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 DiBenedetto, Gianazza and Vespri. In contrast to the exponential change of variables and expansion-of-positivity used in the local case, our technique is nonlocal and relies on comparison arguments and barriers in a variational framework. Indeed, we obtain the stronger conclusion where the forward comparison starts immediately after the reference time, without an additional time gap, which is false in the local case. The latter leads to elliptic-type Harnack estimates for parabolic equations. Our result is new even in the linear case.
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouv...
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, th...
Matheus N. Soares, Weiller F. Chaves Barboza· 0 citations
We consider a general, nonsymmetric Ornstein--Uhlenbeck semigroup $(\mathcal H_t)_{t>0}$. We prove an $L^p$ bound for the jump quasi-seminorms for $1<p<\infty$ and a weak type (1,1) oscillation inequality, both with respect to the invariant measure. These results are established for the order $\varrho=2$. To do so, we...
Valentina Casarino, P. Ciatti, Peter Sjögren· 0 citations
In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1<p\leq2$. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fracti...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
Wen-Guo Liang, Zheng-Ce Zhang· 0 citations
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