ω + inf(α, ν) 2C ∥u∥ 2 1,2,Ω . (3.17)
Then, let us apply Young inequality with weights to the right hand side of (3.16):

Ω f u dx ≤ ∥f ∥ 0,2,Ω ∥u∥ 1,2,Ω ≤ inf(α, ν) 2C ∥u∥ 2 1,2,Ω + C 2 inf(α, ν) ∥f ∥ 2 0,2,Ω . (3.18)
Thus, (3.16) can be rewritten

1 2 ∥ A(k)u∥ 2 0,2,Ω ≤ C 2 inf(α, ν) ∥f ∥ 2 0,2,Ω , (3.19)
which yields (3.15). These estimates will be useful in the following, to show a strong convergence in H 1 for equation (3.1).

A first convergence result: the energy method

This subsection aims at proving the following result.

Theorem 9 (H 1 convergence). Let p ∈]1, +∞[, and (k n ) n∈N be a sequence of functions in W 1,p 0 (Ω). Let us suppose that (k n ) weakly converges to k in W 1,p 0 (Ω). Let u = u(k), and

u n = u(k n ), n ∈ N.
Then a subsequence still denoted (u n ) n∈N strongly converges to u in H 1 (Ω).

This theorem is a consequence of another, which involves A and the previous estimates.

Theorem 10 (L 2 convergence). With the same notations and assumptions as those of Theorem 9, we have:

A(k n )∇u n L 2 -→ n→+∞ A(k)∇u. (3.20) Proof of Theorem 10. As k n W 1,p 0 ⇀ n→+∞ k, the compact embedding W 1,p 0 (Ω) → L 1 (Ω) yields k n L 1 -→ n→+∞ k.
By the Lebesgue inverse theorem, there is a subsequence (still named (k n ))

such that k n ae.

-

→ n→+∞ k. Since A is continuous, we deduce A(k n ) ae. -→ n→+∞ A(k).
We recall that u n and u satisfy

∀φ ∈ H 1 (Ω), ∀n ∈ N, Ω A(k n )∇u n • ∇φ dx + α Γ u n φ dS(x) = Ω f φ dx. (3.21) ∀φ ∈ H 1 (Ω), Ω A(k)∇u • ∇φ dx + α Γ uφ dS(x) = Ω f φ dx. (3.22)
Inequality (3.10) yields that (u n ) is bounded in H 1 (Ω), which ensures the existence of a subsequence still denoted (u n ) n∈N such that

u n H 1 ⇀ n→+∞ w and u n L 2 -→ n→+∞ w, hence ∇u n L 2 ⇀ n→+∞ ∇w, (3.23)
where w ∈ H 1 (Ω). This yields in particular

∀φ ∈ H 1 (Ω), Γ u n φ dS(x) -→ n→+∞ Γ wφ dS(x). (3.24)
Then, in order to show the convergence of the first term in (3.21), an integration lemma is needed.

Lemma 11 (Convergence of integrals). Let (f n ) n∈N and (g n ) n∈N two sequences of L 2 (Ω) functions, and f, g ∈ L 2 (Ω) satisfying f n L 2 -→ n→+∞ f and g n L 2 ⇀ n→+∞ g. Then we have Ω f n g n dx -→ n→+∞ Ω f g dx. (3.25)
This result will be proved after finishing the proof of Theorem 10.

The assumptions of Lemma 11 have to be checked.

• As the sequence (A(k n )) n∈N converges almost everywhere to A(k), the domination

A(k n )|∇φ| ≤ µ|∇φ| and the Lebesgue theorem yield A(k n )∇φ L 2 -→ n→+∞ A(k)∇φ.
• By (3.23):

∇u n L 2 ⇀ n→+∞ ∇w.
Therefore, Lemma 11 yields

∀φ ∈ H 1 (Ω), Ω A(k n )∇u n • ∇φ dx -→ n→+∞ Ω A(k)∇w • ∇φ dx. ( 3
A(k n )∇u n L 2 ⇀ n→+∞ A(k)∇u. (3.27)
In order to obtain the strong convergence of this sequence, the convergence of the norms remains to be proved. Let us write (3.21) with φ = u n and (3.22) with φ = u.

∀n ∈ N, Ω A(k n )|∇u n | 2 dx + α Γ |u n | 2 dS(x) = Ω f u n dx. (3.28) Ω A(k)|∇u| 2 dx + α Γ |u| 2 dS(x) = Ω f u dx. (3.29)
The strong convergence of (u n ) in L 2 directly yields

Ω f u n dx -→ n→+∞ Ω f u dx. (3.30)
And since

H 1/2 (Γ) → L 2 (Γ) is compact: (u n ) |Γ L 2 -→ n→+∞ u |Γ . In particular Γ |u n | 2 dS(x) -→ n→+∞ Γ |u| 2 dS(x). ( 3
Ω A(k n )|∇u n | 2 dx -→ n→+∞ Ω A(k)|∇u| 2 dx. (3.32)
The result of Theorem 10 is then obtained.

Finally, Lemma 11 needs to be proved.

Proof of Lemma 11. Let us decompose the difference between the two integrals:

Ω (f n g n -f g) dx = Ω f (g n -g) dx + Ω (f n -f )g n dx. (3.33)
On one hand, the weak convergence of (g n ) yields: Ω f (g n -g) dx -→ n→+∞ 0.

On the other hand, (g n ) n∈N weakly converges, then is bounded in L 2 (Ω). We deduce

Ω (f n -f )g n dx ≤ ∥f n -f ∥ 0,2,Ω ∥g n ∥ 0,2,Ω ≤ M ∥f n -f ∥ 0,2,Ω -→ n→+∞ 0. ( 3
√ ν∥∇u n -∇u∥ 0,2,Ω ≤ A(k n )∇u n -A(k n )∇u 0,2,Ω ≤ A(k n )∇u n -A(k)∇u 0,2,Ω + A(k n ) -A(k) ∇u 0,2,Ω -→ n→+∞ 0, (3.35)
by Theorem 10 and Lebesgue Theorem.

Analysis of the second equation

From now until the end, N = 3. In this section, we consider the system:

-div(B(k)∇k) = A(k)|∇u(k)| 2 in Ω, k = 0 on Γ. (4.1)
In order to lighten the writing, we define:

D(k) = A(k)|∇u(k)| 2 . (4.2)

Variational formulation and description of the method

Let us define two function spaces:

V = 1<r<N ′ W 1,r 0 (Ω) and W = r>N W 1,r 0 (Ω).
Before stating the variational formulation, we need a result.

Proposition 12. Let k ∈ V and p ∈ W . Then B(k)∇k • ∇p ∈ L 1 (Ω) and D(k)p ∈ L 1 (Ω).
Proof. Let k ∈ V and p ∈ W . By applying the Hölder inequality with r > N such that p ∈ W 1,r 0 (Ω), we obtain

Ω B(k)|∇k • ∇p| dx ≤ µ Ω |∇k • ∇p| dx ≤ µ∥∇k∥ 0,r ′ ,Ω × ∥∇p∥ 0,r,Ω . (4.3) Since r > N , r ′ < N ′ , thus k ∈ W 1,r ′ 0 (Ω)
. This yields the first result.

The second result comes from (3.15), that yields:

∥D(k)∥ 0,1,Ω ≤ C∥f ∥ 0,2,Ω inf(α,ν) 2 , then D(k) ∈ L 1 (Ω) and p ∈ L ∞ (Ω), by Sobolev embedding. Hence D(k)p ∈ L 1 (Ω).
It is important to notice that if N = 3, V → L 2 (Ω). Indeed, we have the equivalence

W 1,q (Ω) → L 2 (Ω) ⇔ q * > 2 ⇔ 1 q < 1 2 + 1 N ⇔ q > 2N N + 2 .
Then the condition that needs to be satisfied is:

N ′ > 2N N +2 , which is equivalent to N < 4.
We are now able to state the variational formulation of (4.1) ; k is called a weak solution of (4.1) if

   k ∈ V ∀p ∈ W, Ω B(k)∇k • ∇p dx = Ω D(k)p dx. (4.4)
As a priori D(k) ∈ L 1 (Ω), it is not possible to solve directly (4.4). Indeed, the method that will be used consists in bringing (4.4) back to a Lax-Milgram problem, which requires being in a Hilbert space. Yet, W 1,p (Ω) with p ̸ = 2 is not a Hilbert space. And D(k) is not in L 2 (Ω).

More precisely, the method is divided into three steps:

• Regularizing the equation, by changing the source term.

• Linearizing it to solve the regularized problem with a fixed-point theorem.

• Taking the limit when the regularizing parameter goes to zero.

Two first steps: solving the regularized problem

In order to set the regularized version of (4.4), several tools must be defined.

Definition 1. If w ∈ L r (Ω) with r ∈ [1, +∞], w ∈ L r (R N ) refers to the extension by zero w : x → w(x) if x ∈ Ω 0 if x / ∈ Ω.
Definition 2 (Mollifier). Let ρ : R N → R a function such that Supp(ρ) = B(0, 1), ρ ≥ 0 and R N ρ(y)dy = 1. We define the family (ρ ε ) ε>0 by

∀x ∈ R N , ρ ε (x) = 1 ε N ρ x ε .
Let ε > 0, and D ε (k) given by: D

ε (k) = A(k) ∇u(k) * ρ ε 2
, and the following regularized problem:

   k ∈ H 1 0 (Ω) ∀p ∈ H 1 0 (Ω), Ω B(k)∇k • ∇p dx = Ω D ε (k)p dx. (4.5)
Linearizing is only useful if the assumptions of the Lax-Milgram theorem are satisfied. This is what will be proved now, beginning by a continuity result on the right hand side of (4.5).

Theorem 13. Let ε > 0 and D ε (k) defined as in (4.5). Then D ε (k) ∈ L 2 (Ω) and there exists a constant C > 0 only depending of the data of the problem such that

∀k ∈ H 1 0 (Ω), ∥D ε (k)∥ 0,2,Ω ≤ C ε 2 . (4.6)
The proof of Theorem 13 is based on a general result.

Lemma 14. Let v ∈ H m (Ω), ℓ ∈ N and (ρ ε ) ε>0 a mollifier defined as previously. For ε > 0 sufficiently close to 0, we have

∥v * ρ ε ∥ m+ℓ,2 ≤ C(m, ℓ) ε ℓ ∥v∥ m,2 . (4.7)
Proof of Lemma 14. Let us recall the definition of the Sobolev norm

∥v * ρ ε ∥ 2 m+ℓ,2 = |α|≤m |β|≤ℓ ∥∂ α+β (v * ρ ε )∥ 2 0,2 . Let α, β ∈ N d such that |α| ≤ m and |β| ≤ ℓ. Plancherel theorem yields ∥∂ α+β (v * ρ ε )∥ 2 0,2 = 1 (2π) N ∥F(∂ α+β (v * ρ ε ))∥ 2 0,2 , (4.8)
where F refers to the Fourier transform operator. Since v ∈ H m (Ω), the following derivation formula stands

∂ α+β (v * ρ ε ) = (∂ α v) * (∂ β ρ ε ).

And by Fourier transform, we get

F ∂ α+β (v * ρ ε ) = F(∂ α v)F(∂ β ρ ε ).
(4.9)

Then with norms:

∥F(∂ α+β (v * ρ ε ))∥ 2 0,2 = R N |F(∂ α v)(ξ)| 2 • |F(∂ β ρ ε )(ξ)| 2 dξ. (4.10)
The definition of ρ ε with respect to ρ yields, by a change of variables

F(ρ ε )(ξ) = 1 ε N R N e -ixξ ρ x ε dx = y=x/ε R N e -iεyξ ρ(y)dy = F(ρ)(εξ). (4.11)
Then we obtain

|F(∂ β ρ ε )(ξ)| 2 = |ξ β F(ρ ε )(ξ)| 2 = |ξ β F(ρ)(εξ)| 2 = 1 ε 2|β| |(εξ) β F(ρ)(εξ)| 2 = 1 ε 2|β| |F(∂ β ρ)(εξ)| 2 . (4.12) Yet, ∂ β ρ ∈ D(R N ) ⊂ L 1 (R N ), which yields ∀ξ ∈ R N , |F(∂ β ρ)(ξ)| ≤ ∥∂ β ρ∥ 0,1 .
This reported in the inequality (4.10) ensure the existence of a constant c 0 > 0 independent of v such that

∥F(∂ α+β (v * ρ ε ))∥ 2 0,2 ≤ c 0 ε 2|β| ∥F(∂ α v)∥ 2 0,2 . (4.13)
Plancherel Theorem then yields

∥∂ α+β (v * ρ ε )∥ 2 0,2 ≤ c 0 ε 2|β| ∥∂ α v∥ 2 0,2 . (4.14)
By addition for |α| ≤ m:

|α|≤m ∥∂ α+β (v * ρ ε )∥ 2 0,2 ≤ c 0 ε 2|β| ∥v∥ 2 m,2 . (4.15) Taking ε ∈]0, 1[ yields 1 ε 2|β| ≤ 1 ε 2ℓ . Thus, adding for |β| ≤ ℓ brings the inequality ∥v * ρ ε ∥ 2 m+ℓ,2 ≤ c(ℓ) ε 2ℓ ∥v∥ 2 m,2 , (4.16) hence (4.7).
We can now apply this result to prove Theorem 13.

Proof of Theorem 13. The definition of D ε yields

∥D ε (k)∥ 0,2,Ω = Ω | A(k) ∇u(k) * ρ ε | 4 dx 1/2 = A(k) ∇u(k) * ρ ε 2 0,4,Ω . (4.17)
And since N = 3, the Sobolev embedding

H 1 (Ω) → L 6 (Ω) → L 4 (Ω) yields A(k) ∇u(k) * ρ ε 0,4,Ω ≤ c 1 A(k) ∇u(k) * ρ ε 1,2,Ω ≤ c 2 ε A(k)∇u(k) 0,2,Ω . (4.18)
Finally, (3.15) yields the result.

In order to solve the regularized problem (4.5), we first linearize it, as said before.

   q = q(k) ∈ H 1 0 (Ω) ∀p ∈ H 1 0 (Ω), Ω B(k)∇q • ∇p dx = Ω D ε (k)p dx. (4.19)
We have already shown that [p → Ω D ε (k)p dx] is a continuous linear form on H 1 0 (Ω), and then it is clear that [(q, p) → Ω B(k)∇q • ∇p dx] is a bilinear form on H 1 0 (Ω).

• Let q, p ∈ H 1 0 (Ω):

Ω B(k)∇q • ∇p dx ≤ µ∥∇q∥ 0,2,Ω ∥∇p∥ 0,2,Ω ≤ µ∥q∥ 1,2,Ω ∥p∥ 1,2,Ω . (4.20)
Then continuity is proved.

• Let p ∈ H 1 0 (Ω): Ω B(k)|∇p| 2 dx ≥ ν∥∇p∥ 2 0,2,Ω ≥ ν 2 ∥∇p∥ 2 0,2,Ω + 1 c 2 Ω ∥p∥ 2 0,2,Ω ≥ ν 2 inf 1, 1 c 2 Ω ∥p∥ 2 1,2,Ω . (4.21)
Where c Ω refers to the Poincaré-Friedrichs constant for Ω. Then coercivity is proved.

Until the end, c Ω keeps the same meaning than in (4.21).

Consequently, problem (4.19) is a Lax-Milgram problem, hence by the same-named theorem, (4.19) has a unique solution q = q(k) ∈ H 1 0 (Ω). Our goal is applying the Leray-Schauder fixed-point theorem on the operator [k → q(k)].

Taking p = q(k) in (4.19) yields Ω B(k)|∇q(k)| 2 dx = Ω D ε (k)q(k) dx, hence ν∥∇q(k)∥ 0,2,Ω ≤ c Ω ∥D ε (k)∥ 0,2,Ω . (4.22)
Then Theorem 13 yields

∥∇q(k)∥ 0,2,Ω ≤ c Ω ∥D ε (k)∥ 0,2,Ω ν ≤ c Ω C νε 2 =: M. (4.23)
The constant M > 0 does not depend on k. Since Ω is bounded, the norms ∥∇ • ∥ 0,2,Ω and ∥ • ∥ 1,2,Ω are equivalent.

The end of this subpart aims at showing the existence of a solution to (4.5). This needs the Leray-Schauder fixed-point theorem (see [START_REF] Zeidler | Nonlinear functional analysis and its applications I: fixed-point theorems[END_REF]).

Theorem 15 (Leray-Schauder). Let X a Banach space, and B a convex closed bounded nonempty subset of X. Let f : B → B a continuous function with f (B) compact.

Then f admits at least one fixed point in B.

In our problem, we set X = H 1 0 (Ω), B = B(0, M ) and the function

Q : B → B k → q(k) .
It is clear that B is a convex closed bounded nonempty subset of H 1 0 (Ω), and

Q(B) ⊂ B. Remains to be proved that Q is continuous and Q(B) is compact. Let us show that Q(B) is compact. Let (k n ) n∈N a sequence of elements of B. Since (k n ) is bounded in H 1 (Ω)
, a subsequence still named (k n ) can be extracted, which satisfies:

k n H 1 ⇀ n→+∞ k.
By theorem 10 with p = 2, this weak convergence is sufficient to obtain (3.20). The Young convolution inequality consequently yields

∥ A(k n )∇u(k n ) * ρ ε -A(k)∇u(k) * ρ ε ∥ 0,2 ≤ ∥ A(k n )∇u(k n ) -A(k)∇u(k)∥ 0,2 × ∥ρ ε ∥ 0,1 =1 -→ n→+∞ 0. (4.24)
This strong convergence in L 2 and the Lebesgue inverse theorem ensure the existence of a subsequence of (k n ) n∈N still named the same way satisfying

A(k n )∇u(k n ) * ρ ε ae. -→ n→+∞ A(k)∇u(k) * ρ ε , hence D ε (k n ) ae. -→ n→+∞ D ε (k). (4.25)
By theorem 9, (u(k n )) n∈N strongly converges in H 1 (Ω), then (∇u(k n )) n∈N strongly converges in L 2 (Ω) N . Thus, the Lebesgue inverse theorem yields the existence of

h ∈ L 2 (Ω) such that ∀n ∈ N, |∇u(k n )| ≤ h ae. Ω, (4.26)
where (k n ) n∈N refers to another subsequence. Therefore, we obtain

| ∇u(k n ) * ρ ε (x)| ≤ h * ρ ε (x). ( 4

.27)

And since h does not depend on n and h * ρ ε is continuous with compact support, D ε (k n ) is bounded by a constant that does not depend on n, then (4.25) and Lebesgue theorem yield The assumptions of Lemma 11 are satisfied, so that it yields

∀p ∈ H 1 0 (Ω), Ω D ε (k n )p dx -→ n→+∞ Ω D ε (k)p dx. ( 4
∀p ∈ H 1 0 (Ω), Ω ∇q(k n ) • ∇p dx -→ n→+∞ Ω ∇q(k) • ∇p dx, hence ∇q(k n ) L 2 ⇀ n→+∞ ∇q(k). (4.31)
We deduce by compact embedding q(k n )

L 2 -→ n→+∞ q(k). We now intend to show

Ω D ε (k n )q(k n ) dx -→ n→+∞ Ω D ε (k)q(k) dx. (4.32)
Another decomposition is needed:

∥D ε (k n )q(k n ) -D ε (k)q(k)∥ 0,1,Ω ≤ ∥(D ε (k n ) -D ε (k))q(k)∥ 0,1,Ω + ∥D ε (k n )(q(k n ) -q(k))∥ 0,1,Ω . (4.33)
On the one hand, the Lebesgue theorem yields

∥(D ε (k n ) -D ε (k))q(k)∥ 0,1,Ω -→ n→+∞ 0.
On the other hand, there exist two constants c 1 and c 2 due to the domination of (D ε (k n )) and the embedding L 2 (Ω) → L 1 (Ω) such that

∥D ε (k n )(q(k n ) -q(k))∥ 0,1,Ω ≤ c 1 ∥q(k n ) -q(k)∥ 0,1,Ω ≤ c 2 ∥q(k n ) -q(k)∥ 0,
B(k n )|∇q(k n )| 2 dx -→ n→+∞ Ω B(k)|∇q(k)| 2 dx. (4.35)
We need a last decomposition: Let N B be defined by

Ω B(k n )|∇q(k n )| 2 -B(k)|∇q(k)| 2 dx = Ω B(k) |∇q(k n )| 2 -|∇q(k)| 2 dx + Ω (B(k n ) -B(k))|∇q(k n )| 2 dx. ( 4
B(k n )∇q(k n ) L 2 -→ n→+∞ B(k)∇q(k).
N B : f ∈ H 1 0 (Ω) → ∥B(k)∇f ∥ 0,2,Ω . (4.40)
The two norms ∥ • ∥ 1,2,Ω and N B are equivalent, and (4.39) ensures that (q(k n )) n∈N strongly converges for the norm N B . Thus, it also strongly converges in H 1 0 (Ω).

We obtain q(k n )

H 1 -→ n→+∞ q(k). (4.41)
Therefore, we have shown that Q(B) is compact, and by a very similar way, the continuity of Q is deduced. Consequently, Theorem 15 ensures the existence of a fixed-point of Q in B. Which means there exists a function k ∈ B that satisfies (4.5).

Last step: Taking the limit when ε goes to zero

Let us define a family of intermediary solutions with respect to what has just been done. For ε > 0, the intermediary solution k ε satisfies

∀p ∈ H 1 0 (Ω), Ω B(k ε )∇k ε • ∇p dx = Ω D ε (k ε )p dx. (4.42)
Our next goal is showing that (k ε ) is bounded uniformly with respect to ε. For this we use the washer lemma, (Lemma 1). But a general theorem from Stampacchia (see [START_REF] Stampacchia | Equations elliptiques du second ordre à coefficients discontinus[END_REF]) is needed before.

Theorem 16 (Regularity and truncation function). Let H ∈ W 1,∞ (R) having a finite number of discontinuities, and such that H(0) = 0. Let q ∈ [1, +∞[ and v ∈ W 1,q 0 (Ω). Then H(v) ∈ W 1,q 0 (Ω) and we have:

∇H(v) = H ′ (v)∇v.
Until the end of this chapter, for j ∈ N, H j refers to the truncation function defined by

- - -(j + 1) -j 0 j j + 1 1 -1 H j (x)
This function satisfies the assumptions of 16, so that we can take p = H j (k ε ) in (4.42) to obtain

∀j ∈ N, Ω B(k ε )|∇k ε | 2 H ′ j (k ε ) dx = Ω D ε (k ε )H j (k ε ) dx. ( 4

.43)

On one hand, by writing

B ε j = {x ∈ Ω / j ≤ |k ε (x)| < j + 1}: Ω B(k ε )|∇k ε | 2 H ′ j (k ε ) dx = B ε j B(k ε )|∇k ε | 2 dx ≥ ν B ε j |∇k ε | 2 dx. (4.44)
One the other hand, the Young convolution inequality yields

Ω D ε (k ε ) H j (k ε ) ≤1 dx ≤ A(k ε ) ∇u(k ε ) * ρ ε 2 0,2 ≤ A(k ε )∇u(k ε ) 2 0,2,Ω . ( 4
∃C > 0, ∀ε > 0, ∀j ∈ N, B ε j |∇k ε | 2 dx ≤ C. ( 4

.46)

This corresponds to the assumptions of lemma 1. Then for any exponent q ∈]1, N ′ [, we have: ∥k ε ∥ 1,q,Ω ≤ A. And given that C does not depend on ε, this is the same for A (as a consequence of lemma 1).

Therefore, (k

ε ) ε>0 is bounded in W 1,q (Ω) for q ∈]1, N ′ [. Banach-Alaoglu theorem yields the existence of a sequence (ε n ) n∈N such that ε n -→ n→+∞ 0 and k εn W 1,q 0 ⇀ n→+∞ k. Therefore, k ∈ V .
We need to show that k is a solution of (4.4), which means we need to show To add the term B(k n ), it seems useful to state a variant of Lemma 11.

∀p ∈ H 1 0 (Ω), Ω B(k)∇k • ∇p dx = Ω D(k)p dx. ( 4
Lemma 17 (Double convergence in V and W ).

Let Ω be an open subset of R N , f ∈ 1<r<N ′ L r (Ω)
and g ∈ L q (Ω), where q > N . Let us consider two sequences (f n ) n∈N and (g n ) n∈N such that

     ∀r ∈]1, N ′ [, f n L r -→ n→+∞ f g n L q ⇀ n→+∞ g (4.50)
Then we have the convergence

Ω f n g n dx -→ n→+∞ Ω f g dx. (4.51)
Proof. The proof of Lemma 17 is similar to the proof of Lemma 11. We write

Ω (f n g n -f g) dx = Ω (f n -f )g n dx + Ω f (g n -g) dx. ( 4

.52)

On one hand, the Hölder inequality yields

Ω (f n -f )g n dx ≤ ∥f n -f ∥ 0,q ′ ,Ω ∥g n ∥ 0,q,Ω ≤ M ∥f n -f ∥ 0,q ′ ,Ω -→ n→+∞ 0, (4.53)
where q ′ refers to the conjugate of q, which is in ]1, N ′ [. On the other hand, the weak convergence of (g n ) n∈N yields We have shown that the function k given by the washer lemma satisfies (4.4). Moreover, since k ∈ V , there exists q ∈]1, N ′ [ such that q * > 2 (for example by taking q = 4 3 , q * = 12 5 > 2).

Ω (g n -g)f dx -→ n→+∞ 0. ( 4
As a result, the embedding V → L 2 (Ω) is compact. Then (k εn ) strongly converges to k in L 2 (Ω). This allows us to conclude.

Conclusion: solution of the coupled problem

Let us consider the sequence (ε n ) mentioned previously, after all the extractions that were done. We have shown that (k εn ) strongly converges in L 2 (Ω) to a function k.

Moreover, (k εn ) weakly converges to k in every space W 1,q 0 (Ω) with q ∈]1, N [. Therefore, theorem 9 yields that (u(k εn )) strongly converges to u(k) in H 1 (Ω).

All of this ensures the existence of a couple (u; k) of solutions of (1.1) at the distribution sense, with u ∈ H 1 (Ω) and k ∈ V (in particular k ∈ L 2 ).

τ

  Let us show it rigorously by studying the variations of F : x > 0 → a + bx γ -x. This function is C ∞ on ]0; +∞[ and

  the Lebesgue inverse theorem, a subsequence still named the same way is dominated.That means |∇q(k n )| 2n∈N is dominated by a L 1 function. Then the Lebesgue theorem yieldsΩ (B(k n ) -B(k))|∇q(k n )| 2 dx -→ ) |∇q(k n )| 2 -|∇q(k)| 2 dx -→

1 -

 1 .54) Hence the result. Let us now check if the assumptions of Lemma 17 are satisfied. The compact embedding V → L 1 (Ω) yields k εn L where (ε n ) refers to a subsequence. Hence, by countinuity of B: B(k εn ) And the domination B(k εn )|∇p| ≤ µ|∇p| stands. Consequently, the Lebesgue theorem yields ∀r ∈]1, N ′ [, ∀p ∈ W, B(k εn )∇p convergence of (k εn ) n∈N is already established. Then, Lemma 17 yields the convergence of the left hand side ∀p ∈ W, Ω B(k εn )∇k εn • ∇p dx -→ n→+∞ Ω B(k)∇k • ∇p dx. (4.56) Convergence of the right hand side: We aim at applying the Lebesgue inverse theorem on (D εn (k εn )), so we need to prove ∀p ∈ W, D εn (k εn )p is clear that for n ∈ N and p ∈ W ∥D εn (k εn )p -D(k)p∥ 0,1,Ω = ∥D εn (k εn ) -D(k)∥ 0,1,Ω ∥p∥ 0,∞,Ω . (4.58)Then we write a decomposition.Ω |D εn (k εn ) -D(k)| dx = R N A(k εn )| ∇u(k εn ) * ρ εn | 2 -A(k)| ∇u(k)| 2 dx ≤ Ω |A(k εn ) -A(k)| • |∇u(k)| 2 dx + R N A(k εn ) | ∇u(k εn ) * ρ εn | 2 -| ∇u(k)| 2 dx.

( 4 .

 4 59)The same argument as in the proof of (4.55) yieldsΩ |A(k εn ) -A(k)| • |∇u(k)| 2 dx -→

2 ≤ 2 + 2 . 2 ≤

 2222 ∇u(k εn ) * ρ εn -∇u(k) 0,∇u(k εn ) * ρ εn -∇u(k) * ρ εn 0,∇u(k) * ρ εn -∇u(k) 0,(4.62) On one hand, the Young convolution inequality yields ∇u(k εn ) * ρ εn -∇u(k) * ρ εn 0,∥∇u(k εn ) -∇u(k)∥ 0,2,Ω -→ of theorem 9. On the other hand, a classical convergence result has to be recalled. Theorem 18. Let v a L p (R N ) function with compact support (1 ≤ p < +∞), and (ρ ε ) a mollifier. Then ∥v * ρ ε -v∥ 0,p -→ to ∇u(k) brings the following convergence ∇u(k) * ρ εn -∇u(k) finally yields (4.61).There is a subsequence, still named(ε n ), such that | ∇u(k εn ) * ρ εn | 2 -→ n→+∞ | ∇u(k)| 2 and | ∇u(k εn ) * ρ εn | 2 ≤ g ∈ L 2 (R N) almost everywhere (g can be taken with compact support). Therefore, Lebesgue theorem can be used for the last term of (4.59) to obtainR N A(k εn ) | ∇u(k εn ) * ρ εn | 2 -| ∇u(k)| 2 dx -→ ∥D εn (k εn ) -D(k)∥ 0,1,Ω -→58) yields the convergence of the right hand side ∀p ∈ W, Ω D εn (k εn )p dx -→ n→+∞ Ω D(k)p dx. (4.68)

  .36) Given that the sequence B(k n )|∇q(k n )| 2n∈N is bounded, (4.35) yields for another subsequence still named (k n ) n∈N :