Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients
Lyudmila Korobenko, Cristian Rios, Eric T. Sawyer, Ruipeng Shen · Memoirs of the American Mathematical Society · 2021
We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form inhomogeneous equations, and also continuity of weak solutions to homogeneous equations. For example, we consider the family { f σ } σ > 0 \left \{ f_{\sigma }\right \} _{\sigma >0} with f σ ( x ) = e − ( 1 | x | ) σ , − ∞ > x > ∞ , \begin{equation*} f_{\sigma }\left ( x\right ) =e^{-\left ( \frac {1}{\left \vert x\right \vert }\right ) ^{\sigma }},\ \ \ \ \ -\infty >x>\infty , \end{equation*} of infinitely degenerate functions at the origin, and show that all weak solutions to the associated infinitely degenerate quasilinear equations of the form d i v A ( x , u ) g r a d u = ϕ ( x ) , A ( x , z ) ∼ [ I n − 1 a m