ABSTRACT
Let U be a domain in https://www.w3.org/1998/Math/MathML"> R ∗ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003440116/5c34fd59-3787-4510-b376-2cd023dd9bdb/content/math3_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
A multiplicative linear differential operator of second order for the function https://www.w3.org/1998/Math/MathML"> u = u ( x 1 , x 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003440116/5c34fd59-3787-4510-b376-2cd023dd9bdb/content/math3_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is given by https://www.w3.org/1998/Math/MathML"> L ( u ) = a ⋅ ∗ u x 1 x 1 ∗ ∗ + ∗ e 2 ⋅ ∗ b ⋅ ∗ u x 1 x 2 ∗ ∗ + ∗ c ⋅ ∗ u x 2 x 2 ∗ ∗ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003440116/5c34fd59-3787-4510-b376-2cd023dd9bdb/content/math3_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where the coefficients https://www.w3.org/1998/Math/MathML"> a , b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003440116/5c34fd59-3787-4510-b376-2cd023dd9bdb/content/math3_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and c are supposed to be nonnegative continuously multiplicative differentiable and not simultaneously multiplicative vanishing functions of x 1 and x 2 in the domain https://www.w3.org/1998/Math/MathML"> U . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003440116/5c34fd59-3787-4510-b376-2cd023dd9bdb/content/math3_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
