Günter Hellwig

The Potential Equation

The problem n n

Elliptic and Elliptic-Parabolic Type

Delta _n u = 0quad wherequad Delta _n u equiv sumlimits_{i = 1}^n {u_{x_i x_i } ,}

Differential Equations of the Second Order

n n(3.1) n n n n

Equations of Hyperbolic Type in Two Independent Variables

begin{gathered} uleft( {{x_1},{x_2},...,{x_{n - 1}},0} right) = {u_0}left( {{x_1},{x_2},...,{x_{n - 1}}} right) hfill {u_{{x_n}}}left( {{x_1},{x_2},...,{x_{n - 1}},0} right) = {u_1}left( {{x_1},{x_2},...,{x_{n - 1}}} right) hfill end{gathered}

Boundary and Initial-Value Problems for Equations of Hyperbolic and Parabolic Type in Two Independent Variables

n n(3.2) n nis called the initial-value problem for the potential equation. By way of examples, we shall show that for this initial-value problem the third requirement cannot, in general, be satisfied, and that the second requirement can in general be satisfied only if the u i , i = 0, 1 are analytic functions in R n-1.

Systems of Differential Equations of the First Order

We consider the most general linear elliptic system of two first-order partial differential equations in two unknown functions. After the considerations in Section 11–2.6 and with suitable hypotheses on the coefficients, we may assume that the system is in the integrable normal form (II-2.65): ( begin{gathered} U_{x_1 }^1 - U_{x_2 }^1 + A_1^1 (x)U^1 + A_2^1 (x)U^2 + C^1 (x) = 0, hfill U_{x_2 }^1 - U_{x_1 }^2 + A_1^2 (x)U^1 + A_2^2 (x)U^2 + C^2 (x) = 0, hfill end{gathered} ) where U 1(x) and U 2(x) are the unknown functions.