{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:55:57Z","timestamp":1777449357285,"version":"3.51.4"},"reference-count":19,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2016,7,11]],"date-time":"2016-07-11T00:00:00Z","timestamp":1468195200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2016,7,11]]},"abstract":"<jats:p>\n                    We address the homogenization of the stationary diffusion equation in a composite medium with two components\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">M<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    having respectively\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b1<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    as diffusivity coefficients. We assume periodic distribution with size\n                    <jats:italic toggle=\"yes\">\u03b5<\/jats:italic>\n                    of the \u201choles\u201d\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    but no periodicity is assumed on the matrices\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . The high contrast between the two components is characterized by the assumption that the sequence\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b1<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    decreases towards zero. We study the three regimes corresponding to the limits\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"italic\">\u03b1<\/mml:mi>\n                        <mml:mo>:<\/mml:mo>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mo movablelimits=\"false\">lim<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\u2192<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mfrac>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"italic\">\u03b1<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:mfrac>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . It is shown in particular that in the case\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"italic\">\u03b1<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , the inclusions\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    behave as holes on the macroscopic diffusion process.\n                  <\/jats:p>","DOI":"10.3233\/asy-161370","type":"journal-article","created":{"date-parts":[[2016,7,12]],"date-time":"2016-07-12T12:25:23Z","timestamp":1468326323000},"page":"309-324","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Diffusion through a composite medium with a semi-periodic structure and high contrasting diffusivity"],"prefix":"10.1177","volume":"98","author":[{"given":"Ali","family":"Sili","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Marseille","place":["France"]},{"name":"Universit\u00e9 de Toulon","place":["France"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2016,7,11]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1137\/0523084"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1993-7201"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/0521046"},{"key":"e_1_3_1_5_2","unstructured":"A.Bensoussan J.-L.Lions and G.Papanicolaou Asymptotic Analysis for Periodic Structures North-Holland Amsterdam 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