편미분방정식 맛보기

(a) If $u(x,y) = x^3-3xy^2$, then

$$\begin{aligned} u_x&= 3x^2-3y^2 \qquad& u_{xx}&= 6x \\ u_y&= -6xy \qquad& u_{yy}&= -6x \end{aligned}$$

So $u_{xx}(x,y)+u_{yy}(x,y)=6x-6x=0$ and $x^3-3xy^2$ is harmonic.

(b) If $u(x,y) = x^3-y^3$, then

$$\begin{aligned} u_x&= 3x^2 \qquad& u_{xx}&= 6x \\ u_y&= -3y^2 \qquad& u_{yy}&= -6y \end{aligned}$$

So $u_{xx}(x,y)+u_{yy}(x,y)=6x-6y$ is not identically zero and $x^3-y^3$ is not harmonic.

(c) If $u(x,y) = \sin(x)\,\cos(y)$, then

$$\begin{aligned} u_x&= \cos(x)\,\cos(y) \qquad& u_{xx}&= -\sin(x)\,\cos(y) \\ u_y&= -\sin(x)\,\sin(y) \qquad& u_{yy}&= -\sin(x)\,\cos(y) \end{aligned}$$

So $u_{xx}(x,y)+u_{yy}(x,y)=-2\sin(x)\,\cos(y)$ is not identically zero and $\sin(x)\,\cos(y)$ is not harmonic.

(d) If $u(x,y) = e^{7x}\,\cos(7y)$, then

$$\begin{aligned} u_x&= 7\,e^{7x}\,\cos(7y) \qquad& u_{xx}&= 49\,e^{7x}\,\cos(7y) \\ u_y&= -7\,e^{7x}\ \sin(7y) \qquad& u_{yy}&= -49\,e^{7x}\,\cos(7y) \end{aligned}$$

So $u_{xx}(x,y)+u_{yy}(x,y)=49\,e^{7x}\,\cos(7y)-49\,e^{7x}\,\cos(7y)=0$ and $e^{7x}\,\cos(7y)$ is harmonic.

(e) If $u(x,y) = \ln(x^2+y^2)$, then

$$\begin{aligned} u_x&= \frac{2x}{x^2+y^2} \qquad& u_{xx}&= \frac{2}{x^2+y^2}- \frac{4x^2}{{(x^2+y^2)}^2}\\ u_y&= \frac{2y}{x^2+y^2} \qquad& u_{yy}&= \frac{2}{x^2+y^2}- \frac{4y^2}{{(x^2+y^2)}^2} \end{aligned}$$

So

$$\begin{aligned} u_{xx}(x,y)+u_{yy}(x,y)&=\frac{2}{x^2+y^2}- \frac{4x^2}{{(x^2+y^2)}^2} +\frac{2}{x^2+y^2}- \frac{4y^2}{{(x^2+y^2)}^2} \\ &=\frac{4}{x^2+y^2} - 4\frac{x^2+y^2}{{(x^2+y^2)}^2} \\ &=\frac{4}{x^2+y^2}-\frac{4}{x^2+y^2} \\ &=0 \end{aligned}$$

and $\ln(x^2+y^2)$ is harmonic.