Abstract
We study the lower semicontinuous envelope in Lp(Ω), F, of a functional F of the form F(u)=∫ΩA ∇u∇udx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)=∫Ω B∇u∇udx with B=√AP √A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.
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Casado-díaz, J. Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix. Potential Analysis 11, 39–76 (1999). https://doi.org/10.1023/A:1008650917875
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DOI: https://doi.org/10.1023/A:1008650917875