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The Baire Method for the Prescribed Singular Values Problem
Author(s) -
De Blasi F. S.,
Pianigiani G.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005599
Subject(s) - mathematics , bounded function , baire category theorem , dirichlet distribution , boundary (topology) , pure mathematics , dirichlet boundary condition , dirichlet problem , singular solution , mathematical analysis , combinatorics , discrete mathematics , boundary value problem
The paper investigates the vectorial Dirichlet problem defined by S j (∇ u(x))=1,x∈n O ; a.e., j=1,…,n, u(x)=\ϕ(x),\,x\in| O . end{cases} Here O is an open bounded subset of R n with boundary | O , and Σ j (A) (j=1,…,n) denote the singular values of the gradient ∇ u(x). The existence of solutions is established under one of the following assumptions: ϕ: O − R n is continuous on O and locally contractive on O , or ϕ: | O − R n is contractive on | O . This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.