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\n Consider the degenerate elliptic operator\n \n \n \n \n \n \n L<\/mml:mi>\n \n \u03b4\n \n <\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n :=<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u2202\n \n <\/mml:mi>\n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n \n \u2212\n \n <\/mml:mo>\n \n \n \n \u03b4\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mfrac>\n \n \n \u2202\n \n <\/mml:mi>\n y<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\mathcal {L_\\delta } := -\\partial ^2_x-\\frac {\\delta ^2}{x^2}\\partial ^2_y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on\n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n :=<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n \u00d7\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n l<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Omega := (0, 1)\\times (0, l)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n \n \u03b4\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n l<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\delta >0, l>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove well-posedness and regularity results for the degenerate elliptic equation\n \n \n \n \n \n \n L<\/mml:mi>\n \n \u03b4\n \n <\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n u<\/mml:mi>\n =<\/mml:mo>\n f<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {L_\\delta } u=f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n u<\/mml:mi>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n \u2202\n \n <\/mml:mi>\n \n \u03a9\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n u|_{\\partial \\Omega }=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n using weighted Sobolev spaces\n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msubsup>\n \\mathcal {K}^m_a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In particular, by a proper choice of the parameters in the weighted Sobolev spaces\n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msubsup>\n \\mathcal {K}^m_a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we establish the existence and uniqueness of the solution. In addition, we show that there is no loss of\n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msubsup>\n \\mathcal {K}^m_a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -regularity for the solution of the equation. We then provide an explicit construction of a sequence of finite dimensional subspaces\n \n \n \n \n V<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n V_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for the finite element method, such that the optimal convergence rate is attained for the finite element solution\n \n \n \n \n \n u<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n \n V<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n u_n\\in V_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , i.e.,\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n <\/mml:mrow>\n u<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n u<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n C<\/mml:mi>\n \n {dim}<\/mml:mtext>\n <\/mml:mrow>\n (<\/mml:mo>\n \n V<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n )<\/mml:mo>\n \n \n \u2212\n \n <\/mml:mo>\n \n m<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:msup>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n <\/mml:mrow>\n f<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n H<\/mml:mi>\n \n m<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n ||u-u_n||_{H^1(\\Omega )}\\leq C\\textrm {{dim}}(V_n)^{-\\frac {m}{2}}||f||_{H^{m-1}(\\Omega )}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n independent of\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-08-02179-0","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"713-737","source":"Crossref","is-referenced-by-count":20,"title":["A-priori analysis and the finite element method for a class of degenerate elliptic equations"],"prefix":"10.1090","volume":"78","author":[{"given":"Hengguang","family":"Li","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,9,2]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","doi-asserted-by":"crossref","first-page":"161","DOI":"10.4171\/dm\/208","article-title":"Sobolev spaces on Lie manifolds and regularity for polyhedral domains","volume":"11","author":"Ammann, Bernd","year":"2006","journal-title":"Doc. 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