{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T03:08:54Z","timestamp":1787022534487,"version":"build-2736575974"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"name":"National Science Foundation","award":["DMS-1015984, DMS-1217262"],"award-info":[{"award-number":["DMS-1015984, DMS-1217262"]}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research, USAF","doi-asserted-by":"crossref","award":["FA99550-12-0358"],"award-info":[{"award-number":["FA99550-12-0358"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We construct mollification operators in strongly\nLipschitz domains that do not invoke non-trivial extensions, are\n                    <jats:italic>\n                      L\n                      <jats:sup>p<\/jats:sup>\n                    <\/jats:italic>\n                    stable for any real number\n                    <jats:inline-formula id=\"eq1_w2aab3b7c18b1b6b1aab1c14b1b3Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_74250eda3533e8fcabc158ef7d8f08d8.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>p<\/m:mi>\n                            <m:mo>\u2208<\/m:mo>\n                            <m:mo>[<\/m:mo>\n                            <m:mn>1<\/m:mn>\n                            <m:mo>,<\/m:mo>\n                            <m:mi>\u221e<\/m:mi>\n                            <m:mo>]<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${p\\in [1,\\infty ]}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and commute with the differential operators \u2207,\n                    <jats:inline-formula id=\"eq2_w2aab3b7c18b1b6b1aab1c14b1b5Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_f85c2aa6e8c810b14e589639dd7dab7b.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>\u2207<\/m:mi>\n                            <m:mo>\u00d7<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\nabla {\\times }}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula id=\"eq3_w2aab3b7c18b1b6b1aab1c14b1b7Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_f8498dff56eebe66a14bf8eabb8f6a70.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>\u2207<\/m:mi>\n                            <m:mo>\u00b7<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\nabla {\\cdot }}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We also construct mollification operators\nsatisfying boundary conditions and use them to characterize the\nkernel of traces related to the tangential and normal trace of\nvector fields. We use the mollification operators to build\nprojection operators onto general\n                    <jats:italic>H<\/jats:italic>\n                    <jats:sup>1<\/jats:sup>\n                    -,\n                    <jats:inline-formula id=\"eq4_w2aab3b7c18b1b6b1aab1c14b1c12Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_2ff82c33e8d364121af7615ef89dcc27.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>H<\/m:mi>\n                            <m:mo>(<\/m:mo>\n                            <m:mi>curl<\/m:mi>\n                            <m:mo>)<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${{H}(\\mathrm {curl})}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    - and\n                    <jats:inline-formula id=\"eq5_w2aab3b7c18b1b6b1aab1c14b1c14Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_68316a161c9b57bf65be48122f1b439e.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>H<\/m:mi>\n                            <m:mo>(<\/m:mo>\n                            <m:mi>div<\/m:mi>\n                            <m:mo>)<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${{H}(\\mathrm {div})}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -conforming\nfinite element spaces, with and without\nhomogeneous boundary conditions. These operators commute with the\ndifferential operators \u2207,\n                    <jats:inline-formula id=\"eq6_w2aab3b7c18b1b6b1aab1c14b1c16Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_f85c2aa6e8c810b14e589639dd7dab7b.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>\u2207<\/m:mi>\n                            <m:mo>\u00d7<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\nabla {\\times }}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula id=\"eq7_w2aab3b7c18b1b6b1aab1c14b1c18Aa\">\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0034_f8498dff56eebe66a14bf8eabb8f6a70.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>\u2207<\/m:mi>\n                            <m:mo>\u00b7<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\nabla {\\cdot }}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , are\n                    <jats:italic>\n                      L\n                      <jats:sup>p<\/jats:sup>\n                    <\/jats:italic>\n                    -stable, and have optimal approximation\nproperties on smooth functions.\n                  <\/jats:p>","DOI":"10.1515\/cmam-2015-0034","type":"journal-article","created":{"date-parts":[[2015,12,12]],"date-time":"2015-12-12T23:13:46Z","timestamp":1449962026000},"page":"51-75","source":"Crossref","is-referenced-by-count":47,"title":["Mollification in Strongly Lipschitz Domains with Application to Continuous and Discrete De Rham Complexes"],"prefix":"10.1515","volume":"16","author":[{"given":"Alexandre","family":"Ern","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vall\u00e9e cedex 2, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Luc","family":"Guermond","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2015,12,15]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/16\/1\/article-p51.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0034\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0034\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:33:40Z","timestamp":1680284020000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0034\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1,1]]},"published-print":{"date-parts":[[2016,1,1]]}},"alternative-id":["10.1515\/cmam-2015-0034"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2015-0034","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2015,12,15]]}}}