ABSTRACT
Consider the nonlinear problem nonlinear equations https://www.w3.org/1998/Math/MathML"> L u ( s ) = f ( s , u ( s ) ) , s ∈ Ω , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/math7_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> u ( s ) = 0 , s ∈ ∂ Ω , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/math7_2.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> with L a self-adjoint elliptic operator over Ω, https://www.w3.org/1998/Math/MathML"> L u ( s ) ≡ − ∑ i , j = 1 d ∂ ∂ s i ( a i , j ( s ) ∂ u ( s ) ∂ s j ) + γ ( s ) u ( s ) , s ∈ Ω . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/math7_3.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> As before, Ω is to be an open, simply–connected, and bounded region in https://www.w3.org/1998/Math/MathML"> R d https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/inline-math7_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> with a smooth boundary ∂Ω that is homeomorphic to https://www.w3.org/1998/Math/MathML"> S d − 1 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/inline-math7_2.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The functions a i,j (s), 1 ≤ i, j ≤ d, are assumed to be several times continuously differentiable over https://www.w3.org/1998/Math/MathML"> Ω ¯ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/inline-math7_3.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and the d × d matrix [a i,j (s)] is to be symmetric and to satisfy https://www.w3.org/1998/Math/MathML"> ξ T A ( s ) ξ ≥ α ξ T ξ , s ∈ Ω ¯ , ξ ∈ ℝ d , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/math7_4.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> for some α > 0. Also assume the coefficient https://www.w3.org/1998/Math/MathML"> γ ∈ C ( Ω ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344374/3221f19a-ec82-4614-b75a-37e8652c7af7/content/inline-math7_4.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Note that because the right-hand function f is allowed to depend on u, any multiple of u can be added to each side of (7.1). For analyses of the existence and uniqueness of a solution to (7.1)-(7.2), see Zeidler [107].
