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Chang-Yu Guo

Publications and source records attributed to Chang-Yu Guo.

44 records · Page 3Linked to original sources
arXiv2016

Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension

We show that all the common definitions of quasiregular mappings $f\colon M\to N$ between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if $f$ is $1$-quasiregular according to one of the definitions, then it is also $1$-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of $1$-quasiconformal mappings. Our main results answer affirmatively a few open questions from the recent research. The main new ingredient in our proofs is the distortion estimates for particular local extensions of the horizontal metrics. These extensions are named "Popp extensions", and based on these extensions, we introduce a new natural and invariant definition of quasiregularity in the equiregular subRiemannian setting. The analysis on Popp extensions and on the implied distortion is also of independent interest.

math.DG↗
arXiv2016

Inverse problems for $p$-Laplace type equations under monotonicity assumptions

We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $σ_1 \geq σ_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.

math.AP↗
arXiv2016

Quantitative uniqueness estimates for $p$-Laplace type equations in the plane

In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1 \max\{p,2\}$ or $q=p>2$, if $\|u\|_{L^\infty(\mathbb{R}^2)}\leq C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1} |u(z)| \geq e^{-CR^{1-\frac{2}{q}}\log R}, \] where $C$ depends only on $p$, $q$, $\tilde{M}$ and $C_0$. When $q=\max\{p,2\}$ and $p\in (1,2]$, under similar assumptions, we have \[ \inf_{|z_0|=R} \sup_{|z-z_0|<1} |u(z)| \geq R^{-C}, \] where $C$ depends only on $p$, $\tilde{M}$ and $C_0$. As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equation. We also prove the SUCP for the weighted $p$-Laplace equation with a locally positive locally Lipschitz weight.

math.AP↗
arXiv2015

Mappings of finite distortion: boundary extensions in uniform domains

In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are exponentially integrable with quantitative bounds. This extends previous results of Chang and Marshall on analytic functions, Poggi-Corradini and Rajala and Akkinen and Rajala on mappings of bounded and finite distortion.

math.CV↗
arXiv2014

Sharp capacity estimates in s-John domains

It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney cube $Q_0$ in terms of the Hausdorff $q$-content of $E$ in an $s$-John domain $Ω$. Moreover, we construct several examples to show the essential sharpness of our estimates.

math.CV↗
arXiv2014

Fractional Sobolev-Poincare inequalities in irregular domains

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, we establish (essentially) sharp fractional (q,p)-Sobolev-Poincare inequality in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tends to the results for the usual derivative. Furthermore, we verified that those domains that support the fractional (q,p)-Sobolev-Poincare inequality together with a separation property are s-diam John domains for certain s, depending only on the associated data. We also point out an inaccurate statement in [2].

math.FA↗