Abstract:We study the Cauchy--Dirichlet problem for parabolic complex Hessian equations on Hermitian manifolds and on bounded strictly m-pseudoconvex domains. In the smooth setting, we prove global existence and uniqueness of classical solutions under the presence of an admissible parabolic subsolution, by establishing a priori estimates up to the parabolic boundary. The estimates combine parabolic boundary techniques for complex Hessian equations with interior second order estimates and a blow-up argument.
We then develop a general pluripotential framework for degenerate right-hand sides with L^p densities, p>n/m, and bounded Cauchy--Dirichlet data. Since the usual automorphism and Walsh-type arguments do not directly apply in a variable Hermitian background, we use approximation by smooth data, balayage, parabolic Perron envelopes, and a continuous obstacle approximation based on Harvey--Lawson--Plis subequation theory. The resulting solution is continuous for positive time, locally uniformly Lipschitz and semi-concave in time, and continuous up to the initial slice when the initial datum is continuous. We also prove a parabolic comparison principle via time regularization, Riemann sum approximations, and mixed Hessian inequalities.
From: Haoyuan Sun [view email]
[v1]
Thu, 18 Jun 2026 02:24:09 UTC (68 KB)
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