Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects |
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Authors: | Marc Briane Juan Casado–Díaz |
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Institution: | (1) Centre de Mathématiques, I.N.S.A. de Rennes & I.R.M.A.R., Rennes Cedex, France;(2) Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain |
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Abstract: | In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type
, where A
n
is a symmetric positive definite matrix-valued function and μ
n
is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A
n
we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ
n
and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of 11,23] in which A
n
is assumed to be uniformly bounded. It is also based on the compactness result of 7] obtained for sequences of two-dimensional
diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear.
However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates,
the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an
explicit formula for the limit energy specifying the kernel of the nonlocal term. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35J70 35B40 35B50 35B65 35B27 |
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