On a Stefan Problem in a Concentrated Capacity

The Stefan problems in a “concentrated capacity” appear in the heat diffusion phenomena involving two adjoining three-dimensional bodies Ω and Ω ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1565.tif"/> , such that in Ω ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1566.tif"/> a change of phase takes place and the thermal conductivity along the direction v normal to the boundary Γ of Ω is much greater than in the other directions. Assuming to be infinite, the body Ω ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1567.tif"/> , as far as heat diffusion is concerned, behaves like a manifold of dimension less tham three and can be identified with a subset of Γ. The mathematical formulation and many related results in some particular important geometrical situations have been first give in [11] (cfr. also [1], [19] and the references therein). In [14], [15] I studied the case where Ω is a regular bounded set of ℜⁿ and Ω ˜ = Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1568.tif"/> in a framework of Hilbert spaces. In the present paper I shall consider the case Ω ˜ = Γ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1569.tif"/> where Γ₁ is a subset of Γ, open and regular in Γ, and on the remaining part of Γ the “thermal flux” coming from Ω is given. This case presents many different and more difficult questions that the previons one. The exact formulation and the main result are described in section 1. I express my thanks to G. Savare’ for the helpful discussions on the subject of the present paper.