Maximum Principles

These are very important and useful tools in analysis, here, restricted to $2^{nd}$-order elliptic equations, although they exist for higher order elliptic equations as well. The $2^{nd}$ derivative gives information on a functions' extrema $\therefore$ maximum principles are useful tools in the analysis of solutions of $2^{nd}$ -order elliptic equations. Two theorems:

I. Theorem (Max Value) let $L\phi=a\phi_{xx}+2b\phi_{xy}+c\phi_{yy}$     , $a,c>0$ and $b^2<ac$ i.e. $L$ is an elliptic operator. If $U$ satisfies $LU\ge 0$ on a bounded domain $\Omega\Rightarrow U$ has its maximum on $\partial \Omega$.

Remark: In 1D, recall that if $U''>0$ on some closed interval in $x\Rightarrow\max$ value $U$ is at interval ends (convince yourself).

II. Theorem (Max/Min): If elliptic equation

$$a U_{xx}+2b U_{xy}+c U_{yy}+d_1 U_x+d_2 U_y+e U=0$$

holds on $\Omega$     $a,c>0$ , $e\le 0\Rightarrow U(x,y)$ cannot have a positive local maximum or a negative local minimum in interior of $\Omega$.

Proof (from Strikwerda): Prove only I when $LU>0$ and II when $e<0$. Cases when $LU\ge 0$ and $e\le 0$ take a little more effort. Assume $U\in C^3$

I: $U\in C^3$ has local max at $(x_0, y_0)\Rightarrow$ gradient of $U$ at $(x_0, y_0)$

$$U_x(x_0, y_0)=U_y(x_0, y_0)=0$$

using Taylor's with $U_{xx}^{0}\equiv U_{xx}(x_0, y_0)$, etc. $\cdots$

$$U(x_0+\Delta x, y_0+\Delta y) = U(x_0, y_0) +\frac12\left(\Delta x^2 U^0_{xx} +2\Delta x\Delta y U^0_{xy}+\Delta y^2 U^0_{yy}\right)$$

$$+ {\cal O} \left(\max (\Delta x, \Delta y)^3\right)$$

Since $U(x_0+\Delta x, y_0+\Delta y)\le U(x_0, y_0)$ for sufficient small $\Delta x$ and $\Delta y$ then

$$\Delta x^2 U^0_{xx} +2\Delta_x\Delta y U^0_{xy}+0y^2 U^0_{yy}\le 0$$

Since expression is homogeneous of degree 2 in $\Delta x$ and $\Delta y$

$$\alpha^2 U^0_{xx}+2\alpha\beta U^0_{xy}+\beta^2 U^0_{yy}\le 0$$ (158)

$$\forall$$    real $$\alpha, \beta.$$

Now, prove $I$ for $LU>0$. Apply (159) twice. First with $\alpha=\sqrt{a^0}$     $\beta=b^0/\sqrt{a^0}$, and then with $\alpha = 0$ and $\beta^2 = C^0-(b^0)^2/a^0$, we have

$$LU = a^0 U^0_{xx}+2b^0U^0_{xy}+c^0_{yy}$$

$$= \left(\sqrt{a^0}\right)^2 U^0_{xx}+2\sqrt{a^0}\left(\frac{b^0} {\... ...t{c^0}}\right)^2 U^0_{yy} +\left(c^{0}-\frac{(b^0)^2}{a^0}\right) U^0_{yy}\le 0$$

Since this contradicts assumption that $LU>0\Rightarrow$ theorem $I$ is proved.

Proof of Theorem II: only when $e(x,y)<0$ proof: From Theorem I if $U$ has maximum at $(x_0, y_0)$ then $LU\le 0\therefore$

$$-LU(x_0, y_0)=e(x_0, y_0) U(x_0, y_0)\ge 0$$

Since $e<0\Rightarrow U(x_0, y_0)\le 0$ at an interior local maximum. Similarly by considering $-U(x_0, y)$ can show that $U$ is not negative at a local minimum. $\Box$

Some uses:

(1)

Physical: theorems state that hottest and coldest temps for steady temperature distribution occur at boundaries.

(2)

Mathematical: Can use principles to prove uniqueness of solutions to many elliptic equations.

Comments on Boundary Conditions for elliptic equations: Look at Poisson only and consider

$$U=b_1 \mbox { on } d\Omega\quad \mbox { Dirichlet}$$

$$\frac{\partial U }{\partial n}=b_2 \mbox { on } d\Omega \mbox { Neumann}$$

$$\frac{\partial U}{\partial U} + \alpha U=b_3 \mbox { on } d\Omega \mbox { Robin}$$

if $\partial \Omega$ is smooth, a unique solution exists with dirichlet boundary condition. It also exists for Neumann, if integrability condition is satisfied (note: solution is unique, to within an additive constant).

Some general remarks on local behavior:

(1)

If Dirichlet is enforced along smooth portion of boundary $\Rightarrow$ normal derivative at $\partial \Omega$ will be as well behaved as the derivative of the boundary data in the direction of boundary. If boundary data is discontinuous $\Rightarrow$ normal derivative of solution will have unboundedness of discontinuities.

(2)

If either Neumann or Robin are enforced at $d\Omega\Rightarrow$ solution differentiable and $1^{st}$ derivative as well behaved as the boundary data function.

Serious difficulty occurs at points on boundary where boundary condition change from Dirichlet to Newmann or Robin type $\Rightarrow$ one gets unbounded derivatives for $u$ at these points.

(3)

Similar difficulties arise in reentrant corners: where local angle is greater than 180$^{\circ}$, as measured from inside: second derivative may be unbounded, although solution and $1^{st}$ derivative bounded (see Figure 31).

Figure 31: Domain with reentrant corner.

In summary: When boundary conditions change type, or when boundary is not smooth, expect derivatives of solution to have unbondedness.