PhD. Abdelfettah Elkchirid

IN SOCIAL WORK ACADEMIC PROGRAMS 1 Improving Diversity in Social Work Academic Programs: From Teaching to Attaining Diversity

Cultural competence can be studied within the curriculum and the knowledge level which requires experience through learning amid diverse students and educators. With demographic alters in education, there is a need for improvement of academic programs and curricula to emphasize multicultural diversity. The purpose of this research is to determine whether social work programs place adequate emphasis on teaching diversity to offer interpositions to attain diversity from classroom to practice in an undergraduate environment. It was found that most schools and programs in social work incorporate diversity regarding knowledge, skills, and values. However, it is imperative for a student or educator to develop new competencies in new areas of diversity as required by the advanced needs of the practice situation.

Introduction

Countries of the western world such as the United States and Canada have become pluralistic societies where multiculturalism plays a significant impact in many areas, e.g., politics and institutions. It is estimated that, by 2050, demographic populations in the United States will change so dramatically that European American population will decrease from 69% to 55% and diverse populations will expand (Kivisto, & Rundblad, 2000). However, multiculturalism is significant today not only because of the changing demographics, but changes in sociopolitical grounds also have a significant impact on social issues regarding oppression, racism, and discrimination [START_REF] Sue | Multicultural counseling competencies and standards: A call to the profession[END_REF].

With demographic alters in education, there is a need for improvement of academic programs and curricula to emphasize such changes. The Council on Social Work Education (CSWE) and the National Association of Social Workers (NASW) have already mandated standards on social work education in incorporating diversity in undergraduate and graduate programs. For example, CSWE of 1973 has modified the Standard 1234 to stress that a school must present "continual efforts to reach for diversity in student body, instructional personnel, and other educational supports" (Association for Counselor Education and Supervision, 1973). However, most social work programs only emphasize diversity content within the curriculum, but yet mandate their aggressiveness in reaching for diverse students, faculty, and interventions to improve the modern social work education.

Aims of the Present Research

The core purpose of this research is to determine whether social work programs put adequate emphasis on teaching diversity and to offer interpositions to attain diversity within the curriculum, student population, faculty, and intervention measures from classroom to practice in an undergraduate environment of colleges and universities. Social work education is necessary to endeavor that prepares students to serve clients of the diverse population starting from a curriculum which reflects on existing social changes to incorporating of contemporary changes into practice. This research paper is organized to provide historical and current reviews, limitations and future implications in social work education.

Historical Review of Social Diversity

In the past two decades, philosophers, writers, artists, educators and other thinkers have been debating new ways of understanding cultures and their legacies of knowledgebuilding, interpretation, and regeneration [START_REF] Martinez-Brawley | Social work, postmodernism and higher education[END_REF]. In a forceful argument backed by developments in other disciplines, Heineman-Pieper, (1985, as cited in Martinez-Brawley, 1999) debunked the primacy of empirical observations suggesting that the five senses, so much treasured by the empiricists, "observe only indirect effects; that is, they register not pristine facts but rather sensory experience interpreted by beliefs, expectations, language and culture" (p. 4). The perceptions, assessments, and interventions of social workers, among others, are influenced as much by the unintended learnings of daily exposure to market forces, the press and television as by professional preparation [START_REF] Brawley | Promoting social justice in partnership with the mass media[END_REF]. If service to others is to continue being the hallmark of social work, service is always provided in particular situations. Tronto (1987, as cited in Martinez-Brawley, 1999) has suggested that in order for an ethic of care to develop, individuals must experience, in personal ways, caring for others and being cared for by others. In offering examples to the developing world, social work cannot afford to draw from the professional school model of the early 20th century that suppressed practitioners as the hand-maiden of scientists and created the normative curriculum (Schön, 1983: 34-41 as cited in Martinez- [START_REF] Martinez-Brawley | Social work, postmodernism and higher education[END_REF]. Social work must exemplify the real world's struggle with what is not normative, or fixed or predetermined. [START_REF] Howard | Teaching evidence-based practice: toward a new paradigm for social work education[END_REF] noticed that, during those times, social workers relied primarily on the advice of their colleagues and supervisors, personal experiences, relevant theory, and authoritative texts for practice direction. For the most part, these information sources have served the profession well over the past century. However, a more significant role for scientific evidence in practice decision making would undoubtedly increase the effectiveness and enhance the credibility of the profession. As the demand for evidence-based practice increased, the issue of diversity and difference was also growing.

The subject of diversity could only be well understood through systemic research.

Timely at this point, [START_REF] Dyeson | Cultural diversity and populations at risk: social work education and practice[END_REF] observed that the social work profession is preparing its practitioners to serve those who experience discrimination and oppression. He further described how issues of cultural diversity and populations at risk were infused throughout the social work curriculum leading to the bachelor and master degrees in social work. Five primary content areas are identified in social work education as human behavior and social environment, social welfare policy, social work practice, research and field internships. It is expected that social work programs integrate the concepts of values and ethics, diversity, populations at risk, and social and economic justice throughout the five primary content areas. Human behavior and the social environment represent a content area that examines the interrelationships between biological, psychological, social, and cultural determinants of human behavior.

One of the learning objectives of a course in this content area is for students to demonstrate an understanding and appreciation of human diversity including groups distinguished by ethnicity, race, culture, class, gender, sexual orientation, religion, physical or mental ability, and age. Students should also demonstrate an understanding of the impact of diversity on human development and behavior with particular reference to specified diverse groups in the practice environment. Social welfare policy is a content area that provides a historical perspective of social welfare history and policy. One of the learning objectives of a course in this content area is for students to develop an awareness of how sexism, racism, classism, ageism, and homophobia have permeated North American society and social welfare practice and policy, in which cultural competence plays an important role.

Cultural Competence and Teaching Diversity

The importance of cultural competence in social work education has gained tremendous research attention. In the early 1900s, the emphasis on cultural competence in social work started within the Black populations who needed assistance in dealing with social oppression (Fox, 1983). Although cultural competence was a well-known concept derived from racism issues from the past, this concept has been incorporated in the social work education. The NASW defines cultural competence as an ability to understand racism, oppression, discrimination, and to practice within diverse populations regardless of race, age, sex, ethnicity, gender, sexual orientation, and physical ability (National Association of Social Workers, 2001). Also, the CSWE also added standards stating that social work programs should be flexible and "add competencies consistent with their missions and goals" (CSWE, 2008, p. 3). Social work programs are required to present nondiscrimination policies and data that demonstrate their efforts to accomplish diversified student-faculty composition.

However, the attainment of a diverse population among students, faculty, and school interventions is yet to be explored. Social issues have influenced modern education throughout the past, but new eloquent voices stressed the importance of diversity in educational programs leading to students' academic improvements. For instance, in 2003, in the case of Grutter v. Bollinger, the Supreme Court ruled in favor of the University of Michigan in contribution to diversity in universities. As stated by the Associate Justice, Sandra Day O'Connor, "student body diversity promotes learning outcomes, and better prepares students for an increasingly diverse workforce, and society and better prepares them as professionals" (Grutter v. Bollinger, et al., 2003, p. 18).

Cultural Competence and Teaching Experience

Cultural competence cannot only be studied within the curriculum and the knowledge level. It requires cultural competence experience through learning amid diverse students, diverse educators, and through the application of social work skills to clients of diverse backgrounds. In 1975, Armitage & Clark (as cited in Gingerich, Kaye, & Bailey, 1999) stated that social work education was based upon 3 premises as 1) the ultimate purpose of a profession is practice; 2) the purpose of professional education is to teach practice behaviors effectively, and 3) practice behaviors can be specified as the operational objectives of social work education.

Current social work educators attempt to respond to educational and practice diversity differences, and their practice is considered to be evidence-based. Social work educators attempt to demonstrate the attainments of measurable educational outcomes so that the measures demonstrate students' knowledge in their practice (Jani, Ortiz, Pierce, & Sowbel, 2001). The Educational Policy and Accreditation Standards (EPAS) has created opportunities for evidence-based-practice where social work educators not only measure their students' competence knowledge but also evaluate students' skills when working with diverse populations (EPAS, 2008).

A guideline for assessing outcomes of learning in social work education was presented by [START_REF] Gingerich | Assessing quality in social work education: focus on diversity[END_REF]. The guidelines incorporate seven points, which can be applied to this review as guidelines: 1) learning outcomes determine the curriculum; 2) outcomes are defined as abilities; 3) course objectives are tied to the abilities; 4) assessment occurs throughout the educational process; 5) self-assessment becomes part of learning; 6) assessment leads to continuous program improvement; and 7) focus on learning rather than teaching.

Affirmation and respect for diversity and difference in providing multicultural education are core principles in the social work profession. Indeed, social work education and practice have been grounded on these principles of commitment to and respect for students with diverse identities and difference in background. Demonstrating an ongoing commitment through academic programs is possible. For example, the CSWE (2008) educational policies and accreditation standards specify there requirement for social work education programs to affirm their commitment to diversity by "age, class, color, culture, disability, ethnicity, gender, gender identity and expression, immigrant status, political ideology, race, religion, sex and sexual orientation" (p.2). Social work education programs are required to continuously reflect this commitment through providing a learning environment defined as inclusive of the institution, program structure, and activities, faculty and students, curriculum and field education, which is conducive to respect and celebrate diversity and difference [START_REF] Vakalahi | Diversity and Difference: What's the Social Work Commitment[END_REF].

Methodology

A literature search was conducted using multiple databases such as SAGE Journals Online, ERIC, PsycARTICLES, PsycINFO, ScienceDirect, MEDLINE, and JSTOR. A large number of articles were found, and the majority of them was published in the past decade thus were given special consideration. Even though the majority of research was published within the past decade, this review also considers articles published more than ten years. However, some researches were excluded due to their irrelevance to the aims of the present research. A qualitative analysis was carried out to review the articles as to the standards set by the NASW.

Limitations

The gathered articles were reviewed and analyzed based on the guidelines for assessing the outcomes of learning in social work education as presented by [START_REF] Gingerich | Assessing quality in social work education: focus on diversity[END_REF] as mentioned above. Since some of the articles are not able to present their related programs, they were analyzed according to the levels of competency as presented in 1997 by the Mandel School Faculty [START_REF] Gingerich | Assessing quality in social work education: focus on diversity[END_REF]. Their final ability statements for valuing a diverse world includes six developmental levels regarding knowledge, skills, and values needed. The levels move from the foundation level (self-awareness, knowledge of diversity, skills of engagement) to the advanced levels (developing new competencies, personal/professional identity, and self-assessment). The reviewed articles for this research were, in one way or another, seen to have enumerated or discussed their findings fit the guidelines presented above.

Discussion

Meaning of diversity. A controversy at the core of this discussion is the meaning of diversity. Diversity education dilemma occurs when exposed to information concerning status hierarchies in a classroom. A status hierarchy is a relative ranking of individuals based on shared valuations of specific characteristics, such as occupational position, country of origin, gender, or physical attractiveness (Ridgeway, 2001;Ridgeway, & Walker, 1995cited in Amoroso, et al., 2010). Discussions and activities centered on the characteristics of socially excluded demographic groups (e.g., race/ethnicity, gender, disability) in a variety of management courses may highlight or draw attention to these status characteristics. This may inadvertently reinforce group boundaries as individuals identify themselves with-and are identified as belonging to-higher or lower status groups.

The "real world" of the status hierarchy is strengthened within a classroom. Although status differences among students can also be inadvertently reinforced through non-diversity related discussions and activities, we argue that the negative consequences are exacerbated when learning objectives are directly linked to diversity-related issues. A diversity education dilemma exists when discussions and experiential activities that focus on socially excluded demographic groups (e.g., a discussion of the history of illegal workplace discrimination against African Americans) inadvertently strengthens the existing status hierarchy. In an effort not to reinforce status hierarchies in a classroom, educators may avoid altogether discussions of group memberships that imply status differences.

This option is untenable because awareness is a fundamental aspect of diversity education. Awareness of inequalities resulting from unearned privilege and disadvantage is an essential part of learning how to better structure work systems and to encourage interactions that reduces potentially discriminatory processes within a workplace [START_REF] Linnehan | Diluting diversity: Implications for intergroup inequality in organizations[END_REF]. Many students enter colleges with limited exposure to diversity (Avery & Thomas, 2004;Orfield, Bachmeier, James, & Eitle, 1997as cited in Amoroso et al. 2010); and increasing awareness and learning to value differences move students from "a relatively egocentric and cognitively simple state to a more other-centered and cognitively complex way of viewing themselves and the world in which they live" (Garcia, 1994, p. 429 as cited in [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF].

Cultural competence in students. [START_REF] Guy-Walls | Exploring cultural competence practice in undergraduate social work education[END_REF] examined the effectiveness of bachelor of social work (BSW) in multicultural curricula at two universities in the mid-south of America. Data were collected using survey research designed to explore how well students were prepared for multicultural practice. The study finds that senior undergraduate of social work students had higher scores on cultural competence than entry-level social work students and senior students of other disciplines. What alarming is when asked, "At this time in your life, how would you rate yourself regarding understanding how your cultural background has influenced the way you think and act?" only 50% of participants responded "always." As the students understood that their worldview impact showed how they interact with others, they should further understand that their worldview may impact how they deliver services (Carter, 1990;Garcia&VanSoest, 1999as cited in Guy-Walls, 2007). The researcher posed a significant challenge to social work educators to have a substantive knowledge base and empathic appreciation of multiple perspectives, as well as a conscious awareness of how their attitudes and values impact the worker-client relationship. [START_REF] Krentzman | Review of multidisciplinary measures of cultural competence for use in social work education[END_REF] explored measures of cultural competence from many disciplines as databases. These databases access scholar work from medicine, psychology, nursing, education, sociology, political science, pharmacy, criminal justice and social work.

Most authors from these disciplines agree that cultural competence is made up of values, skills, and knowledge; and thus the knowledge base necessary to practice competently may vary from profession to profession. It seems scales are discipline specific to some degree.

Given the multidisciplinary concern for culturally competent practice, cross-disciplinary collaboration seems promising in working toward a socially just world. It was revealed that scales themselves are cultural artifacts, shaped by the country, locale, and era in which they were developed.

While current conceptualizations of cultural competence are inclusive of all types of human difference, most scales measure competence with diverse races and ethnicities only [START_REF] Kitaoka | Multicultural counseling competencies: Lessons from assessment[END_REF]. Some scales address more than one type of diversity adequately by including several items for each type. For example, Talbot's scale (1992 as cited in [START_REF] Krentzman | Review of multidisciplinary measures of cultural competence for use in social work education[END_REF] comprehensively addresses gender, sexual orientation, and race. Lum's scale (2003) addresses six diverse groups, but this scale primarily emphasizes knowledge rather than skills or values. The development of a more comprehensive model of cultural competence that integrates or transcends individual types of difference would be advantageous. Further attention to this issue would help measure cultural competence more comprehensively. This may enhance understanding and measurement of cultural competence; and ultimately, advance effort to create a socially just and culturally competent world.

In 2009, a group from the University of Southern Maine and University of Louisville looked into assessing cultural competence in graduating students (Kohli, Kohli, Huber, & Faul, 2009). Relationships were examined within cultural competence, a latent variable with Racial diversity in students. Human diversity education should be guided toward awareness of practitioners and respect of differences among people (Kohli et al., 2009).

Moreover, attitudes and beliefs about working with people with a difference should be more open and respectful than a conservative way of thinking. This implies that the more conservative and less liberal the students, the more fixed their ideas will be. Research also shows that the total number of years of education of the respondents was inversely related to students' attitudes and beliefs. This implies that the higher the number of years of education they received, the more conservative their thinking will be (Kohli et al., 2009). This finding is in contrast to general assumptions that more education leads to increased cultural competence.

In this sample, Caucasian students perceived themselves to be less skilled in working with diverse populations than students from other ethnic groups. This supports the finding that non-White counselors were more skilled than their White counterparts (Mitchell, 1999as cited in Kholi, et al., 2009). [START_REF] Denson | Racial diversity matters: the impact of diversity-related student engagement and institutional context[END_REF] Evidence that many students have a fundamental lack of awareness is provided by numerous studies indicating the belief in many Americans that America is a meritocracy (Kluegel, & Smith, 1982;Ladd, & Bowman, 1998;Rasmussen Report, 2007 as cited in [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF]. Given that teacher and student, expectations are partially based on their status group memberships; we are concerned that attention to status differences among students exacerbates inequalities in learning opportunities and outcomes. Research has demonstrated that people are aware of differential expectations of competency afforded to various groups; and as a result, may perform worse if low expectations are expected from them (i.e., stereotype threat; Steele, & Aronson, 1995 cited in [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF].

The exercise may effectively educate the White students at the expense of the nonwhite students. Though it feels particularly egregious to affirm negative stereotypes in a classroom, Cheryan and Bodenhausen (2000 as cited in [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF] found that making positive stereotypes salient can depress performance of stereotyped students. In their study of female Asian American undergraduates who indicated that math performance was significant to them, Cheryan and Bodenhausen found that, when others' high-performance expectations (based on ethnic stereotypes) were made salient, the students had diminished ability to concentrate and depressed math performance.

Groups should not be designed based on salient group categories. Miller, Brewer, and Edwards (1985 as cited in [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF] found experimentally that minority students demonstrated more bias toward majority group members when the groups had been composed based on salient group categories. By using interventions such as those discussed above to minimize the "real-world" hierarchy in our classrooms, instructors can increase learning and ultimately may reduce discrimination in the workplace as the students shall compose future workforce [START_REF] Amoroso | The diversity education dilemma: exposing status hierarchies without reinforcing them[END_REF].

Roles of educators in diversity.

There is widespread agreement among North Americans that a crucial facet of college and university education is the preparation of students to participate effectively in a diverse society (Ford Foundation, 1998as cited in Amosoro, Loyd, & Hoobler, 2010). Educators play an essential role in exposing students to many diversity-related topics not only in courses primarily focused on managing diversity in organizations but also in other courses that highlight demographic differences between members of the workforce. Since the presence of status differences has been shown to impede learning and depress performance of members of lower status groups (e.g., Cohen, 1982Cohen, , 1994;;Cohen, & Lotan, 1995;Lovaglia, Lucas, Houser, Thye, & Markovsky, 1998;Ridgeway, 1982as cited in Amoroso, et al., 2010), classroom learning objectives are undermined.

To deal with diversity issues, one needs to realize that race is a critical part of identity formation. While race may have no medical or biological significance, it is not less as a social fact. What 'race' is constituted and how it is defined are a consequence of the environment. Thus, environment-not genotype-should play a significant role in determining how social work educators define identity relative to biracial Americans [START_REF] Hall | Eurocentrism in social work education from race to identity across the lifespan as biracial alternative[END_REF]. As a result of being biracial requires two processes.

On the one hand, it may precipitate a conscious distancing from the stigmatized biracial group. On the other hand, it may involve the creation of a new identity-based in part upon an inability to be accepted without reservation by monorace populations. Extended from ecology is the ecological perspective of social work education utilized as a mechanism for integrating theories of human behavior in a social environment (Teall, 2000;Greene andEphross, 1991 as cited in Hall, 2005). This mechanism is tantamount to a metaphor enabling the organization of people and their environments into a collective entity where one is defined in the context of its relationship to another (Mackey, & O'Brien, 1999 as cited in [START_REF] Hall | Eurocentrism in social work education from race to identity across the lifespan as biracial alternative[END_REF]. The relationship is differentiated by reciprocity in which people and their environments continuously manipulate and define, and occasionally alter, one another making the relationship a transactional phenomenon. Relevant to the biopsychosocial apparatus of an organism is the concept of adaptation. Adaptation is frequently confused with the adjustment but in fact is distinguishable, in which adjustment refers to a passive effort to accommodate the environment.

In contrast with earlier views, human development across lifespan construct stresses the impact of historical events on the processes of personal identity, the processes observed during an individual's lifespan, and the lifespan of family members within a community or in a social setting (Gatson, 1994as cited in Hall, 2005). Human development across the lifespan construct is also an effort to encompass variations among individuals as they move in time through social space, community, culture, and history. Moreover, biracial Americans are not passive recipients but with the potential to shape circumstances and contexts.

Social acceptance and teaching diversity.

In the interest of mental and emotional health, those who perceive themselves as biracial must counter-define the social and information universe (Long, 1991 as cited in [START_REF] Hall | Eurocentrism in social work education from race to identity across the lifespan as biracial alternative[END_REF]. In the face of two powerful barriers-racism and culture-it characterizes the viability of their existence. In the social work emphasis upon racial diversity, the process of self-acknowledgment and the proclamation of existence are the first critical step in personal and, later, social acceptance of what is different. The recent trend of diversity in social work education has facilitated assertions on the part of biracial Americans to define an identity for themselves. Social work education is, then, an active process of acquiring, assessing and producing knowledge in the environment of tolerance and respect for various populations (Germain, & Gitterman, 1980as cited in Hall, 2005). Therefore, educators are compelled to seek and embrace new forms of knowledge and commit to social justice in the study of, and service to diverse populations without limitation by race. In turn, educators in the present apply cultural content to solve problems and transform profession for succeeding generations.

Ortiz & Jani (2011) published an article with a critical race theory (CRT): A Transformational Model for Teaching Diversity. The critique addresses causes and personal distress while pursuing transformational change. In conceptualizing diversity, social workers need to address a broad social context that includes institutional/structural arrangements, recognize the intersection of multiple identities, and integrate an explicit social justice orientation.

Teaching about diversity in higher education involves more than merely presenting a mosaic of different people in the faculty and student bodies. Also, it cannot be taught as the sole enterprise of the curriculum in isolation from the overall ethos of its host institution, or as a single add-on course saddled with the responsibility of teaching (Lee, &Greene, 2003as cited in Ortiz, & Jani, 2011). Instead, diversity education requires attention to the institutional and social context because of the centrality of race in the U.S. society. The teaching of diversity content in social work, therefore, is more effective when institutional arrangements and social location of researchers, teachers, learners, and clients are considered. In other words, attention to diversity needs to be integrated throughout both the implicit and explicit curriculums.

CRT addresses this need that it does not assume the existence of universal truths and rejects master narratives but attempts to encompass all phenomena or dictate the construction of lives. Instead, it is based on the following assumptions: race is a social construction, race permeates all aspects of social life, and race-based ideology is threaded throughout society.

Proponents of CRT are also committed to social justice locating the voice of the marginalized and employing the concept of intersectionality (Delgado, & Stefancic, 2001;Solórzano, & Yosso, 2001 as cited in [START_REF] Jani | Access to intersectionality, content to competence: deconstructing social work education diversity standards[END_REF]. In sum, CRT provides social work educators with an opportunity to employ several different ways to enhance the teaching of diversity and to conceptualize transformational social work practice. Also, CRT makes a significant contribution to curriculum content itself. More than merely teaching students about culture, it requires them to analyze the institutional arrangements of society, assesses how dominant cultural assumptions shape them, and recognize how they may disadvantage members of non-dominant cultural groups. Effective teaching of diversity, therefore, requires a thorough examination and critique of social institutions. Such an evaluation needs to be applied to all areas of the curriculum; otherwise, diversity content runs the risk of being ghettoized and having its institutional nature denied [START_REF] Abrams | Critical race theory and the cultural competence dilemma in social work education[END_REF]as cited in Ortiz, & Jani, 2011). Merely appreciating and understanding marginalized and non-dominant group culture, however, is not sufficient for CRT.

CRT is a race-based critical paradigm that assesses power differentials at all levels.

One lesson social work educators can take from its critique is that teaching diversity is more complicated than trying to attend to differences among people in a society and the resulting "isms" [START_REF] Jani | Access to intersectionality, content to competence: deconstructing social work education diversity standards[END_REF]. Effective teaching of diversity requires a thorough examination of social structures, institutions, and ways of knowing and being. Without such examination, discussions of diversity evolve into polite (or, in some cases, impolite) conversations that do little to transform the institutions that perpetuate diversity mainly as the basis for maintaining differential accesses to social opportunities and rewards. This is a subtle but significant distinction from the discussion of intersectionality in new cultural competence frameworks because it uses the concept as an overarching perspective rather than a single element (CSWE, 2008;Lum, 2007as cited in Jani, et al., 2011). In contrast, recent publications have pointed out an increasing movement among social work educators toward teaching diversity content from a postmodern perspective to enhance students' ability to engage in culturally competent practice [START_REF] Abrams | Critical race theory and the cultural competence dilemma in social work education[END_REF]Nicotera, & Kang, 2009;Ortiz, & Jani,2010as cited in Ortiz, & Jani, 2011). This is significant because EPAS influences not only the content of what social work educators teach but also, more subtly, the manner in which it is taught. The need for certainty is contraindicated when teaching content on challenging topics such as diversity and social justice, as implied in the 2008 EPAS.

A movement toward a postmodern approach to diversity and difference needs to be accompanied, therefore, by revised methods of assessing whether students can apply curriculum content to their practice [START_REF] Jani | Access to intersectionality, content to competence: deconstructing social work education diversity standards[END_REF]. If a program, however, chooses to teach from a perspective that promotes access rather than justice and advocates generalizations over context, it will be insufficient. The challenge is to move social work education and practice to a new level of working with diversity and difference.

Roles of school in diversity.

With all campaign and attention to cultural competence, one may ask a question: Are school social work courses responding to the changing context? [START_REF] Berzin | Educating today's school social workers: are school social work courses responding to the changing context[END_REF] examined syllabuses from school social work courses for content and attention to these shifts. The most common topics covered were special education, confidentiality and ethics, history and role of school social workers, and collaboration. Calls from research, theoretical literature, and policy changes suggest that the following factors are related to large-scale shifts in educational landscape that affects school social work: evidence-based practice, response to intervention, positive behavior supports, No Child Left Behind, shifts in mental health prevalence, bullying, and school choice.

Syllabuses show inconsistent incorporation of these factors (Allen-Meares, 1994;Hoagwood, & Erwin, 1997;Kellyetal, 2008as cited in Berzinan, & O'Connor, 2010). Subject matter related to multilevel practice is limited with a strong focus on content aimed at clinical practice. Preparation of school social workers to meet changes in education and calls from scholars to support multi-tiered approaches to helping students are discussed. Although theoretical developments support multiple practice levels, research consistently finds that individual counseling is the primary practice choice in school social work practice.

A study found that course content was highly driven toward clinical preparation with significantly less emphasis on embedding school social workers in the educational context (Berzinan, & O'Connor, 2010). It examines school social work syllabuses from MSW programs to determine whether courses incorporated content related to relevant trends in education and school social work. Although attention to defining the roles of school social work was apparent, inconsistencies remain between school social work practice, theory, and education. A significant finding of the study was that the content of MSW courses on school social work was widely driven toward outlining the roles of school social workers and concentrating on clinical work.

Another study provides the first insight into how MSW programs are preparing school social workers through specific school social work courses (Slovaketal, 2006as cited in Berzinan, & O'Connor, 2010). Although school social workers have indicated that taking school social work course better prepares them for practice than the MSW degree alone, understanding the content included in these courses helps in assessment of their relevance. As found in this study, current school social work syllabuses include inconsistent and limited content addressing the changing educational landscape; and thus it is essential to consider how calls from the research, theory and policy arenas can be better incorporated.

Costin (1969 as cited in Berzinan, & O'Connor, 2010) stated that "it is a matter of concern that the main body of tasks and goals in school social work is not attuned to the urgent problems of School children and youth today" (p .274). School social work continues to face the same challenge as we prepare school social workers to address the changing educational context. MSW programs, through school social work courses, have an opportunity to provide school social workers with coursework that is relevant and attuned to the needs of diverse students.

Country wise attention to diversity. Diversity has been a part of EPAS for more than 35 years [START_REF] Ortiz | Critical race theory: a transformational model for teaching diversity[END_REF]. During this period, coverage of content related to specific groups has been either mandated or recommended in two ways-through curriculum or educational policies or through specific accreditation standards. The EPAS guidelines generally have been coupled with requirements that programs treat such groups in a nondiscriminatory manner in their policies and procedures. Defining diversity in specific ways advantages some people and disadvantages others based on their proximity or distance from the dominant culture [START_REF] Ortiz | Critical race theory: a transformational model for teaching diversity[END_REF].

The EPAS understanding is critical to relate the issue of diversity to the goal of social justice. In their attempts to satisfy the EPAS requirement to "engage diversity and difference in practice" (CSWE, 2008, pp.4-5), the social work programs may strive to demonstrate that students have acquired cultural competence. Perhaps because of how the concept is interpreted, the notion of cultural competence appears to imply that individuals can interact seamlessly between cultures purely through the expansion of their knowledge, values, and skills.

The operationalization of cultural competence often indicates learning about the shared history and characteristics of various groups, an objective that some educators believe can be measured through positivist means (Bonder, Martin, & Miracle, 2001as cited in Jani, et al., 2011). Understanding a group's history and changes in its cultural practices over time enables social workers to recognize the commonalities and differences among and between people with similar and different cultural identities. The 2008 EPAS competency of engaging diversity and difference in practice presents a possible conundrum for the cultural competence paradigm.

International arena and diversity. One might find it essential to examine the status of social work education and practice in the international arena. In the United States, the CSWE, 2005 accreditation guidelines and the NASW, 1999 Code of Ethics have been emphasizing that practitioners must be professionally trained to practice competently in our cultural and social diversity. This point of view is supported by the assumption that teaching diversity leads to knowledge, understanding, acceptance, and sensitivity to cultural and human diversity, which are prerequisites for empowering clients from diverse cultural and social environments (Chau, 1990 as cited in [START_REF] Kohli | Cross-cultural differences towards diversity issues in attitudes of graduating social work students in India and the United States[END_REF].

A meta-analysis of articles in the Indian Journal of Social Work revealed that only two articles discussed the need for including diversity in the social work curriculum [START_REF] George | Towards an inclusive paradigm in social work: The diversity framework[END_REF][START_REF] Kulig | Multicultural issues within universities: Identifying trends and challenges[END_REF]. There is no mandate on the integration of diversity content in social work curricula in India thus far. Despite the awareness of and importance given to understanding and including diversity issues in other countries, not many studies have been conducted in India to appraise cross-cultural differences in the attitudes of graduating students towards diversity issues as well as their perceived practice competence (Boyle, & Springer, 2001;Carrillo, et al., 1993as cited in Kohli, & Faul, 2005), and how the country in which these students are educated influences their attitudes and perceived competence.

It is therefore understandable that social work students in India are not sensitive enough to ethnic and religious differences and how these differences can affect their practice with clients (Gurumurthy, 1994 as cited in [START_REF] Kohli | Cross-cultural differences towards diversity issues in attitudes of graduating social work students in India and the United States[END_REF]. In India, due to the lack of professionalization and unity, the social work profession has not gained respect, trust, and popularity among human service profession. Religion, ethnic affiliations and patriarchy were and still are the primary institutions that govern the dominant reality and the way in which different social issues are being dealt with. Hence, diversity problems seemed to be viewed by social work students as personal, and hence not in their jurisdiction of service delivery.

In Japan, social work education seeks to recognize the competence of care professionals and relate this to practice fitness (Sewpaul, & Jones, 2004as cited in Saito, & Johns, 2009). The emphasis on international social work varies with choices regarding content, focus and intensity of international education influenced by a variety of factors, only some of which originate in the needs, wants and educational preferences of faculty and students. It is hard to find evidence of Japanese social work education making concerted and continuous efforts to enrich educational experience by reflecting cultural and ethnic diversity in its programmes. Nakamura and Ichibangase (2000a as cited in [START_REF] Saito | Japanese students' perceptions of international perspectives in social work[END_REF]) admit that Japan is less experienced than other countries regarding support for its ethnic minorities.

A study analyzes using comparative research methodology at the cross-nation level from Japan to compare critical thinking, knowledge of the history, philosophy of social work and social provisions of various countries (Nakamura, & Ichibangase, 2000bas cited in Saito, & Johns, 2009). More specifically, the study focuses on social work internationally through placements abroad to deepen students' knowledge. They exposed students to different cultural norms and enhanced their experience in being an outsider in which they were compelled to a re-evaluation of their own and the host country's values and norms.

International placements make it more evident what social work is about and underlines similarities between codes of ethics; for instance, highlighting the importance of treating service users as individuals, students may observe that there are more similarities between Japan and Western countries than between Japan and countries and cultures in Southeast Asia. Supporting these findings, Gibbs and Grambrill (1999 as cited in [START_REF] Saito | Japanese students' perceptions of international perspectives in social work[END_REF] stressed the importance of self-knowledge as a link to values development.

Studying abroad enhanced students' self-awareness, especially concerning recognizing and, at times, challenging their values, biases, beliefs, and ways of thinking about their own country. As one said, "Having something to compare my own practice with would highlight both my strengths and weaknesses and give me great opportunities to think about how my own practice could be improved" (Saito, & Johns, 2009, p. 60).

A study conducted by [START_REF] Beecher | International students' views about transferability in social work education and practice[END_REF] explored international students' views on the comparison between the US social work education and the Asian-Pacific indigenous practice. Some students recognized indigenous practice and valued multiculturalism; and regretted not having agency-specific knowledge and skills. The views of international students of their CSWE-based educational training and practicum experience in their home countries are rare topics in the literature of international social work. Another study examined social work education programmes and practice in South Africa, England and Canada [START_REF] Spolander | Social work education in canada, england and south africa: a critical comparison of undergraduate programmes[END_REF]. It acknowledges a need for flexibility in assessment of social work education and its application across cultural and structural contexts and offers a starting point in the dialogue regarding the impacts of labor mobility. Its concept of universal standards attempts to ensure a standard level of expectation about social work education globally; and offers a framework to recognize transnational mobility of social workers and acknowledge the impact of global events on local practice (Lyons, 2006as cited in Spolander et al., 2011). At the same time, these standards cannot ensure standardized education and practice due to the diversity of spoken languages internationally, the differences in economic and geographical situations, and the various cultural norms. Global standards are not always internationally relevant, which makes their application challenging (Healy, 2004;Yip, 2004[START_REF] Spolander | Social work education in canada, england and south africa: a critical comparison of undergraduate programmes[END_REF].

Migrants and diversity.

In the context of global migratory patterns, providing international standards for education and qualification of social workers is a valid task. [START_REF] Spolander | Social work education in canada, england and south africa: a critical comparison of undergraduate programmes[END_REF] advocated for the conceptualization of transnational social work to lay the groundwork for transnational criteria, standards, and processes to facilitate the migration of social work professionals who will continue to be on the move for professional, personal and political reasons.

Cultural competence is essential especially that it: (a) reduces the risk of crosscultural misunderstanding and miscommunication and (b) increases the likelihood of positive outcomes to planned interventions [START_REF] Bowie | Integrating diversity into graduate social work education: A 30-year retrospective view by MSW-level African American social workers[END_REF]. The more significant issue is the general preparation of professionals to grant them the skills and abilities to effectively interface and work with constituencies, clients, patients and consumers in culturally diverse settings across the spectrum of disciplines. This greater issue transcends academic disciplines, work settings, and even organizational purposes.

Many studies indicate that the most significant differences were in the content areas of social systems, family structure, poverty, class, and discrimination [START_REF] Bowie | Integrating diversity into graduate social work education: A 30-year retrospective view by MSW-level African American social workers[END_REF]. These findings suggest that, while progress is being made sporadically in some areas, its current progress is plodding; especially when considering the facts that the original standard 1234A was established over three decades ago; and that the numbers of citizens and residents from non-White cultures have been increasing at an astronomical rate particularly those of Latino and Asian. Most importantly, human behavior areas are found lacking in diversity and multiculturalism contents involving individuals, families, and multiplicity of external variables that play a role in their well-being.

Conclusion

It is becoming increasingly clear that diversity-related effort in colleges and universities appreciably enhances the quality of undergraduate education. The effort appears to improve students' experience and to learn by cultivating essential behavior and knowledge and by providing a unique educational context. Regardless of the context of diversity, a comprehensive framework of diverse values, ethics, knowledge, and skills are necessary for effective multicultural social work in educational practice [START_REF] Canda | Spiritual diversity in social work practice: the heart of helping (2 nd ed)[END_REF].

Providing multicultural social work education that prepares social workers to a diverse society is a way of responding to an ethical obligation to fight against social injustice and respond to complex cultural relations [START_REF] Abrams | Critical race theory and the cultural competence dilemma in social work education[END_REF]. [START_REF] Abrams | Reframing multicultural education: Teaching white privilege in the social work curriculum[END_REF] support the need to create dynamic social work education programs with a cutting-edge curriculum that reflects changing times. Fundamentally, it is imperative to ensure congruency between professional social work identity and social equality, in which social work values, ethics, and practice are reflective of the ever-changing demographics and cultural landscape. [START_REF] Clifford | Equality, diversity, ethics and management in social work education[END_REF] More research, however, is needed to uncover more details regarding educational context, which will provide a clearer understanding of what institutions can do to apply diversity to student learning and development. There is a need to establish and maintain congruency between social equality and professional identity [START_REF] Vakalahi | Diversity and Difference: What's the Social Work Commitment[END_REF]. Social work education, ethics, and practice must be adapted to current demographic and cultural transformation. Although race and ethnic-based diversity have been part of history as well as current conversations, it continues to be a point of resistance and controversy. While current conceptualizations of the construct are inclusive of all types of human differences, most scales measure competence with diverse races and ethnicities alone.

The development of cultural competence that integrates or transcends individual types of difference would be advantageous. Further attention to this issue may help measure cultural competence more comprehensively [START_REF] Krentzman | Review of multidisciplinary measures of cultural competence for use in social work education[END_REF].

  three indicators as knowledge, attitudes, and beliefs, and skills along with other background variables of the respondents. The cross-sectional research aimed to raise understanding about the development of cultural competence in graduate social work students. The Cross-Cultural Inventory (CCI) was developed to measure the three components of cultural competence: (a) knowledge; (b) attitudes and beliefs; and (c) skills. A factor analysis confirmed the effectiveness of these three factors and the amount of education in human diversity was strongly correlated with all of the three components. Therefore, education about diversity issues is an essential construct in the process of understanding the development of cultural competence.

  revealed that the consideration of racial diversity extended beyond student composition and included social and curricular engagement. Although institutional measures do not point to precisely what institutions are doing to facilitate greater racial diversity-related engagement among their student body, achieving higher levels of student body engagement does not occur by accident but requires intentional effort. While all institutions should theoretically have the potential of exposing students to diversity throughout their college careers (Alger, 1997 as cited in Densen, & Chang, 2009), the case that academic skills outcome has positive effects of cross-racial interaction varies by institution. Other studies have also found variations related to diversity across institutions. Hu and Kuh (2003 as cited in Densen, & Chang, 2009) found that students attending large doctoral-extensive universities and liberal arts colleges had more experiences with diversity than students of other institutional types.

  argue the need for social work educators to reflect on their social work professional values and ethics about their existence in an institutional context; and the use of pedagogy oppressive to ethnic minorities. Social work educators are challenged to contemplate the congruency between their professional values and practice in academia. The authors further argue that the teaching excellence must acknowledge the complexity and challenge in adverse 21-century society. Indeed, reflecting diverse perspectives in academia will lead to effective scientific advancement which in turn results in effective social change and relevant anti-oppressive social work practice in individuals and communities with diversity and difference.


Luc Robbiano
email: luc.robbiano@uvsq.fr

Qiong Zhang
email: zhangqiong@bit.edu.cn

Logarithmic Decay of a Wave Equation with Kelvin-Voigt Damping *

Keywords: Carleman estimate, wave equation, Kelvin-Voigt damping, logarithmic stability AMS 2010 subject classification: Primary 93B05, Secondary 93B07, 35B37

In this paper we analyze the long time behavior of a wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-differential calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective.

Introduction

In this paper, we consider a wave equation with local Kelvin-Voigt damping and analyze long time behaviour for the -solution of the system. Let Ω ⊂ R d be a bounded domain with smooth boundary Γ = ∂Ω. Denote by ∂ n the unit outward normal vector on boundary Γ. The PDE model is as follows.

         y tt (t, x)div [∇y(t, x) + a(x)∇y t (t, x)] = 0 in (0, ∞) × Ω, y(t, x) = 0 on (0, ∞) × Γ, y(0, x) = y 0 , y t (0, x) = y 1 in Ω,

where the coefficient function a(•) ∈ L 1 (Ω) is nonnegative and not identically null. The natural energy of system (1.1) is

E(t) = 1 2 Ω | ∇y(t) | 2 + | y t (t) | 2 dx . (1.2)
A direct computation gives that

d dt E(t) = - supp a a(x) | ∇y t (t) | 2 dx. (1.3)
Formula (1.3) shows that the only dissipative mechanism acting on the system is the viscoelastic damping div [a∇y t ], which is only effective on supp a. To rewrite the system as an evolution equation, we set the energy space as

H = H 1 0 (Ω) × L 2 (Ω), (1.4)
with norm (Ω), div (∇y 1 + a∇y 2 ) ∈ L 2 (Ω) . Let Y (t) = (y(t), y t (t)). Then system (1.1) can be written as d dt Y (t) = AY (t), ∀ t > 0, Y (0) = (y 0 , y 1 ).

Y H = ∇y 1 2 L 2 (Ω) + y 2 2 L 2 (Ω) , ∀ Y = y 1 , y 2 ∈ H. ( 1
(1.6)

It is known from [START_REF] Liu | Exponential stability for the wave equation with local Kelvin-Voigt damping[END_REF] that if supp a is non-empty, the operator A generates a contractive C 0 semigroup e tA on H and iR ⊂ ρ(A), the resolvent of A. Consequently, the semigroup e tA is strongly stable. Moreover, if the entire medium is of the viscoelastic type (i.e. supp a = Ω), the damping for the wave equation not only induces exponential energy decay, but also restricts the spectrum of the associated semigroup generator to a sector in the left half plane, and the associated semigroup is analytic ( [START_REF] Huang | On the mathematical model for linear elastic systems with analytic damping[END_REF]). When the Kelvin-Voigt damping is localized on a subdomain of Ω, the properties of system is quite complicated. First, it has been proved that properties of regularity and stability of 1-d system (1.1) depend on the continuousness of coefficient function a(•). More precisely, assume that Ω = (-1, 1) and a(x) behaviours like x α with α > 0 in supp a = [0, 1]. Then the solution of system (1.1) is eventually differentiable for α > 1, exponentially stable for α ≥ 1, polynomially stable of order 1 1-α for 0 < α < 1, and polynomially stable of optimal decay rate 2 for α = 0 (see [START_REF] Liu | Eventual differentiability of a string with local Kelvin-Voigt damping, to appear in[END_REF][START_REF] Liu | Stability of a string with local Kelvin-Voigt damping and non-smooth coefficient at interface[END_REF][START_REF] Renardy | On localized Kelvin-Voigt damping[END_REF][START_REF] Zhang | Exponential stability of an elastic string with local Kelvin-Voigt damping[END_REF]). For the higher dimensional system, the corresponding semigroup is exponentially stable when a(•) ∈ C 2 (Ω) and supp a ⊃ Γ ( [START_REF] Liu | Frequency domain characterization of rational decay rate for solution of linear evolution euqations[END_REF]). However, when the Kelvin-Voigt damping is local and the material coefficient a(•) is a positive constant on supp a, the energy of system (1.1) does not decay exponentially for any geometry of Ω and supp a ( [START_REF] Chen | Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping[END_REF][START_REF] Zhang | On the lack of exponential stability for an elastic-viscoelastic waves interaction system[END_REF]). The reason is that the strong damping and non-continuousness of the coefficient function lead to reflection of waves at the interface γ .

= ∂(supp a) \ Γ, which then fails to be effectively damped because they do not enter the region of damping. It turns out that the Kelvin-Voigt damping does not follow the principle that "geometric optics" condition implies exponential stability, which is true for the wave equation with local viscous damping ( [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]).

Recently, [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF] proves the polynomial stability of system (1.1) when a(•) ≡ a 0 > 0 on supp a and supp a satisfies certain geometry conditions. Then, a natural problem is: how about the decay rate if supp a = ∅ is arbitrary? In [START_REF] Ammari | Stabilization for the wave equation with singular Kelvin-Voigt damping[END_REF], a is assumed discontinuous along a (d -1)-manifold, supp a is arbitrary and the rate of the decay of semi-group is estimated by (log t) -k for a data in D(A k ). In this paper, we analyze the logarithmic decay properties of the solution to (1.1) when a is smooth and supp a is arbitrary. The main result reads as follows.

Theorem 1.1. Suppose that the coefficient function a(•) ∈ C ∞ 0 (Ω) is nonnegative and supp a ⊂ Ω is non-empty. Then the energy of the solution of (1.1) decays at logarithmic speed. More precisely, one has that there exists a positive constant C such that

e tA Y 0 H ≤ C [log(t + 2)] 4k 5 Y 0 D(A k ) , ∀ t > 0, Y 0 = (y 0 , y 1 ) ∈ D(A k ). (1.7)
Our approach is based on the results duo to [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF], which reduced the problem of determining the rate of energy decay to estimating the norm of the resolvent operator along the imaginary axis, see also [START_REF] Duyckaerts | Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF], etc. Our argument divides naturally three steps. First, in Section 2, we show some preliminaries including definitions and classical results about symbol, pseudo-differential calculus, and commutator estimate, etc. Then in Section 3, we prove corresponding Carleman estimates. Finally, in Section 4, we present a resolvent estimate and obtain Theorem 1.1. This theorem is a consequence of a resolvent estimate. The proof is given in Section 4. The method was developed in ( [START_REF] Bellassoued | Distribution of resonances and decay rate of the local energy for the elastic wave equation[END_REF][START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Bellassoued | Decay of solutions of the elastic wave equation with a localized dissipation[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF][START_REF] Robbiano | Fonction de coût et contrôle des solutions des équations hyperboliques[END_REF] and the references cited therein).

Throughout this paper, we use • V and (• | •) V to denote the norm and inner product on L 2 (V ), where V ⊂ R d if there is no further comments. When writing f g (or f g), we mean that there exists a positive constant C such that f ≤ Cg (or f ≥ Cg). For j = 1, 2, • • • , define operators

D j = -i∂ xj , D = (D 1 , • • • , D d ), D 2 = d j=1 D 2 j and Da(x)D = d j=1 D j a(x)D j .

Preliminaries

We shall prove Theorem 1.1 by Weyl-Hörmander calculus, which was introduced Hörmander ( [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF]). In this section, some definitions and results on the class of symbol and pseudo-differential calculus are given.

Symbol and Symbolic calculus

For any (x, ξ) ∈ R d × R d , λ ∈ R and τ > 0, we introduce the metric

g = g x,ξ = λdx 2 + µ -2 dξ 2 , where µ 2 = µ(τ, ξ) 2 = τ 2 + | ξ | 2 , (2.1)
and the weight

ν = ν(x, λ) = 1 + λ 2 a(x) 2 . (2.2)
Note that and g

x,ξ (X, Ξ) = λ | X | 2 + µ -2 (τ, ξ) | Ξ | 2 for all X, Ξ ∈ R d .
Then we have the following results.

Lemma 2.1. Assume that there exist positive constants C and λ 0 such that λ ≥ λ 0 and τ ≥ max{ Cλ, 1 }. It holds

(i)
The metric g = g x,ξ defined by (2.1) is admissible, i.e., it is slowly varying and temperate.

(ii) The weight ν = ν(x, λ) defined by (2.2) is admissible, i.e., it is g-continuous and g-temperate.

Proof. (i) From Definition 18.4.1 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF], the metric g x,ξ defined by (2.1) is slowly varying if there exist δ > 0 and C > 0 such that

g x,ξ (y -x, η -ξ) ≤ δ implies g y,η (X, Ξ) ≤ Cg x,ξ (X, Ξ), ∀ x, y, ξ, η, X, Ξ ∈ R d ,
where the constants δ and C are independent on the parameters λ and τ . Suppose 0 < δ ≤ 1/4 and

g x,ξ (y -x, η -ξ) = λ | y -x | 2 + (τ 2 + | ξ | 2 ) -1 | η -ξ | 2 ≤ δ.
Then, we have

τ 2 + | ξ | 2 ≤ τ 2 + 2 | ξ -η | 2 + 2 | η | 2 ≤ τ 2 + 2δ(τ 2 + | ξ | 2 ) + 2 | η | 2 .
This implies that τ

2 + | ξ | 2 ≤ 4(τ 2 + | η | 2 ). Consequently, g y,η (X, Ξ) = λ | X | 2 + (τ 2 + | η | 2 ) -1 | Ξ | 2 ≤ λ | X | 2 + 1 4 (τ 2 + | ξ | 2 ) -1 | Ξ | 2 ≤ g x,ξ (X, Ξ).
Therefore, g is slowly varying. For a given metric g x,ξ , the associated metric g σ x,ξ is defined by

g σ x,ξ = (τ 2 + | ξ | 2 )dx 2 + λ -1 dξ 2 . The metric g x,ξ is temperate if there exist C > 0 and N > 0, such that g x,ξ (X, Ξ) ≤ Cg y,η (X, Ξ) 1 + g σ x,ξ (x -y, ξ -η) N , ∀ x, y, ξ, η, X, Ξ ∈ R d , (2.3)
where the constants C and N are independent on the parameters λ and τ (Definition 18.5.1 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]).

For the metric g = g x,ξ defined by (2.1), (2.3) is equivalent to

λ | X | 2 + (τ 2 + | ξ | 2 ) -1 | Ξ | 2 ≤ C λ | X | 2 + (τ 2 + | η | 2 ) -1 | Ξ | 2 1 + (τ 2 + | ξ | 2 ) | x -y | 2 + λ -1 | ξ -η | 2 N . (2.4) First, assume that τ 2 + | η | 2 ≤ 4(τ 2 + | ξ | 2 ). It follows that (τ 2 + | η | 2 ) ≤ C(τ 2 + | ξ | 2 ) 1 + λ -1 | ξ -η | 2 N , C > 0, N > 0. (2.5)
Then it is easy to obtain (2.4) from (2.5). Secondly, consider the case

τ 2 + | η | 2 > 4(τ 2 + | ξ | 2 ). Then | η | > 2 | ξ |, | η | > √ 3τ, (2.6)
and

| ξ -η | > 1 2 | η | > √ 3 2 τ > √ 3 2 Cλ. (2.7)
It follows from (2.6) and (2.7) that

λ -1 | ξ -η | 2 > √ 3 2 C | ξ -η | > √ 3 4 C | η | .
Consequently,

1 + λ -1 | ξ -η | 2 2 > 3 16 C 2 | η | 2 > 3 32 C 2 ( | η | 2 + 3τ 2 ).
This together with τ 2 + | ξ | 2 ≥ 1 yields that there exists a positive constant C such that (2.5) holds with N = 2.

(ii) It is known from Definition 18.4.2 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF] that a weight ν(x) is g-continuous if there exist δ > 0 and C > 0 such that

g x,ξ (y -x, η -ξ) ≤ δ implies C -1 ν(x) ≤ ν(y) ≤ Cν(x), ∀ x, y, ξ, η ∈ R d .
where the constants δ and C are independent on the parameters λ and τ . Since the weight ν(x) defined by (2.2) does not depend on ξ, the above condition is reduced to

λ | x -y | 2 ≤ δ implies C -1 ν(x) ≤ ν(y) ≤ Cν(x), ∀ x, y ∈ R d . The weight ν(x) is g-temperate if there exist C > 0 and N > 0 such that ν(y) ≤ Cν(x) 1 + g σ y,η (x -y, ξ -η) N , ∀ x, y, ξ, η ∈ R d , (2.8)
where the constants C and N do not depend on the parameters λ and τ (Definition 18.5.1 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]). The weight ν(x) is admissible if it is g-continuous and g-temperate. When a weight is admissible, all the powers of this weight are g-continuous and g-temperate. Therefore, it suffice to prove that

1 + λa(x) is admissible. Let s ∈ [0, t] and t ∈ [0, 1]. Define f (s) = λa(x + s(y -x)) and F (t) = sup s∈[0,t] f (s) where x, y ∈ Ω satisfying λ | x -y | 2 ≤ δ. It is clear that f ′ (s) = λ a ′ (x + s(y -x)) (y -x).
Combining this with the following inequality

| a ′ (x) | 2 ≤ 2a(x) a ′′ L ∞ (Ω) , ∀ x ∈ Ω, (2.9)
we obtain

| f ′ (s) | ≤ λ | a ′ (x + s(y -x)) | | y -x | ≤ 2λ a ′′ 1 2 ∞ [a(x + s(y -x))] 1 2 | y -x | .
The proof of (2.9) will be given later. Consequently,

sup s∈[0,t] | f ′ (s) | ≤ 2λ 1 2 a ′′ 1 2 L ∞ (Ω) F (t) 1 2 | y -x | . Since f (t) ≤ f (0) + t sup s∈[0,t] | f ′ (s) | , F is non-decreasing and λ | x -y | 2 ≤ δ, we obtain that for all t ∈ [0, 1], f (t) ≤ f (0) + Cλ 1 2 F (t) 1 2 | y -x | ≤ f (0) + C √ δF (t) 1 2 ≤ f (0) + C √ δF (α) 1 2
,

where C = 2 a ′′ 1 2 L ∞ (Ω) and α ∈ [t, 1]. Note that f (0) = F (0). It follows that F (α) = sup t∈[0,α] f (t) ≤ F (0) + C √ δF (α) 1 2
.

This yields

1 + F (α) ≤ 1 + F (0) + C √ δ 1 + F (α) 1 2 ≤ 1 + F (0) + C √ δ 1 + F (α) . (2.10)
By choosing δ sufficiently small such that C √ δ ≤ 1/2, one can deduce from (2.10) that

1 + F (α) ≤ 2(1 + F (0)), ∀ α ∈ [t, 1].
In particular, we have

1 + λa(y) ≤ 2(1 + λa(x)).
The above inequality remains true if we exchange x and y. Therefore, the weight 1 + λa(x) is g-continuous.

On the other hand, note that 1 + λa(x) is independent to ξ. Then, to obtain the weight 1 + λa(x) is σ-temperate, it is sufficient to prove that

1 + λa(y) ≤ C(1 + λa(x))(1 + τ 2 | x -y | 2 ) N .
(2.11)

In fact, it is clear that 1 + λa(y)

≤ 1 + λ(a(x) + C | x -y | ) where C = a ′ L ∞ (Ω)
. Therefore, there exists positive constant C ′ = CC -1 such that

1 + λa(y) ≤ (1 + λa(x))(1 + C ′ τ | x -y | ) ≤ (1 + λa(x))(2 + 2(C ′ τ | x -y | ) 2 ) 1 2 .
Thus, we obtain (2.11) with N = 1 2 , C = 2 max{1, C ′ }.

Remark 2.1. We claim that (2.9) holds for any compactly supported and nonnegative function a ∈ C 2 (Ω). In fact, from the following identity

a(x + h) = a(x) + a ′ (x)h + 1 0 (1 -t)a ′′ (x + th)h 2 dt, ∀ h ∈ R, one can get a(x) + a ′ (x)h + 1 2 a ′′ L ∞ (Ω) | h | 2 ≥ 0.
Let h = y a ′ (x), where y ∈ R and x ∈ Ω are arbitrary. It follows from the above inequality that

a(x) + | a ′ (x) | 2 y + 1 2 a ′′ L ∞ (Ω) | a ′ (x) | 2 y 2 ≥ 0, ∀ x ∈ Ω, y ∈ R. Then, | a ′ (x) | 4 -2a(x) a ′′ L ∞ (Ω) | a ′ (x) | 2 ≤ 0,
and (2.9) is proved.

Definition 2.1. (Section 18.4.2 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]) Assume the weight m(x, ξ) is admissible and the metric g is defined by (2.1). Let q(x, ξ, λ, τ ) be a C ∞ function with respect to (x, ξ) and λ, τ be parameters satisfying conditions in Lemma 2.1. The symbol q(x, ξ, λ, τ ) is in class S(m, g) if for all α, β ∈ N d there exists C α,β independent of τ and λ such that

| ∂ α x ∂ β ξ q(x, ξ, λ, τ ) | ≤ C α,β m(x, ξ) λ | α | /2 µ(τ, ξ) -| β | . Remark 2.2. (i) It is clear that µ = τ 2 + | ξ | 2 ∈ S(µ, g) since | ∂ β ξ µ(τ, ξ) | µ 1-| β | for all β ∈ R d .
(ii) Let ν be the weight defined by (2.2). It is easy to get that λa ∈ S(ν, g). In fact, if

| α | ≥ 2, it holds that | ∂ α x (λa(x)) | ≤ C α λ ≤ C α λ | α | /2 ν(x) , where C α > 0. For the case | α | = 1, it follows from (2.9) that | ∂ α x (λa(x)) | ≤ √ 2 a ′′ 1 2 L ∞ (Ω) λ 1 2 λa(x) 1 2
Note that | λa(x) | < Cν 2 (x) for some C > 0. This together with the above inequality, we have that

| ∂ α x (λa(x)) | < √ 2C a ′′ 1 2 L ∞ (Ω) λ 1 2 ν(x).
(iii) It is known from Lemma 18.4.3 of [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF] that if the metric g and weights m 1 , m 2 are admissible, symbols a ∈ S(m 1 , g) and b ∈ S(m 2 , g), then ab ∈ S(m 1 m 2 , g). In particular, (λa) j µ k ∈ S(ν j µ k , g) for all j, k ∈ N ∪ {0}. where r ∈ S(λ

1 2 µ -1 m 1 m 2 , g). (ii) for commutator i[Op(b), Op(c)] = Op(f ), it holds that f ∈ S(λ 1 2 µ -1 m 1 m 2 , g) and f (x, ξ) = {b, c}(x, ξ) + r(x, ξ) where r ∈ S(λµ -2 m 1 m 2 , g).
The operators in S(ν j µ k , g) act on Sobolev spaces adapted to the class of symbol. Let b ∈ S(ν j µ k , g), where µ and g are defined by (2.1). Then there exists C > 0 such that

Op(b)u R d ≤ C ν j Op(µ k )u R d , ∀ u ∈ S (R d ).
By symbolic calculus, the above estimate is equivalent to Op(µ -k ν -j ) Op(b) acts on L 2 (R d ) since the operators associated with symbol in S(1, g) act on L 2 (R d ). In particular, if b ∈ S(ν j µ, g), then for any λ ≥ λ 0 , τ ≥ max{ Cλ, 1 } and u ∈ S (R d ), it holds

Op(b)u R d ≤ Cτ ν j u R d + C ν j Du R d ,
where C > 0 depends on positive constants λ 0 and C.

Commutator estimate

In this subsection, we suppose that λ = 1 since the symbol does not depend on λ. The metric in (2.1) becomes

g = dx 2 + µ -2 dξ 2 ,
where µ is defined by (2.1).

(2.12)

To get the commutator estimate, we shall use the following Gårding inequality ([12, Theorem 18.6.7]).

Lemma 2.4. Let b ∈ S(µ 2k , g) be real valued. µ and g are defined by (2.12). We assume there exists C > 0 such that b(x, ξ, τ ) ≥ Cµ 2k . Then there exist C > 0 and τ 0 > 0 such that

Re(Op(b)w | w) R d ≥ C Op(µ k )w 2 R d , ∀ w ∈ S (R d ) and τ ≥ τ 0 . (2.13) Definition 2.3. Let V be a bounded open set in R d . We say that the weight function ϕ ∈ C ∞ (R d ; R) satisfies the sub-ellipticity condition in V if | ∇ϕ | > 0 in V and there exists constant C > 0, Ô ϕ (x, ξ, τ ) = 0, ∀ (x, ξ) ∈ V × R d , τ > 0 ⇒ {Õ 2 , Õ 1 }(x, ξ, τ ) ≥ C( | ξ | 2 + τ 2 ) 3/2 , (2.14)
where Ô ϕ (x, ξ, τ ) = |ξ + iτ ∇ϕ(x)| 2 = Õ 2 (x, ξ, τ ) + iÕ 1 (x, ξ, τ ) and Õ 1 , Õ 2 are real valued.

Lemma 2.5.

([12]) Let V be a bounded open set in R d and ψ ∈ C ∞ (R d ; R) be such that | ∇ψ | > 0 in V .
Then, for γ > 0 sufficiently large, ϕ = e γψ fulfills the sub-ellipticity property in V .

Lemma 2.6. Assume that ϕ satisfies the sub-ellipticity in Definition 2.3. For all w ∈ C ∞ 0 (V ), there exist C 1 , C 2 > 0 and τ 0 > 0 such that the following inequality holds for all τ ≥ τ 0 ,

C 1 τ 3 w 2 V + C 1 τ Dw 2 V ≤ Re Op({|ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x)})w | w V + C 2 τ -1 Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 )w 2 V + C 2 τ -1 Op(2τ ξ • ∇ϕ(x))w 2 V .
(2.15)

Proof. First, by homogeneity in (ξ, τ ), compactness arguments and sub-ellipticity condition, we claim that there exist constants C, δ > 0 such that

C | 2ξ • ∇ϕ(x) | 2 + µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 + |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x) ≥ δµ 2 . (2.16)
The proof of (2.16) is classical. In fact, set

K = {(x, ξ, τ ) ∈ R d × R d × R : x ∈ V , | ξ | 2 + τ 2 = 1, τ ≥ 0},
and for (x, ξ, τ

) ∈ K, κ > 0, G(x, ξ, τ, κ) = κ | 2ξ • ∇ϕ(x) | 2 + µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 + {|ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x)}. If | 2ξ •∇ϕ(x) | 2 +µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 = 0 for (x, ξ, τ ) ∈ K
, it is clear that there exists a positive constant δ such that (2.16) holds due to the fact that φ is sub-elliptic.

When | 2ξ • ∇ϕ(x) | 2 + µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 > 0, there exists a positive constant κ x,ξ,τ such that G(x, ξ, τ, κ) > 0 for every κ ≥ κ x,ξ,τ since {|ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x)} is bounded on K.
By continuity of G(x, ξ, τ, κ), there exists a neighborhood of (x, ξ, τ ), denoted by V x,ξ,τ , such that G(x, ξ, τ, κ) > 0 for all (x, ξ, τ ) ∈ V x,ξ,τ and κ ≥ κ x,ξ,τ . Since K is compact, there exist finite sets V j = V xj ,ξj ,τj and corresponding constants κ j = κ xj ,ξj ,τj (j = 1, 2, • • • , n), such that K ⊂ ∪ n j=1 V j and G(x, ξ, τ, κ) > 0 for all (x, ξ, τ ) ∈ V j and κ > κ j . Let κ = max{κ j : j = 1, 2, • • • , n}. It follows that G(x, ξ, τ, κ) > 0 for all (x, ξ, τ ) ∈ K and κ ≥ κ. Finally, using the compactness of K again, we conclude that there exists δ > 0 such that G(x, ξ, τ, κ) ≥ δ. Thus, (2.16) is reached since g is a homogeneous function of degree 2 with respect to variables (ξ, τ ).

By Gårding inequality (2.13), there exists a constant C > 0 such that, for τ ≥ τ 0 with τ 0 sufficiently large,

C Op(µ)w 2 V ≤ Re Op | 2ξ • ∇ϕ(x) | 2 + µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 + |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x) w w V .
(2.17)

Now we are going to estimate the terms τ Op(

| 2ξ • ∇ϕ(x) | 2 ) and τ Op(µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 ).
Firstly, it follows from Lemma 2.3 that

τ -1 Op( | 2τ ξ • ∇ϕ(x) | 2 ) = τ -1 Op(2τ ξ • ∇ϕ(x)) * Op(2τ ξ • ∇ϕ(x)) + τ Op(r 1 ), (2.18)
where r 1 ∈ S(µ, g) and g is defined by (2.12). Therefore, for any ε > 0, there exists a positive constant C ε such that

τ -1 Op( | 2τ ξ • ∇ϕ(x) | 2 )w | w V ≤ τ -1 Op(2τ ξ • ∇ϕ(x))w 2 V + τ | (Op(r 1 )w | w) V | ≤ τ -1 Op(2τ ξ • ∇ϕ(x))w 2 V + ε τ Op(µ)w 2 V + C ε τ w 2 V .
(2.19) Substituting (2.19) into (2.17) and choosing ε small enough, we have

Cτ Op(µ)w 2 V ≤ Re Op τ µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 + τ |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x) w w V +τ -1 Op(2τ ξ • ∇ϕ(x))w 2 V + C ε τ w 2 V .
(2.20)

Secondly, by symbolic calculus, we have that

τ Op(µ -2 ( |ξ| 2 -τ 2 |∇ϕ(x)| 2 ) 2 ) = τ Op(r 0 ) Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 ) + τ Op(r 2 ), (2.21)
where r 0 (x, ξ) = µ -2 (|ξ| 2τ 2 |∇ϕ(x)| 2 ) ∈ S(1, g) and r 2 ∈ S(µ, g). Therefore, for all ε > 0, there exists C ε > 0 such that

| (τ Op(r 0 ) Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 )w | w) V | ≤ C ε τ -1 Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 )w 2 V + ετ 3 w 2 V .
(2.22)

We choose ε small enough and combine (2.21)-(2.22) with (2.20) to get

Cτ Op(µ)w 2 V ≤ Re τ Op |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2ξ • ∇ϕ(x) w w V +C ε τ -1 Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 )w 2 V +τ -1 Op(2τ ξ • ∇ϕ(x))w 2 V + C ε (τ + ετ 3 ) w 2 V .
(2.23)

Finally, it is clear that there exist positive constant C such that

τ 3 w 2 V + τ Dw 2 V ≤ Cτ Op(µ)w 2 V . (2.24)
Thus, we obtain (2.15) by using (2.23)-(2.24), choosing ε small enough and letting τ > τ 0 large enough.

Carleman Estimate

In this section, we shall prove several Carleman inequalities. Define the operator

P (x, D, λ) = D 2 + iλDa(x)D -λ 2 .
Let the weight function ϕ ∈ C ∞ (R d ; R). The associated conjugate operator of P (x, D, λ) is P ϕ (x, D, λ) = e τ ϕ P (x, D, λ)e -τ ϕ . Then,

P ϕ = (D + iτ ∇ϕ(x)) 2 + iλ(D + iτ ∇ϕ(x))a(x)(D + iτ ∇ϕ(x)) -λ 2 .
By setting

Q 2 = 1 2 (P ϕ + P * ϕ ) and Q 1 = 1 2i (P ϕ -P * ϕ ), we have P ϕ = Q 2 + iQ 1 .
We denote by p(x, ξ, λ), p ϕ (x, ξ, λ) the associated symbol of P (x, D, λ), P ϕ (x, D, λ), respectively.

Let the metric g and weight ν be defined by (2.1) and (2.2). Due to the results Remark 2.2, we know that D + iτ ∇ϕ(x) is an operator with symbol in S(µ, g) class, λa is in S(ν, g), and (1 + iλa(x))|ξ + iτ ∇ϕ(x)| 2 , the principal symbol of P ϕ belongs to S(νµ 2 , g). Furthermore, from Lemma 2.2 and 2.3, one has that

P ϕ = Op (1 + iλa(x))|ξ + iτ ∇ϕ(x)| 2 -λ 2 + R 3 , Q 2 = Op(q 2 ) -λ 2 + R 2 , Q 1 = Op(q 1 ) + R 1 , (3.1) V τ 3 ν(x)v 2 V + τ D(ν(x)v) 2 V . (3.5)
It follows from (3.4) and (3.5) that

C ′ 1 τ 3 ν(x)v 2 V + τ ν(x)Dv 2 V ≤ Re Op( |ξ| 2 -τ 2 ϕ(x) 2 , 2τ ξ • ∇ϕ(x) )ν(x)v | ν(x)v V +C 2 τ -1 Op(|ξ| 2 -τ 2 |∇ϕ(x)| 2 )ν(x)v 2 V +C 2 τ -1 Op(2τ ξ • ∇ϕ(x))ν(x)v 2 V , C ′ 1 , C 2 > 0. (3.6)
Now, we estimate the first term on the right side hand of (3.6). Note that

2 Re(Q 2 v | iQ 1 v) V = (Q 2 v | iQ 1 v) V + (iQ 1 v | Q 2 v) V = (i[Q 2 , Q 1 ]v | v) V , (3.7)
where the principal

symbol of i[Q 2 , Q 1 ] is {q 2 , q 1 }. Due to Lemma 2.3, we obtain that i[Q 2 , Q 1 ] = Op({q 2 , q 1 }) + R 4 , (3.8)
where {q 2 , q 1 } ∈ S(λ 1 2 ν 2 µ 3 , g) and the symbol of R 4 is in S(λν 2 µ 2 , g). A direct computation gives that

q 2 , q 1 = (1 + a 2 (x)λ 2 ) |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x) + |ξ| 2 -τ 2 |∇ϕ(x)| 2 , λa(x) -λa(x) 2τ ξ.∇ϕ(x), λa(x) (|ξ| 2 -τ 2 |∇ϕ(x)| 2 ) -2τ ξ • ∇ϕ(x) λa(x) λa(x), |ξ| 2 -τ 2 |∇ϕ(x)| 2 + λa(x), 2τ ξ • ∇ϕ(x) .
From the definition of q 1 and q 2 , we have

q 2 (x, ξ) + λa(x)q 1 (x, ξ) = (1 + λ 2 a(x) 2 )(|ξ| 2 -τ 2 |∇ϕ(x)| 2 ), q 1 (x, ξ) -λa(x)q 2 (x, ξ) = 2τ ξ • ∇ϕ(x)(1 + λ 2 a 2 (x)). (3.9)
Consequently,

q 2 , q 1 = ν 2 (x) |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x) +ν -2 (x)(q 2 (x, ξ) + λa(x)q 1 (x, ξ)) |ξ| 2 -τ 2 |∇ϕ(x)| 2 , λa(x) -λa(x) 2τ ξ • ∇ϕ(x), λa(x) -ν -2 (x)(q 1 (x, ξ) -λa(x)q 2 (x, ξ)) λa(x) λa(x), |ξ| 2 -τ 2 |∇ϕ(x)| 2 + λa(x), 2τ ξ • ∇ϕ(x) .
Then, it follows from (3.8) and the above equation that

i[Q 2 , Q 1 ] = ν(x) Op |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x) ν(x) + B 1 ν -1 (x) Op q 2 (x, ξ) + λa(x)q 1 (x, ξ) -B 2 ν -1 (x) Op q 1 (x, ξ) -λa(x)q 2 (x, ξ) + R 5 ,
where

B 1 = Op ν -1 (x) |ξ| 2 -τ 2 |∇ϕ(x)| 2 , λa(x) -λa(x) 2τ ξ • ∇ϕ(x), λa(x) , B 2 = Op ν -1 (x) λa(x) λa(x), |ξ| 2 -τ 2 |∇ϕ(x)| 2 + λa(x), 2τ ξ • ∇ϕ(x) , symbols of B 1 and B 2 belong to S(λ 1 2 νµ, g) and the symbol of R 5 is in S(λν 2 µ 2 , g). Combining this with (3.1) yields i[Q 2 , Q 1 ] = ν(x) Op |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x) ν(x) + B 1 ν -1 (x) Q 2 (x, ξ) + λa(x)Q 1 (x, ξ) + λ 2 -B 2 ν -1 (x) Q 1 (x, ξ) -λa(x)Q 2 (x, ξ) -λ 3 a(x) + R 5 , (3.10)
We refer to Section 2 where the rules on symbolic calculus are given and precise. Therefore, by the continuity of pseudo-differential operator, we have that for j = 1, 2, k = 0, 1 and ℓ = 1, 2,

| (B j ν(x) -1 (λa(x)) k Q ℓ v | v) V | = | (ν(x) -1 (λa(x)) k Q ℓ v | B * j v) V | Q ℓ v V B * j v V ≤ 1 10 Q ℓ v 2 V + Cλτ 2 ν(x)v 2 V + Cλ ν(x)Dv 2 V , C > 0, (3.11)
| (B j ν -1 (x)λ 2 (λa(x)) k v | v) V | λ 5 2 τ ν(x)v V + ν(x)Dv V ν(x)v V λ 5 2 τ ν(x)v 2 V + λ 5 2 τ -1 ν(x)Dv 2 V , (3.12)
and

| (R 5 v | v) V | λτ 2 ν(x)v 2 V + λ ν(x)Dv 2 V . (3.13)
Due to (3.10)-(3.13), there exists a positive constant C such that

Op( |ξ| 2 -τ 2 |∇ϕ(x)| 2 , 2τ ξ • ∇ϕ(x) )ν(x)v | ν(x)v V ≤ (i[Q 2 , Q 1 ]v | v) V + 1 4 ℓ=1,2 Q ℓ v 2 V + C λτ 2 ν(x)v 2 V +λ ν(x)Dv 2 V + λ 5 2 τ ν(x)v 2 V + λ 5 2 τ -1 ν(x)Dv 2 V . (3.14)
Remark 3.2. Since a(•) is nonnegative and not identically null, there exists δ > 0 such that {x ∈ Ω : a(x) > δ} = ∅. We introduce several sets as follows.

W 0 = Ω \ supp a, W 1 = Ω \ O supp a , W 2 = Ω \ {x ∈ Ω : a(x) ≥ δ} ∪ O(Γ) , W 3 = Ω \ {x ∈ Ω : a(x) ≥ δ 2 },
where O(Γ) means the neighborhood of Γ.

It is known that there exists a function ψ ∈ C ∞ (Ω) such that ( [START_REF] Fursikov | Controllability of evolution equations[END_REF])

1) ψ(x) = 0 for x ∈ ∂Ω.
2) ∂ n ψ(x) < 0 for x ∈ ∂Ω.

3) ∇ψ(x) = 0 for x ∈ Ω \ {x ∈ Ω : a(x) ≥ δ}.

Let ϕ = e γψ . It follows from Lemma 2.5 that ϕ satisfies the sub-ellipticity condition on x ∈ Ω \ {x ∈ Ω : a(x) ≥ δ} if γ > 0 is sufficiently large. Then, in Theorem 3.2 and 3.3, one can choose V as Ω \ {x ∈ Ω : a(x) ≥ δ}.

The following result is a classical Carleman estimate and corresponding to the Laplacian with Dirichlet boundary condition ( [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], Proposition 2). We shall use it to deal with the terms on Ω\supp a. Lemma 3.1. Suppose ϕ is chosen as in Remark 3.2. Then, there exist C > 0, λ 0 > 0, such that for all u ∈ C ∞ (Ω) satisfying supp u ⊂ W 1 and u = 0 on Γ, it holds

τ 3 e τ ϕ u 2 Ω + τ e τ ϕ Du 2 Ω e τ ϕ (D 2 -λ 2 )u 2 Ω , (3.32)
where λ ≥ λ 0 and τ ≥ max{Cλ, 1}.

Theorem 3.4. Suppose ϕ is chosen as in Remark 3.2. Let u ∈ C ∞ (Ω) and satisfy

P (x, D, λ)u = f 0 + d j=1 ∂ xj f j in Ω, u = 0 on Γ,
where

f j ∈ L 2 (Ω), supp f 0 ⊂ Ω and supp f j ⊂ O supp a for j = 1, • • • , d.
Then, there exist K > 0, λ 0 > 0, such that for all λ ≥ λ 0 and τ ≥ max{ K|λ| First, it is clear that

P χ 2 u = χ 2 f 0 + d j=1 ∂ xj (χ 2 f j ) -f j ∂ xj χ 2 + [P, χ 2 ]u.
Since [P, χ 2 ] is a first order operator, we have there exist a 0 , a 1 ,

• • • , a d and b 0 , b 1 , • • • , b d such that [P, χ 2 ]u = d j=0 a j ∂ xj u + λ d j=0 b j ∂ xj u,
where supp a j ⊂ {x ∈ Ω :

δ 2 < a(x) < δ} ∪ [Ω \ (O(supp a) ∪ O(supp ∂Ω))], supp b j ⊂ {x ∈ Ω : δ 2 < a(x) < δ} for j = 0, 1, • • • , d.
Then, applying Theorem 3.3 with χ 2 u instead of u, Ω \ {x ∈ Ω : a(x) ≥ δ} instead of V, we obtain Consequently, due to (1 + λ 2 a(x) 2 ) ≥ 1, τ ≥ {Cλ, 1} and λ ≥ λ 0 , we obtain

τ (1 + λ 2 a(x) 2 ) 1 2 e τ ϕ χ 2 u 2 Ω + τ -1 (1 + λ 2 a(x) 2 )
τ (1 + λ 2 a(x) 2 ) 1 2 e τ ϕ χ 2 u 2 Ω + τ -1 (1 + λ 2 a(x) 2 ) 1 2 e τ ϕ Dχ 2 u 2 Ω d j=0 e τ ϕ f j 2 Ω + e τ ϕ u 2 Ω\supp a + λ e τ ϕ u 2 {x∈Ω : a(x)≥ δ 2 } . (3.33)
On the other hand, by using χ 1 u instead of u in Lemma 3.1, we obtain

τ e τ ϕ (χ 1 + χ 2 )u 2 Ω + τ -1 e τ ϕ D(χ 1 + χ 2 )u 2 Ω d j=0 e τ ϕ f j 2 Ω + e τ ϕ u 2 Ω\supp a + λ e τ ϕ u 2 {x∈Ω : a(x)≥ δ 2 } + τ -2 e τ ϕ [P, χ 1 ]u 2 Ω .
Note that [P, χ 1 ] is a first order operator and supported on supp χ 1 \ {x ∈ Ω : χ 1 ≡ 1}, which is an subset of {x ∈ Ω : χ 1 + χ 2 = 1}. Then, for τ sufficiently large, we have

τ e τ ϕ (χ 1 + χ 2 )u 2 Ω + τ -1 e τ ϕ (χ 1 + χ 2 )Du 2 Ω d j=0 e τ ϕ f j 2 Ω + λ e τ ϕ u 2 {x∈Ω : a(x)≥ δ 2 } .
Then as 1 ≤ χ 1 + χ 2 ≤ 2 on W 3 , we obtain the statement of Theorem 3.4.

Resolvent estimate

In this section, we shall prove the main result. From the results in [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF][START_REF] Duyckaerts | Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface[END_REF], the logarithmic decay of the energy in Theorem 1.1 can be obtained through the following theorem. Let λ be a real number such that | λ | is large enough. Consider the resolvent equation:

F = (A -iλ)Y, where Y = y 1 , y 2 ∈ D(A), F = f 1 , f 2 ∈ H, (4.2)
which yields

         div (∇y 1 + a∇y 2 ) + λ 2 y 1 = iλf 1 + f 2 ,
in Ω,

y 2 = iλy 1 + f 1 in Ω, y 1 | Γ = 0 (4.3)
In what follows, we shall prove that there exists a positive constant C such that

(y 1 , y 2 ) H e C | λ | 5 4 (f 1 , f 2 ) H .
First, let η > 0 and χ ∈ C ∞ 0 (Ω) be real valued function such that supp χ = {x ∈ Ω : a(x) > η}) and χ ≡ 1 in {x ∈ Ω : a(x) > 2η}. The following lemma is helpful.

(i) χy 1 Ω ∇y 1 {x∈Ω : a(x)>η} + f 1 H 1 (Ω) + f 2 Ω . (ii) Ω a(x)|∇y 1 | 2 dx f 1 2 H 1 (Ω) + ( f 1 Ω + f 2 Ω ) y 1 Ω .
Proof. (i) Multiplying the first equation in (4.3) by χ 2 y 1 and using y 2 = iλy 1 + f 1 , we obtain,

1 2 f 1 H 1 (Ω) + ( f 1 Ω + |λ | -1 f 2 Ω ) y 1 Ω ≤ 1 4 Ω a(x)|∇y 1 | 2 dx + |λ | -2 f 1 2 H 1 (Ω) + ( f 1 Ω + |λ | -1 f 2 Ω ) y 1 Ω .
The proof of lemma is finished.

. 5 )

 5 Define an unbounded operatorA : D(A) ⊂ H → H by AY = y 2 , div (∇y 1 + a∇y 2 ) , ∀ Y = y 1 , y 2 ∈ D(A),andD(A) = y 1 , y 2 ∈ H : y 2 ∈ H 1 0

Definition 2 . 2 .Lemma 2 . 2 . 1 2Lemma 2 . 3 .

 2222123 Let b ∈ S(m, g) be a symbol and u ∈ S (R d ), we set b(x, D, τ )u(x) = Op(b)u(x) := (2π) -d R d e ix•ξ b(x, ξ, τ ) u(ξ) dξ. It is known that Op(b) : S (R d ) → S (R d ) is continuous and Op(b) can be uniquely extended to S ′ (R d ) continuously. The following two lemmas are consequences of Theorem 18.5.4 and 18.5.10 in [12]. Let b ∈ S(m, g) where m is an admissible weight and g is defined by (2.1). Then there exists c ∈ S(m, g) such that Op(b) * = Op(c) and c(x, ξ) = b(x, ξ) + r(x, ξ) where the remainder r ∈ S(λ µ -1 m, g). Let b ∈ S(m 1 , g) and c ∈ S(m 2 , g) where m j are admissible weights for j = 1, 2 and g is defined by (2.1). Denote by [Op(b), Op(c)] = Op(b) • Op(c) -Op(c) • Op(b) and Poisson bracket {b, c}(x, ξ, τ ) = 1≤j≤d (∂ ξj b ∂ xj c -∂ xj b ∂ ξj c)(x, ξ, τ ). Then, (i) there exists d ∈ S(m 1 m 2 , g) such that Op(b) Op(c) = Op(d) and d(x, ξ) = b(x, ξ)c(x, ξ)+ r(x, ξ)

5 4 ,, it holds τ e τ ϕ u 2 W3 + τ -1 e τ ϕ Du 2 W3 d j=0 e τ ϕ f j 2 Ω

 4222 1}+ λ e τ ϕ u 2 {x∈Ω : a(x)≥δ/2} , where the positive constant δ is defined as in Remark 3.2.Proof. Let χ 1 , χ 2 ∈ C ∞ (Ω) be non-negative and satisfy the following assumption (i) 0 ≤ χ 1 , χ 2 ≤ 1; χ 1 and χ 2 are supported on W 1 and W 2 , respectively.(ii) χ 1 + χ 2 ≥ 1 in W 3 . In particular, χ 1 ≡ 1 on [O(∂Ω) ∩ Ω] \ O(suppa), and χ 2 ≡ 1 on O(supp a) \ {x ∈ Ω : a(x) ≥ δ 2 }.

1 2 e τ ϕ Dχ 2 u 2 Ω d j=0 e τ ϕ f j 2 Ω

 122 + e τ ϕ u 2 {x∈Ω : δ 2 <a(x)<δ}∪(Ω\supp a) + λ e τ ϕ u 2 {x∈Ω : δ 2 <a(x)<δ} .

τ 3 e τ ϕ χ 1 u 2 Ω + τ e τ ϕ Dχ 1 u 2 Ω 2 Ωτ -2 e τ ϕ f 0 2 Ω

 2222 e τ ϕ (P -iλDa(x)D)χ 1 u 2 Ω .(3.34)Since χ 1 ∂ xj f j = 0 for j = 1, • • • , d, we haveP χ 1 u = χ 1 f 0 + [P, χ 1 ]u.Therefore, combining these with (3.34) and O(supp a)∩ supp χ 1 = ∅ yields τ e τ ϕ χ 1 u 2 Ω + τ -1 e τ ϕ Dχ 1 u + e τ ϕ [P, χ 1 ]u 2 Ω .(3.35)Adding up (3.35) and (3.33), using (1 + λ 2 a(x) 2 ) ≥ 1 and τ ≥ 1, we obtain

Theorem 4 . 1 . 1 L

 411 Suppose conditions in Theorem 1.1 hold. Then, for every λ ∈ R with |λ| large enough, there exists C > 0 such that (Aiλ) -

Lemma 4 . 1 .

 41 For y 1 , f 1 , f 2 satisfying (4.3) and |λ| large enough, it holds

λ 2 χy 1 2 Ω = Ω ∇y 1 f 1 χ 2 y 1 dx + Ω f 2 χ 2 y 1

 211 • ∇(χ 2 y 1 )dx + iλ Ω a(x)∇y 1 • ∇(χ 2 y 1 )dx +iλ Ω dx + Ω a(x)∇f 1 • ∇(χ 2 y 1 )dx.

(4. 4 ) 2 {x∈Ω≤ 2 ∇y 1 2 {x∈Ω

 422 Since ∇(χ 2 y 1 ) = χ 2 ∇y 1 + 2y 1 χ∇χ, we have Ω a(x)∇y 1 • ∇(χ 2 y 1 )dx ∇y 1 : a(x)>η} + χy 1 Ω ∇y 1 {x∈Ω : a(x)>η}

(4. 5 )λ 2 χy 1 2 Ω 1 2Ω + λ -2 f 2 2 Ω 2 H 1

 521221 By the similar argument, we can deal with the rest terms on the right hand side of (4+ (|λ| + 1) ∇y 1 2 {x∈Ω : a(x)>η} + f + (λ -2 + 1) ∇f 1 (Ω) .

(4. 6 )

 6 Then, (i) is reached.(ii) Multiplying the first equation in (4.3) by y 1 and using y 2 = iλy 1 + f 1 , we obtain,Ω [-(iλa(x) + 1)|∇y 1 | 2 + λ 2 |y 1 | 2 ]dx = Ω [a(x)∇f 1 • ∇y 1 + (iλf 1 + f 2 )y 1 ]dx. (4.7)Taking the imaginary part of (4.7) yieldsΩ a(x)|∇y 1 | 2 dx = -λ -1 Im Ω [a(x)∇f 1 • ∇y 1 + (iλf 1 + f 2 )y 1 ]dx.(4.8)Thus, by the Cauchy-Schwarz inequality, one can conclude from (4.8) thatΩ a(x)|∇y 1 | 2 dx ≤ |λ | -1 Ω a(x)|∇y 1 | 2 dx

Proof of Theorem 4 . 1 . 1 2 1 2 2 H 1 ( 2 H 1 ( 2 H 1 (Ω) + f 2 2 L 2 (e τ ϕ y 1 2 Ω + τ -1 e τ ϕ ∇y 1 2 Ω λ 2 e τ ϕ f 1 2 Ω + e τ ϕ f 2 2 Ω + d j=1 e τ ϕ a ∂ xj f 1 2 Ω+(λ + τ ) e τ ϕ y 1 2 {x∈Ωe τ ϕ ∇y 1 2 {x∈Ω 5 4 , 1} that τ e 2c1τ y 1 2 Ω + τ -1 e 2c1τ ∇y 1 2 Ω λ 2 e 2c2τ f 1 2 H 1 (Ω) + e 2c2τ f 2 2 Ω + τ e 2c2τ y 1 2 H 1

 411121212122222222252221221 Due to Lemma 4.1 (i) and (ii), we havey {x∈Ω : a(x)>2η} ∇y {x∈Ω : a(x)>η} + f 1 Ω) + ( f 1 Ω + f 2 Ω ) y 1 Ω . Ω) + ( f 1 Ω + f 2 Ω ) y 1 Ω . (4.11)On the other hand, by (4.3), one has that y 1 satisfies div (∇y 1 + iλa(x)∇y 1 ) + λ 2 y 1 = iλf 1 + f 2div(a(x)∇f 1 ). (4.12)Then, applying Theorem 3.4 to y 1 satisfying (4.12), we obtainτ : a(x)≥δ/2} + τ -1 : a(x)≥δ/2} .Let c 1 = min x∈Ω ϕ(x) and c 2 = max x∈Ω ϕ(x). We conclude from the above inequality and τ ≥ max{ K|λ| ({x∈Ω : a(x)≥δ/2}) .

(4. 13 ) 2 Ω 2 Ω τ 2 e 2c2τ f 1 2 H 1 2 Ω

 1322212 Setting η = δ 8 and substituting (4.11) into (4.13), we obtain τ e 2c1τ y 1 + τ -1 e 2c1τ ∇y 1 (Ω) + τ e 2c2τ f 2 + τ e 2c2τ ( f 1 Ω + f 2 Ω ) y 1 Ω .

* This work was supported by the National Natural Science Foundation of China (grants No. 60974033) and Beijing Municipal Natural Science Foundation (grant No. 4182059).

where q 2 = |ξ| 2τ 2 |∇ϕ(x)| 2 -2λτ a(x)ξ • ∇ϕ(x), q 1 = 2τ ξ • ∇ϕ(x) + λa(x)(|ξ| 2τ 2 |∇ϕ(x)| 2 ) belong to S(νµ 2 , g) and the symbols of R j is in S(λ 1 2 νµ, g) for j = 1, 2, 3. It is clear that

In what follows, several Carleman estimates are introduced. First, we give an estimation on the subdomain which is far away from the boundary Γ. Theorem 3.1. Suppose ϕ satisfies sub-ellipticity condition in V ⊂ Ω. Then, there exist positive constants C, K and λ 0 , such that for every v ∈ C ∞ 0 (V ), it holds

where λ ≥ λ 0 and τ ≥ max{ K|λ| 5 4 , 1}.

Proof. Let w = ν(x)v in (2.15). We obtain

Consequently, for τ ≥ Cλ with C > 0 sufficiently large, it holds

Next, we are going to estimate the last two terms on the right side hand of (3.6). It follows from (3.1) and (3.9) that

(3.15)

Combining this with the fact that the symbols of R j are in S(λ 1 2 νµ, g) for j = 1, 2, we have

Consequently, for τ ≥ Cλ with C large enough, it holds

(3.17)

Finally, by (3.6), (3.14) and (3.17), one can choose τ ≥ max{ K|λ| 5 4 , 1} with K sufficiently large such that for some C > 0,

where ε > 0 is arbitrary. Choosing ε small with respect to C, using (3.2), (3.7) and (3.18), we obtain

Remark 3.1. The estimates in (3.12) impose the assumption τ ≥ K|λ| 5 4 . The other remainder terms only impose the condition τ ≥ Cλ. This condition is related with the principal normal condition. Indeed for a complex operator, with symbol p 1 + ip 2 where p 1 , p 2 are both real valued, the Carleman estimate is only true if {p 1 , p 2 } = 0 on p 1 = p 2 = 0. Here the symbol of operator before conjugation by weight is |ξ| 2 -λ 2 +iλa(x)|ξ| 2 , and the Poisson bracket is {|ξ| 2 -λ 2 , λa(x)|ξ| 2 } = 2λ(ξ•∇a(x))|ξ| 2 . We can estimate this term, uniformly in a neighborhood of a(x) = 0, by Cλa Theorem 3.2. Suppose ϕ satisfies sub-ellipticity condition in V ⊂ Ω. Then, there exist positive constants K and λ 0 , such that for every u ∈ C ∞ 0 (V ), it holds

where λ ≥ λ 0 and τ ≥ max{ K|λ| 12 Proof. Set v = e τ ϕ u. From Theorem 3.1, it suffices to prove that (3.19) is equivalent to

First, assume (3.20) holds. Then, Dv = e τ ϕ (Duiτ ∇ϕ u) and e τ ϕ Du = Dv + iτ ∇ϕ v. Then there exist positive constants c 1 , c 2 such that

Combining this with (3.20), we conclude that

On the other hand, (3.19) implies that

Then, we proved (3.20) from the above estimate and (3.21).

Since there is higher order term div(a(x)∇y t ) in system (1.1), it is necessary to deal with the term div(a(x)∇f ) for f ∈ H 1 (Ω) when proving the resolvent estimate. The following result is analogue to the work by Imanuvilov and Puel ([14]).

Theorem 3.3. Suppose ϕ satisfies sub-ellipticity condition on V ⊂ Ω. Then, there exist C, K, λ 0 > 0, such that for all u ∈ C ∞ 0 (V ) satisfying

it holds

where λ ≥ λ 0 and τ ≥ max{ K|λ| Proof. First, from (3.1), we have

Using (3.3), (3.24) and the fact that ν(x

Let u and χ be in C ∞ 0 (Ω) such that χ = 1 on a neighborhood of supp u. Similarly to (3.21), we obtain

Then, combining this with Fourier transform and the following inequality

From Lemma 2.3, we have Op(µ -1 )νχ = νχ Op(µ -1 ) + R 1 , where R 1 has a symbol in S(µ -2 νλ 1 2 , g), and D 2 Op(µ -1 )νχ = νD 2 χ Op(µ -1 ) + R 2 , where R 2 has a symbol in S(νλ 1 2 , g). Then, it follows from (3.26) that

For τ ≥ max{Cλ, 1} with C large enough, one has the following result from the above inequality.

Now, we apply (3.25) to v = χ Op(µ -1 ) u to have

Thus, combining this with (3.27) yields

Finally, note that P ϕ has a symbol in S(νµ 2 , g). Consequently, P ϕ χ Op(µ -1 ) = Op(µ -1 )P ϕ χ+ R, where R has a symbol in S(νλ 1 2 , g). Then, we can deduce from (3.28) that

When τ ≥ Cλ with C large enough, the error term λ ν u 2 V can be absorbed by the left hand side.