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 viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*--to--strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overline\Omega)$ and identify the obstruction when the drift reaches the boundary.
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 affine-invariant improvement-of-flatness argument with a Liouville theorem for globally Lipschitz entire solutions. In the large-slope regime, the shifted Bregman energies converge to an anisotropic stable form of order $sp-p+2>1$. In the bounded-slope regime, the Liouville theorem follows from rigidity of extremal secants, a recurrent blow-down argument, and a directional Morrey-Kato estimate for the singular linearized kernel. An affine Campanato argument controls the variation of the best affine approximations across scales. These estimates yield a scale-invariant decay of the affine excess and hence the local $C^{1,\alpha}$ estimate.
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 Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations \textbf{25} (2006), 139--159] have been available, for more than twenty years, only under the restriction $\frac{2N+2}{N+2}<p<2$, a range that shrinks to the empty set as $N\to\infty$. We remove the dependence on the dimension and prove all of these results for $\frac32<p<2$. Since the strong comparison principle is used as a black box throughout the qualitative theory of $p$-Laplace equations, the improvement propagates. We show in particular that the assumption $\frac{2N+2}{N+2}<p<2$ may be replaced by $\frac32<p<2$ in the resolution of Gibbons'conjecture for $-\Delta_pu=f(u)$ by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ \textbf{382} (2022), 943--974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations \textbf{61} (2022), art.\ 154]; further applications are given in the body of the paper.
We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to $C^{1,p+1}(\overline\Omega)$ in space for every positive time and are $C^\infty$ in time uniformly up to the boundary. Moreover, all their time derivatives belong to $C^{1,p+1}(\overline\Omega)$, and the exponent $p+1$ is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the $C^{1,p+1}(\overline{\Omega})$ topology and an improved convergence rate for the relative error in $C^{p+1}(\overline\Omega)$.
Tian-Ling Jin, Xushan Tu, Jingang Xiong et al.· 0 citations
We develop a fixed-reference weighted $L^2$ theory for terminal-value fully nonlinear parabolic equations. A prescribed uniformly elliptic diffusion starting from a point determines both the linear reference operator and a space--time occupation measure; time weights of order $\alpha$ accommodate the point-start singularity. For the associated linear backward equation, we identify the optimal source-to-intrinsic-Hessian norm $C_w(X)$ and derive bounds on it from a moving-measure Bochner identity. We compute the sharp Brownian benchmark $C_\alpha^{\mathrm{Br}}=\sup_{n\in\mathbb{N}, n\geq 2}2\sqrt{n(n-1)}/(n-1+\alpha)$, prove that it is a universal tangent lower bound, and show that it is attained by regular weights for time-inhomogeneous affine Gaussian diffusions and is the infimum over such weights for normalized state-dependent covariances. Under the standing data and lower-order assumptions, the Hessian estimate yields our main nonlinear result: if the driver's Hessian Lipschitz constant satisfies $C_w(X)L_H<1$, then the equation has a unique weighted Sobolev solution and the Picard iteration on the source converges geometrically, with no smallness condition on the value or gradient channels. For the Brownian model at order $1/2$, counterexamples in dimensions $d\geq7$ show that this strict threshold cannot be uniformly improved within the occupation-Sobolev class. For weights of order below one, additional spatial regularity and unweighted integrability of the data yield an unweighted spatial jet and a unique solution in the regular fixed-reference Markovian class for second-order backward stochastic differential equations (2BSDEs) defined here.
We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only H\"older regular in space---$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$---thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related"refreshing"regime.
Maria Colombo, Elias Hess-Childs, Keefer Rowan· 0 citations
In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally H\"older continuous. Suppose that $Du\in C^{0,\alpha}_{\rm loc}$ with $\alpha\le 1/(p-1)$, and let $\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\Phi(Du)\in C^k_{\rm loc}$ whenever $m>k/\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/\alpha}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate H\"older assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.
Quoc-Hung Nguyen, Le Xuan Truong· 0 citations
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