11 - Characteristics of Second Order Semi-Linear PDEs

Characteristics of Second Order Semi-Linear PDEs

Analogous Properties to the Characteristic Projections of First Order Semi-Linear PDEs

Discontinuities in Second Derivatives

Discontinuities in Second Derivatives of a solution across a given curve $\Gamma$ means

$$\text{curve }\Gamma \text{ is characteristic curve}$$

Proof

Suppose curve $\Gamma$ defined parametrically by

$$(x(s),y(s))$$

is a curve across which there are discontinuities in the second derivative of solution

Let $u_{xx}^{+}$ denote values on one side of $\Gamma$ and $u_{xx}^{-}$ denoting values on the other side of $\Gamma$

Differentiating $u_x(x(s),y(s))$ and $u_y(x(s),y(s))$ along $\Gamma$
with $u,u_x,u_y$ being continuous across the curve then

\begin{alignat*}{3} \frac{du_{x}}{ds} &= \quad && \frac{dx}{ds}u_{x x}^\pm + \quad && \frac{dy}{ds} u_{xy}^\pm \\ \frac{du_{y}}{ds} &= && && \frac{dx}{ds}u_{y x}^\pm + \quad && \frac{dy}{ds} u_{y y}^\pm \\ \end{alignat*}

with

$$f(x,y,u,u_x,u_y)=a(x,y)u_{xx}^{\pm }+2b(x,y)u_{xy}^{\pm }+c(x,y)u_{yy}^{\pm }$$

Subtracting the minus equation from the plus equations we get

\begin{alignat*}{3} 0 &= \quad && \frac{dx}{ds}[u_{x x}]^+_{-} + \quad && \frac{dy}{ds} [u_{xy}]^+_{-} \\ 0 &= && && \frac{dx}{ds}[u_{y x}]^+_{-} + \quad && \frac{dy}{ds} [u_{y y}]^+_{-} \\ 0 &= &&a(x, y)[u_{x x}]^+_{-} + &&2b(x, y)[u_{x y}]^+_{-} + &&c(x, y) [u_{yy}]^+_{-} \end{alignat*}

where $[u_{xx}{]}_{-}^{+}=u_{xx}^{+}-u_{xx}^{-}$ denotes the jump in $u_{xx}$ across $\gamma$, etc

If there are discontinuities in the second derivatives then
the set of equations in the jumps must have a non-zero solution hence

$$a(x,y){\left(\frac{dy}{ds}\right)}^2-2b(x,y)\frac{dx}{ds}\frac{dy}{ds}+c(x,y){\left(\frac{dx}{ds}\right)}^2=0$$

Hence $\Gamma$ is a characteristic