English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this article, we study the order of vanishing and a quantitative form ofLandis' conjecture in the plane for solutions to second-order ellipticequations with variable coefficients and singular lower order terms. Precisely,we let $A$ be real-valued, bounded and elliptic, but not necessary symmetric orcontinuous, and we assume that $V$ and $W_i$ are real-valued and belong to$L^p$ and $L^{q_i}$, respectively. We prove that if $u$ is a real-valued,bounded and normalized solution to an equation of the form $-\nabla \cdot (A\nabla u + W_1 u) + W_2 \cdot \nabla u + V u = 0$ in $B_d$, then under suitableconditions on the lower order terms, for any $r$ sufficiently small, thefollowing order of vanishing estimate holds $$\|u\|_{L^\infty(B_r)} \ge r^{CM},$$ where $M$ depends on the Lebesgue norms of the lower order terms. In anumber of settings, a scaling argument gives rise to a quantitative form ofLandis' conjecture, \[ \inf_{|z_0| = R} \|u\|_{L^\infty(B_1(z_0))} \ge \exp(- CR^\beta \log R), \] where $\beta$ depends on $p$, $q_1$, and $q_2$. Theintegrability assumptions that we impose on $V$ and $W_i$ are nearly optimal inview of a scaling argument. We use the theory of elliptic boundary valueproblems to establish the existence of positive multipliers associated to theelliptic equation. Then the proofs rely on transforming the equations toBeltrami systems and applying a generalization of Hadamard's three-circletheorem.

Landis' conjecture for general second order elliptic equations with singular lower order terms in the plane