Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains

Svitlana Mayboroda; Alberto Pacati

Home/Archive/arXiv

arXiv · 2610.19983

Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains

Svitlana Mayboroda·Alberto Pacati

View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a boundary value and potential theory for elliptic operators \[ L=-\Div(A\nabla)+\B\cdot\nabla \] with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\Div\B$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\Div\B\le0$, we construct a Green's function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\Div\B=0$ or $\B\in L^{n,1}_{\textup{loc}}(Ω)$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green's function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-Δ+\B\cdot\nabla$ with $\B$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Svitlana Mayboroda, Alberto Pacati. 2026-10-14. Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains. https://arxiv.org/abs/2610.19983

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equation in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem \ref{thm1}). Furthermore, by carefully analyzing the asymptotic expansion of the Dirichlet energy $\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2$ as $λ\to 0$, and establishing that \[ \lim_{\substack{λ\to 1/4}} \int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u_λ|^2 \, dV_{\mathbb{B}^2} = +\infty \] for~the ~nonzero~least energy solution, we are able to prove the existence of a critical point for the Moser-Trudinger functional $F(u) = \int_{\mathbb{B}^2} (e^{u^2} - 1) \, dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u|^2 \, dV_{\mathbb{B}^2} = Λ$ for $Λ\in (4π, +\infty)$, up to some Möbius transformation. This result is substantially different from the non-existence of critical points for the Moser-Trudinger functional in the Euclidean disk. Finally, by applying a local Pohozaev identity around the origin and exploiting the uniqueness of solutions to the Cauchy initial value problem for ODEs, we prove that the Moser-Trudinger equation has a unique positive solution up to some Möbius transformation when $λ$ is close to $0$.

math.AP↗

arXiv2026

Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

math.AP↗

arXiv2026

Lipschitz regularity of fractional $p$-Laplacian

In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-Δ_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(Ω)$. Specifically, we prove that $u\in C^{0, γ_\circ}_{\rm loc}(Ω)$ for $γ_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.

math.AP↗