ABSTRACT
Let https://www.w3.org/1998/Math/MathML"> Ω ⊂ IR n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/ieq0700.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be open and bounded. We assume that the boundary Γ of Ω is a C∞-manifold of dimension n-1. Locally, Ω is totally on one side of Γ. With T > 0 put QT.=]0,T[ × Ω, ΣT.= ]0,T[× Γ. For i,j = 1,…,n let be given 00 functions https://www.w3.org/1998/Math/MathML"> a o , a ij ∈ C ∞ ( Ω ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/ieq0701.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and a real number c > 0 such that https://www.w3.org/1998/Math/MathML"> a i j = a j i i , j = 1 , … , n ∑ i , j = 1 n a i j ( x ) ξ i ξ j ⩾ c ∑ i = 1 n ξ i 2 ∀ ξ ∈ ℝ n , ∀ x ∈ Ω . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0544.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> For φ ϵ L2(QT) define https://www.w3.org/1998/Math/MathML"> A φ . = − ∑ i , j = 1 n ∂ ∂ x i a i j ∂ φ ∂ x j + a 0 φ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0545.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Furthermore, let K be a convex subset of https://www.w3.org/1998/Math/MathML"> C ( Ω ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/ieq0702.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> having a nonempty interior and let https://www.w3.org/1998/Math/MathML"> f : [ 0 , + ∞ [ × Ω ¯ × ℝ → ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0546.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be a continuous function such that f(t,x, •) is convex for every https://www.w3.org/1998/Math/MathML"> ( t , x ) ∈ [ 0 , + ∞ [ × Ω ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/ieq0703.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Finally, assume that α, m are positive real constants.
