ELLIPTIC EQUATIONS

Brief overview Refer to Figure 30 Archetypical Equation in 2D:

$$\nabla^2 U=f(x,y)$$    in a domain $$\Omega(x,y).$$

with $\nabla^2\equiv \partial _{xx}+\partial _{yy}$

is called Poisson's Equation. If $f=0$ we call it Laplace's equation. The solutions to Laplace's equations are called harmonic functions and are intimately tied to the theory of complex analysis.

Figure: Domain of definition for Poisson's Problem in 2 dimensions. The $\hat n$ indicates the convention on the unit normal to the boundary $\partial \Omega .$

Boundary Conditions:

(1)

(field specified at boundary) Dirchlet: $U=b_1$ on $\partial \Omega$

(2)

Neumann: $$\frac{\partial U}{\partial n}\equiv\hat n\cdot \nabla U=b_2$$ on $\partial \Omega$

(3)

Robin: mix of the two above: $$\frac{\partial U}{\partial n}+\alpha U=b_3$$ on $\partial \Omega$. Only 1 boundary condition may be specified on $\partial \Omega$. But can have Neumann, say, for one portion of $\partial \Omega$ and Dirichlet for the other, for example.

Only 1 boundary condition may be specified on $\partial \Omega$. But can have Neumann, say, for one portion of $\partial \Omega$ and Dirichlet for the other, for example.

Physically: Poisson equation describes many things. For example, the steady state-temperatue distribution of an object occupying $\Omega$, with heat sources and sinks represented by $f$. The Dirichlet boundary conditions represent the situation when the temperature is specified at boundary and Naumann would be if the flux of temperature is specified at boundary. In particular, if $$\frac{\partial U}{\partial n}=0\vert _{\partial \Omega}$$ we say we have a perfect insulator boundary.

In order for solution to exist, if Newmann B.C. are specified, is the data must satisfy the ``integrability'' condition:

$$\int\int_{\Omega}fdV=\int_{\partial \Omega}b_2ds$$

(to prove: use divergence theorem).

General Quasi-linear 2nd-order elliptic in 2D has the form:

$$a(x,y)U_{xx}+2b(x,y)U_{xy}+c(x,y)U_{yy}+d(x,y,U,U_x,U_y)=f(x,y)$$

with $a,c>0$     $b^2<ac$

A $1^{st}$-order elliptic equation system example:

$$\left\{\begin{array}{l} U_x-V_y=0\qquad \mbox {\lq\lq Cauchy-Riemann'' equations}\\ U_y+V_x=0 \end{array}\right.$$

example of $4^{th}$ order

$$\nabla^4 U=f$$$$\qquad \mbox {\lq\lq Biharmonic Equation.''}$$

An essential feature of elliptic equation solutions is that they are smoother than the data. For example $U$ has 2 more derivatives then $f$ in the Poisson equation. 4 more than $f$ in biharmonic equation. Solutions to Laplace and Cauchy-Riemann Eqs are infinitely differentiable.

For the general $2^{nd}$-order linear constant coefficient elliptic equations, the following ``regularity estimate'' can be proved:

$$\vert\vert U\vert\vert^2_{s+2}\le C_s\left(\vert\vert f\vert\vert^2_s+\vert\vert U\vert\vert^2_0\right)$$

where

$$\vert\vert\cdot\vert\vert^2_s\equiv \sum_{s_1+s_2\le s}\vert\vert\partial ^{s_{1}}_x\partial ^{s_2}_y \cdot \vert\vert^2$$

i.e. if solution exists and is finite in $L_2$, i.e. $\vert\vert U\vert\vert _0$ finite and that $f$ has all derivatives of order up to $s$ in $L_2({\Bbb R}^2)\Rightarrow U$ has $s+2$ in $L_2(R_2)$.

The solution of the elliptic equation is more differentiable than the data and the increase in differentiability $=$ order of equation.

There is an ``interior regularity estimate'' as well. Suppose $\Omega_1 \in \Omega$ whose boundary is wholly contained in $\Omega$. Then

$$\vert\vert U\vert\vert^2_{s+2, \Omega_1}\le C_s(\Omega, \Omega_1)(\vert\vert f\vert\vert^2_{s,\Omega}+\vert\vert U\vert\vert^2_{0, \Omega})$$

For the non-constant coefficient case, we require that coefficients be defined and bounded, and very similar estimates are obtained.