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Yi Ru-Ya Zhang

Publications and source records attributed to Yi Ru-Ya Zhang.

At least 37 records · Page 2Linked to original sources
arXiv2025

A growth estimate for the planar Mumford--Shah minimizers at a tip point: An alternative proof of David--Léger

Let $Ω\subset \mathbb R^2$ be a bounded domain and $u\in SBV(Ω)$ be a local minimizer of the Mumford--Shah problem in the plane, with $0\in \overline{S}_u$ being a tip point and $B_1\subset Ω$. Then there exist absolute constants $C>0$ and $0<r_0<1$ such that $$|u(x)-u(0)|\le C r^{\frac 1 2} \quad \text{ for any } \ x\in B_r \ \text{ and } \ 0<r<r_0. $$ This estimate is a local version of the original one in \cite[Proposition 10.17]{DL2002}. Our result is based on a dichotomy and the John structure of $Ω\setminus \overline{S}_u$, different from the one by David--Léger \cite{DL2002} or Bonnet--David \cite[Lemma 21.3]{BD2001}.

math.AP↗
arXiv2024

A geometric characterization of planar Sobolev extension domains

We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $Ω\subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus Ω$ can be connected with a curve $γ\subset \mathbb R^2 \setminus Ω$ satisfying $$\int_γ dist(z,\partial Ω)^{1-p}\, dz \lesssim |z_1-z_2|^{2-p}.$$ Combined with known results, we obtain the following duality result: a Jordan domain $Ω\subset \mathbb R^2$ is a $W^{1,p}$-extension domain, $1 < p < \infty$, if and only if the complementary domain $\mathbb R^2 \setminus \barΩ$ is a $W^{1,p/(p-1)}$-extension domain.

math.CA↗
arXiv2023

Strong stability of convexity with respect to the perimeter

Let $E\subset \mathbb R^n$, $n\ge 2$, be a set of finite perimeter with $|E|=|B|$, where $B$ denotes the unit ball. When $n=2$, since convexification decreases perimeter (in the class of open connected sets), it is easy to prove the existence of a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) \ge c\,|EΔF|, \qquad c>0. $$ Here we prove that, when $n\ge 3$, there exists a convex set $F$, with $|E|=|F|$, such that $$ P(E) - P(F) \ge c(n) \,f\big(|EΔF|\big), \qquad c(n)>0,\qquad f(t)=\frac{t}{|\log t|} \text{ for }t \ll 1. $$ Moreover, one can choose $F$ to be a small $C^2$-deformation of the unit ball. Furthermore, this estimate is essentially sharp as we can show that the inequality above fails for $f(t)=t.$ Interestingly, the proof of our result relies on a new stability estimate for Alexandrov's Theorem on constant mean curvature sets.

math.OC↗
arXiv2023

Regularity from $p$-harmonic potentials to $\infty$-harmonic potentials in convex rings

The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the $C^1$ and higher-order regularity of $\infty$-harmonic potentials -- a specialized category of $\infty$-harmonic functions. Additionally, it prompts questions regarding their corresponding approximations using $p$-harmonic potentials. It is worth noting that establishing $C^1$ and higher-order regularity for $\infty$-harmonic functions remains a central concern within the realm of $\infty$-Laplace equations and $L^\infty$-variational problems. In this study, we investigate the regularity properties from $p$-harmonic potentials to $\infty$-harmonic potentials within arbitrary convex ring domains $Ω=Ω_0\backslash \overline Ω_1$ in $\mathbb R^n$. Here $Ω_0$ is a bounded convex domain in $\mathbb R^n$ and $\overlineΩ_1\subset Ω_0$ is a compact convex set. We prove the interior $C^1$ and some Sobolev regularity for $\infty$-harmonic potentials.

math.AP↗
arXiv2022

Jacobian determinants for (nonlinear) gradient of planar $\infty$-harmonic functions and applications

In dimension 2, we introduce a distributional Jacobian determinant $\det DV_β(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto |Dv|^β(v_{x_1},-v_{x_2})$ for any $β>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $β|Dv|^{1+β}\in W^{1,2}_{\text{loc}}$. Then for any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. (i) Applying this result with $β=0$, we develop an approach to build up a Liouville theorem, which improves that of Savin [33]. Precisely, if $u$ is $\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$ \liminf_{R\to\infty}\inf_{c\in\mathbb R}\frac1 {R^3}\int_{B(0,R)}|u(x)-c|\,dx<\infty,$$ then $u=b+a\cdot x$ for some $b\in{\mathbb R}$ and $a\in{\mathbb R}^2$. (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_β(Du_p) \to \det DV_β(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_β(Du_p)$ is always quasiregular mappings, but $V_β(Du)$ is not in general.

math.AP↗
arXiv2022

Interior Hölder regularity for stable solutions to semilinear elliptic equations up to dimension 5

Let $2\le n\le 5$. We establish an apriori interior Hölder regularity of $C^2$-stable solutions to the semilinear equation $-Δu=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex in addition,we obtain an interior Hölder regularity, and hence the local boundedness, of $W^{1,2}(Ω)$-stable solutions by locally approximating them via $C^2(Ω)$-stable solutions. In particular, we do not require any lower bound on $f$.

math.AP↗
arXiv2021

Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in $\mathbb R^n$ with $n\ge10$

Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $Ω\subset \mathbb R^n$, we prove that any stable solution to the equation $-Δu=f(u)$ in $Ω$ satisfies a BMO interior regularity when $n=10$, and an Morrey $M^{p_n,4+2/(p_n-2)}$ interior regularity when $n\ge 11$, where $$p_n=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}. $$ This result is optimal as hinted by earlier results, and answers an open question raised by Cabré, Figalli, Ros-Oton and Serra. As an application, we show a sharp Liouville property: Any stable solution $u \in C^2(\mathbb R^n)$ to $-Δu=f(u)$ in $\mathbb R^n$ satisfying the growth condition, i.e.\ $|u(x)|= o\left( \log|x| \right)$ as $|x|\to+\infty$ when $n=10$; or $|u(x)|= o\left( |x| ^{ -\frac n2+\sqrt{n-1}+2 }\right)$ as $x|\to+\infty$ when $n\ge 11$, must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas.

math.AP↗
arXiv2020

Duality of capacities and Sobolev extendability in the plane

We reveal relations between the duality of capacities and the duality between Sobolev extendability of Jordan domains in the plane, and explain how to read the curve conditions involvd in the Sobolev extendability of Jordan domains via the duality of capacities. Finally as an application, we give an alternative proof of the necessary condition for a Jordan planar domain to be $W^{1, q}$-extension domain when $2 < q < \infty$.

math.CV↗
arXiv2019

Strong stability for the Wulff inequality with a crystalline norm

Let $K$ be a convex polyhedron and $\mathscr F$ its Wulff energy, and let $\mathscr C(K)$ denote the set of convex polyhedra close to $K$ whose faces are parallel to those of $K$. We show that, for sufficiently small $ε$, all $ε$-minimizers belong to $\mathscr C(K)$. As a consequence of this result we obtain the following sharp stability inequality for crystalline norms: There exist $γ=γ(K,n)>0$ and $σ=σ(K,n)>0$ such that, whenever $|E|=|K|$ and $|EΔK|\le σ$ then $$ \mathscr F(E) - \mathscr F(K^a)\ge γ|E ΔK^a| \qquad \text{for some }K^a \in \mathscr C(K). $$ In other words, the Wulff energy $\mathscr F$ grows very fast (with power $1$) away from the set $\mathscr C(K).$ The set $K^a \in \mathscr C(K)$ appearing in the formula above can be informally thought as a sort of "projection" of $E$ on the set $\mathscr C(K).$ Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of $\mathscr F(E)+\int_E g$ with small mass are polyhedra with sides parallel to the ones of $K$. In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers. This extends to arbitrary dimensions a two-dimensional result obtained in [9].

math.OC↗
arXiv2019

Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality

Denote by $Δ$ the Laplacian and by $Δ_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {Δv Δ_\infty v } -\frac12[|D^2v|^2-(Δv)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-γ}2}Du\in W^{1,2}_{\rm loc},$$ with $γ<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm loc}$-regularity of $p$-harmonic functions for some $q>2$. 2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when $n=2$ an open question in [17] is completely answered. 3. When $n\ge 1 $ and $p\in(1,2)\cup(2,3)$, the weak/viscosity solutions to parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\rm loc}^{2,2}$-regularity.

math.AP↗
arXiv2019

C^1-Regularity of planar \infty-harmonic functions - REVISIT

In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high technique arguments therein. Our argument is essentially based on a topological lemma of Savin, a flat estimate by Evans and Smart, % \cite{es11a}, $W^{1,2}_{loc}$-regularity of $|Du|$ and Crandall's flow for infinity harmonic functions.

math.AP↗
arXiv2018

Schoenflies solutions of conformal boundary values may fail to be Sobolev

We show that there exists a planar Jordan domains $Ω$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $φ\colon \mathbb D \to Ω$, any homeomorphic extension of $φ$ or $φ^{-1}$ to the entire plane is not in $W^{1,\,1}_{\rm loc}$ (or even not in $BV_{\rm loc}$).

math.CV↗
arXiv2018

An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

Given an arbitrary planar $\infty$-harmonic function $u$, for each $α>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^α$, which is sharp as $α\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\infty$-harmonic functions in whole plane.

math.AP↗
arXiv2018

Some sharp Sobolev regularity for inhomogeneous $\infty$-Laplace equation in plane

Suppose $Ω\Subset \mathbb R^2$ and $f\in BV_{loc}(Ω)\cap C^0(Ω)$ with $|f|>0$ in $Ω$. Let $u\in C^0(Ω)$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$ -Δ_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ Ω. $$ The following are proved in this paper. (i) For $ α> 3/2$, we have $|Du|^α\in W^{1,2}_{loc}(Ω)$, which is (asymptotic) sharp when $ α\to 3/2$. Indeed, the function $w(x_1,x_2)=-x_1^ {4/3} $ is a viscosity solution to $-Δ_\infty w=\frac{4^3}{3^4}$ in $\mathbb R^2$. For any $p> 2$, $|Dw|^α\notin W^{1,p}_{loc}(\mathbb R^2)$ whenever $α\in(3/2,3-3/p)$. (ii) For $ α\in(0, 3/2]$ and $p\in[1, 3/(3-α))$, we have $|Du|^α\in W^{1,p}_{loc}(Ω)$, which is sharp when $p\to 3/(3-α)$. Indeed, $ |Dw|^α\notin W^{1,3/(3-α)}_{loc}(\mathbb R^2)$. (iii) For $ ε> 0$, we have $|Du|^{-3+ε}\in L^1_{loc}(Ω)$, which is sharp when $ε\to0$. Indeed, $|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$. (iv) For $ α> 0$, we have $$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$ Some quantative bounds are also given.

math.AP↗
arXiv2016

A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_α(\mathbb R^n)$ depends on the index $α$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $α$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-Väisälä's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_α (\mathbb R^n)$ for any $(α,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_α(\mathbb R^n)$ is shown to be invariant under inversions for all $0<α<1$.

math.FA↗