We prove a pointwise gradient estimate for the fractional $p$-Laplace equation with measure data. Let $n\ge 2$, $p>1$ and $\max\{n/p,1/p'\}<s<1$, where $p'=p/(p-1)$. Suppose $u\in W^{s,p}(\mathbb R^n)$ is a weak solution of \[ (-\Delta_p)^s u=\mu \quad\text{in }\Omega, \] with $\mu\in\mathcal M_{\mathrm{loc}}(\Omega)$. Set $\gamma=s-(p-1)/p$. If $B_{2R}(x_0)\Subset\Omega$ and $\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)<\infty$, then $u$ is Fr\'echet differentiable at $x_0$, and \[|\nabla u(x_0)|\le C\bigl[\mathcal A(u;x_0,2R)+\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)\bigr], \] where $\mathcal{W}_{\gamma,p}^{|\mu|}$ denotes the truncated Wolff potential, and $\mathcal A$ depends on the local oscillation of $u$ and its nonlocal tail. The proof uses a comparison with fractional $p$-harmonic replacements and an affine decay estimate for homogeneous solutions, with constants independent of the affine slope. In the large slope regime, the affine decay estimate follows from a Schauder estimate for the linear nonlocal equation satisfied by the affine remainder.
Let $n\ge2$, $1p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,\alpha}$ for some $\alpha=\alpha(n,p,s)>0$. This settles the open problem of interior gradient H\"older regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an a...
We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally H\"older $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this visco...
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...
We consider positive weak solutions of $-\Delta_p u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Dam...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study the finite element approximation of the Dirichlet problem for the Bessel $(p,s)$-Laplacian, $\operatorname{div}^{s} \left(|\nabla^{s} u|^{p-2}\nabla^{s} u\right)$, built from the Riesz fractional gradient $\nabla^s$ and posed on the Bessel potential space $X^{s,p}$, obtained by complex interpolation between $L...
J. P. Borthagaray, Jos'e Camilo Rueda Nino· 0 citations
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