Consider the equation

$$-\frac{\partial^2u}{\partial x^2}=-\text{div}\left(\begin{pmatrix}1&0\\0&0\end{pmatrix}\nabla\right)=0\mbox{ in }\mathbb{R}^2.$$ Solving the equation, we can conclude that the solution takes the form $$u(x,y)=c_1(y)x+c_2(y).$$ If the equation is thought of as a degenerate Laplacian, can we conclude that the diffusion due to the degenerate Laplacian happens in the $x$-direction only?

I am interested in the physical interpretation of the solutions to degenerate Laplacians.

More generally, if we have an operator of the form $-\text{div}(A\nabla)$, would it be correct to say that the diffusion due to the operator happens along the eigenvectors of the matrix $A$?

How does one interpret this physically or mathematically?


1 Answer 1


Since many physical applications of second order elliptic equations flow from their maximum principles, this answer will explain the maximum principles which are available for degenerate second order equations. Such equations are also called degenerate elliptic equations, second order equations with non-negative characteristic form and also elliptic-parabolic equations.

These theorems are quoted from the book of Oleĭnik-Radkevič called "Second Order Equations With Nonnegative Characteristic Form". The results are attributed to Pucci, A.D. Aleksandrov and Bony.

Consider the equation of the form $$Lu=-\text{div}(A\nabla u)+b.\nabla u+cu,$$ where $(A\xi,\xi)\geq 0$ for all $\xi\in\mathbb{R}^d$. We assume that the coefficients are bounded and $u\in C^2$.

  1. For a point $x$ in the domain $\Omega$, consider the eigenvectors corresponding to the positive eigenvalues of the matrix $A$. The plane spanned by these eigenvectors is called the the plane of ellipticity at $x$.

  2. A curve $l$ is called the line of ellipticity for the equation if in a neighborhood of each of its points, there is a $C^1$ vector field $Y=(Y_1,Y_2,\ldots,Y_m)$ lying in the plane of ellipticity at that point, $(AY,Y)> 0$, and the curve $l$ is a trajectory of the system of equations given by $\frac{dX}{dt}=Y$.

  3. A set of elliptic connectivity of the given equation is a maximal set satisfying the property that any two points in the set can be joined by finitely many arcs of the lines of ellipticity of the equation.

The following theorem holds for the propagation of zeros:

Theorem: If $u\geq 0$ and $Lu\leq 0$ in $\Omega$ and if $u=0$ at a point $x_0$ then $u=0$ in the set of elliptic connectivity containing the point $x_0$.

The second theorem is a strong maximum principle for the equation:

Theorem: Suppose $Lu\geq 0$ in $\Omega$ and the coefficient $c$ and $M=\sup_\Omega u$ satisfy $Mc\leq 0$. If $u(x_0)=0$ and $x_0\in\Omega$ then either $u=0$ or $u=M$ and $c=0$ in the set of elliptic connectivity containing the point $x_0$.


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