We establish a parameterized Heintze--Karcher-type inequality for positive solutions of nonlinear Dirichlet problems involving the $p$-Laplacian, with $p\geq2$, on bounded Riemannian domains. Under a Ricci curvature lower bound, positive boundary mean curvature, and suitable structural and approximation assumptions, the estimate follows from a regularized Reilly-type identity and a weighted Hessian inequality. We compare the resulting bound with previous estimates and investigate the geometry associated with equality. For $p=2$, an exact deficit identity allows the pointwise condition $f'\leq nk$ to be replaced by a weighted integral condition. As applications, we obtain Heintze--Karcher and Soap Bubble-type rigidity results, with equality forcing the domain to be a metric ball and the solution to be radial.
We establish a sharp Faber-Krahn inequality for the normalized (or game-theoretic) $p$-Laplacian, proving that, for $p\geq 2$, the ball uniquely minimizes the principal Dirichlet eigenvalue among bounded connected Lipschitz domains of prescribed volume satisfying a uniform exterior sphere condition. More generally, we...
We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBen...
We prove a Faber-Krahn inequality for the first eigenvalue of the $p$-Hermite operator (the weighted $p$-Laplacian with Gaussian weight) on Lipschitz domains in $\R^n$ under Robin boundary conditions with positive Robin parameter. The main result states that, among all domains of given Gaussian measure, the first eigen...
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequal...
We prove a scale-invariant boundary Harnack inequality and H\"older continuity up to the boundary of quotients of positive fractional $p$-harmonic functions in Reifenberg flat domains with sufficiently small flatness constant, for $0<s<1$ and $2\le p<2/(1-s)$. The solutions are globally nonnegative, vanish on a common...
We prove a weak $(1,1)$ estimate for the full Poisson Littlewood--Paley $g$-function on every complete Riemannian manifold without boundary, with an explicit universal constant approximately equal to $3.33606$. The estimate requires no curvature restriction, volume doubling, heat kernel bound, or stochastic completenes...
Bo Li, Ji Li, Jin-Xia Li· 0 citations
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