A systematic approach on the second order regularity of solutions to the general parabolic p-Laplace equation
Feng, Y., Parviainen, M., & Sarsa, S. (2023). A systematic approach on the second order regularity of solutions to the general parabolic p-Laplace equation. Calculus of Variations and Partial Differential Equations, 62, Article 204. https://doi.org/10.1007/s00526-023-02537-z
© The Author(s) 2023
We study a general form of a degenerate or singular parabolic equation ut−|Du|γ(Δu+(p−2)ΔN∞u)=0 that generalizes both the standard parabolic p-Laplace equation and the normalized version that arises from stochastic game theory. We develop a systematic approach to study second order Sobolev regularity and show that D2u exists as a function and belongs to L2loc for a certain range of parameters. In this approach proving the estimate boils down to verifying that a certain coefficient matrix is positive definite. As a corollary we obtain, under suitable assumptions, that a viscosity solution has a Sobolev time derivative belonging to L2loc.
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Additional information about fundingOpen Access funding provided by University of Jyväskylä (JYU). The first author was supported by China Scholarship Council, no. 202006020186. The third author was supported by the Academy of Finland, Center of Excellence in Randomness and Structures and the Academy of Finland, project 308759.
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