Degenerate elliptic equation

2020 Mathematics Subject Classification: Primary: 35J70 [MSN][ZBL]

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A partial differential equation \begin{equation} \label{eq1} F(x,Du) = 0 \end{equation} where the real-valued function $F(x,q)$ satisfies the condition \begin{equation} \label{eq2} \sum_{\abs{\alpha} = m} \frac{\partial F(x,Du)}{\partial q_\alpha} \xi^\alpha \geq 0 \end{equation} for all real $\xi$, and there exists a $\xi \neq 0$ for which \ref{eq2} becomes an equality. Here, $x$ is an $n$-dimensional vector $(x_1,\ldots,x_n)$; $u$ is the unknown function; $\alpha$ is a multi-index $(\alpha_1,\ldots,\alpha_n)$; $Du$ is a vector with components $$ D^\alpha u = \frac{\partial^{\abs{\alpha}}u}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}; $$ the derivatives in equation \ref{eq1} are of an order not exceeding $m$; the $q_\alpha$ are the components of a vector $q$; $\xi$ is an $n$-dimensional vector $(\xi_1,\ldots,\xi_n)$; and $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n} $. If strict inequality in equation \ref{eq2} holds for all $x$ and $Du$ and for all real $\xi \neq 0$, equation \ref{eq1} is elliptic at $(x,Du)$. Equation \ref{eq1} degenerates at the points $(x,Du)$ at which inequality \ref{eq2} becomes an equality for any real $\xi \neq 0$. If equality holds only on the boundary of the domain under consideration, the equation is called degenerate on the boundary of the domain. The most thoroughly studied equations are second-order degenerate elliptic equations $$ \sum a^{ik}(x) u_{x_i x_k} + \sum b^i(x) u_{x_i} + c(x)u = f(x), $$ where the matrix $\left[ a^{jk}(x) \right]$ is non-negative definite for all $x$-values under consideration.

See also Degenerate partial differential equation and the references given there.