The connection between the generalized geometric motion of interfaces, interpreted in the viscosity sense, and a singularly perturbed parabolic problem with double obstacle +/- 1 and small parameter epsilon is examined. This approach retains the local character of the limit problem, because the noncoincidence set, where all the action takes place, is a thin transition layer of thickness O(epsilon) irrespective of the forcing term. Zero-level sets are shown to converge past singularities to the generalized motion by mean curvature with forcing, provided no fattening occurs. If the underlying viscosity solution satisfies a nondegeneracy property, namely, its gradient does not vanish, then our results yield interface error estimates and layer width estimates of order O(epsilon). The proofs are based on constructing viscosity subsolutions and supersolutions to the double obstacle problem in terms of the signed distance function and approximate traveling waves dictated by formal asymptotics.

Convergence of double obstacle problems to the generalized geometric motion of fronts / R. Nochetto, C. Verdi. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 26:6(1995), pp. 1514-1526. [10.1137/S0036141093255429]

Convergence of double obstacle problems to the generalized geometric motion of fronts

C. Verdi
Ultimo
1995

Abstract

The connection between the generalized geometric motion of interfaces, interpreted in the viscosity sense, and a singularly perturbed parabolic problem with double obstacle +/- 1 and small parameter epsilon is examined. This approach retains the local character of the limit problem, because the noncoincidence set, where all the action takes place, is a thin transition layer of thickness O(epsilon) irrespective of the forcing term. Zero-level sets are shown to converge past singularities to the generalized motion by mean curvature with forcing, provided no fattening occurs. If the underlying viscosity solution satisfies a nondegeneracy property, namely, its gradient does not vanish, then our results yield interface error estimates and layer width estimates of order O(epsilon). The proofs are based on constructing viscosity subsolutions and supersolutions to the double obstacle problem in terms of the signed distance function and approximate traveling waves dictated by formal asymptotics.
Settore MAT/08 - Analisi Numerica
Article (author)
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/176118
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 9
social impact