By Maksimov V. I.
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Additional info for A Boundary Control Problem for a Nonlinear Parabolic Equation
Sample text
We refrain from giving more details here. Appendix A. Surface divergence Let M be an m-dimensional C k -submanifold of Rn . Then, locally around any point p ∈ M , the submanifold is given by a parametrization of class C k . Hence there is an open parameter region B ⊂ Rm and a C k -function g : B → Rn such that g(B) = M ∩ U for some open neighborhood U ⊂ Rn of p. Furthermore, the set {∂1 g(y), . . , ∂m g(y)} is a linearly independent subset of Rn , and g is a homeomorphism from B onto M ∩ U . t.
Hence there is an open parameter region B ⊂ Rm and a C k -function g : B → Rn such that g(B) = M ∩ U for some open neighborhood U ⊂ Rn of p. Furthermore, the set {∂1 g(y), . . , ∂m g(y)} is a linearly independent subset of Rn , and g is a homeomorphism from B onto M ∩ U . t. γ(0) = p , γ (0) = τ } . Then, Tp M = span{∂1 g(y), . . , ∂m g(y)} for p ∈ M, y = g −1 (p) . , f (p) ∈ Tp M for all p ∈ M . Such an f is a C 1 -vector field on M if f ◦ g is continuously differentiable for every C 1 -parametrization g of M .
Whenever a distinction between the different phases is not necessary, the phase indices + and − are omitted. The phases are separated by a phase boundary Γ(t) at which at least one of the material parameters has a jump-discontinuity. At this interface, the additional jump conditions [ρ(u − uint )] n [ρu ⊗ (u − uint ) − S] n = 0, = σκn + gradΓ σ appear, where uint is the interfacial velocity, gradΓ σ denotes the surface gradient of the surface tension σ (cf. Appendix A) and κ = −divΓ n is the sum of the (local) principal curvatures of the interface.