\(C^1\)-regularity for degenerate diffusion equations. (English) Zbl 1500.35070
Summary: We prove that any solution of a degenerate elliptic PDE is of class \(C^1\) provided the inverse of the equation’s degeneracy law satisfies an integrability criterium, viz. \(\sigma^{- 1} \in L^1 (\frac{1}{\lambda} \mathbf{d} \lambda)\). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate \(u(x)\), near \(x_0\), by an error of order \(\text{o}(| x - x_0 |)\). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the \(C^1\)-regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35J70 | Degenerate elliptic equations |
Keywords:
regularity of solutions; degenerate elliptic equations; differentiability of solutions; fully nonlinear equationsReferences:
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