By E. Zeidler, L.F. Boron

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**Extra resources for Nonlinear functional analysis vol.3: Variational methods and optimization**

**Example text**

A/at - P(a/ax), namely, a distribution E(t, x) that satisfies LE = 0 (t>O) and E(t,x)I,=,=6(x). If we put &t,x)= E(t,x) (t>O) and E”(t,x)=O (t

141 F. ces and linear partial differential equations, Springer, 1967. [ 151 F. Treves, Linear partial differential equations with constant coefficients, Gordon & Breach, 1966. [ 161 F. John, Partial differential equations, Springer, 1971. [ 173 L. Bers, F. John, and M. Schechter, Partial differential equations, Interscience, 1964. [IS] L. Nirenberg and F. Treves, On local solvability of linear partial differential equations I, II, Comm. Pure Appl. , 23 (1970), l-38,459-510. 1195 321 B PDEs (Initial [19] R.

This characteristic manifold is considered as a part of the integral surface with infinitesimal width, and in this case we call it the characteristic strip. The characteristic strip is represented by the equations x = x(n),~=y(l),z=z(l),p=~(l),q=q(l)containing a parameter. On the integral z = z(x, y), we have g=azdx dl ax dl surface I azdY dy dl and dz=pdx+qdy. (19) 320 E Partial Differential 1192 Equations Equation (19) is called the strip condition. The equations (18) evidently satisfy this condition.