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Yi Ru-Ya Zhang

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

At least 19 recordsLinked to original sources
arXiv2026

A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains

Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematically on modern techniques such as those developed in \cite{FZ2022}. We then give two globalization schemes on bounded, connected domains. For bounded John domains, instead of Reshetnyak's original construction \cite{R19762}, which glues Möbius transformations on adjacent Whitney cubes, we use Whitney chains and Bojarski's enlarged-cube estimate to obtain a quantitatively sharp Euclidean stability result. Finally, we extend this idea to general bounded, connected open subsets of $\mathbb{R}^n$ via the Lorentzian representation of the Möbius group, yielding a compactified, weighted stability theorem whose underlying measure is absolutely continuous with respect to Lebesgue measure and has density bounded above by one.

math.CV↗
arXiv2026

An Almgren-type formula for planar $p$-harmonic functions

For a nonconstant planar $p$-harmonic function $u$, with $1 < p < \infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$ N_*(x_0,r) = \frac{r\displaystyle\int_{B_r(x_0)} |D u|^p\,dx} {\displaystyle\int_{\partial B_r(x_0)} |D u|^{p-2}(u-u(x_0))^2\,dS}. $$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.

math.AP↗
arXiv2026

Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

We investigate positive weak solutions of the critical $p$-Laplace equation $$ -Δ_p u = u^{p^*-1} h(u), \qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate $$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $$ for nonnegative weak solutions defined in $B_{3R}$. As a direct corollary, we obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin--Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.

math.AP↗
arXiv2026

Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets

We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity $e^u$ and the family of shifted power-type nonlinearities $(a+u)^p$, where $a>0$ and $p>1$. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.

math.AP↗
arXiv2026

An anisotropic Serrin's problem in general domains

Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more generally rough) domains in the weak formulation has been a long-standing and challenging problem, recently resolved by the authors in~\cite{FZ2025}. In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex $C^{2,γ}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $Δ_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $Ω$. Assuming the Ahlfors--David regularity of $\partial^*Ω$ and a global $β$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $Ω$ is a translate and dilation of {the reflected Wulff shape $-K$}, in which case the solution is unique and explicit. In particular, the result applies to Lipschitz domains. While our approach follows the rough-domain strategy of~\cite{FZ2025} at a high level, the key Laplacian-specific ingredients exploited there have no direct analog for $Δ_H$, necessitating the development of new ideas and techniques.

math.AP↗
arXiv2026

Sharp stability of Alexandrov's theorem for $C^1$ domains in the small-excess regime

We prove a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension for bounded $C^1$ open sets in a small-excess regime. More precisely, if $E\subset \mathbb R^n$ is a bounded $C^1$ open set with the same volume as the unit ball $B$, small excess, and scalar distributional mean curvature $\mathcal H_{\partial E}\in L^2(\partial E)$, then, up to a translation, $$ \operatorname{Exc}(E)+|EΔB|^2+|μ-(n-1)|^2 \le C(n)\|\mathcal H_{\partial E}-μ\|_{L^2(\partial E)}^2 \qquad \forall\,μ\in \mathbb R. $$ In other words, both the excess and the symmetric difference from the ball are controlled by the optimal $L^2$-oscillation of the mean curvature. This yields a sharp stability estimate in a genuinely non-parametric regime. The proof combines a $BV$ version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region. We note that the $C^1$ regularity assumption enters only as a qualitative technical ingredient of the proof, but all constants in the final estimate depend only on the dimension.

math.DG↗
arXiv2026

Serrin's overdetermined theorem and weak Bernoulli laws without Alt--Caffarelli regularity

We study distributional Bernoulli-type conditions in geometrically irregular domains $Ω$. Here the zero extension of $u$ to $\mathbb{R}^n$ satisfies $$ Δu \;=\; c\,\mathcal{H}^{n-1}\!\lfloor_{\partial^*Ω}\;-\;f(u)\,\mathbf{1}_Ω\,dx $$ in the distributional sense. This is a weak version of the one-phase Bernoulli free boundary condition, which avoids the uniform Lipschitz/density assumptions of classical Alt--Caffarelli theory. We prove that for every $n \ge 2$ and every $f \in C^2(\mathbb{R})$ with $f(0)>0$, there exist bounded, non-spherical, finite-perimeter domains $Ω\subset \mathbb R^n$ satisfying this distributional Bernoulli law with $$ 0<\mathcal H^{n-1}(\partial^*Ω)<\infty, \qquad \mathcal H^{n-1}(\partialΩ\setminus\partial^*Ω)=0, $$ yet $$ {\rm ess}\sup_{x\in\partial^*Ω} \sup_{0<r<1} \frac{\mathcal H^{n-1}(B_r(x)\cap\partial^*Ω)}{r^{n-1}} =\infty. $$ This shows the key constraint is not absence of a reduced boundary, but failure of uniform all-scale surface density bounds. For $f \equiv 1$, these results yield counterexamples to the weak Serrin-type overdetermined problems in all dimensions, proving the distributional Bernoulli law alone cannot replace the uniform growth/density conditions core to Alt--Caffarelli theory. On the other hand, we prove a planar rigidity result: Within the Smirnov class, the associated harmonic quadrature identity forces $Ω$ to be a disk. Thus, for the constant-source Serrin/Bernoulli law, Smirnov regularity is the threshold for weak Bernoulli rigidity in $\mathbb R^2$, while uniform upper density bounds form the threshold for $n\ge 3$ according to [23].

math.AP↗
arXiv2026

A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$ -Δ_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin--Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

math.AP↗
arXiv2026

Planar $W^{1,\,1}$-extension domains

We show that a bounded planar simply connected domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $Ω^c$ there exists a curve $γ\subset Ω^c$ connecting $x$ and $y$ with $$ \int_γ\frac{1}{χ_{\mathbb R^2\setminus \partialΩ}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if it is a $BV$-extension domain, and if and only if its complementary domain $\tilde Ω$ is a $W^{1,\,\infty}$-extension domain.

math.FA↗
arXiv2026

Bounded Continuous weak quasiregular mappings that fail to be quasiregular

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain $Ω\subset\mathbb R^n$ and every $1\leq p<nK/(K+1)$, we construct a bounded continuous weakly $K$-quasiregular mapping $$ f\in W^{1,\,p}(Ω;\,\mathbb R^n)\cap C(Ω;\,\mathbb R^n) \cap L^\infty(Ω;\mathbb R^n) $$ which fails to be quasiregular. We further construct weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the maximal size permitted by their Sobolev regularity. These examples show that, the almost-everywhere sign condition on the Jacobian is too weak to serve as an orientation-preserving hypothesis below $W^{1,n}$. In contrast, we show that, for $n-1<p<n$, quasiregularity follows once this condition is replaced by a one-sided condition on the distributional degree (together with boundedness).

math.CV↗
arXiv2026

Geometric properties of Euclidean domains supporting trace inequalities

We investigate the geometric behavior of $τ(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $τ(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key ingredient in proving a quantitative isoperimetric inequality with the optimal exponent. We first show that for every $ε>0$ one can find a bounded open set $Ω\subset \mathbb R^n$ that is very close to the unit ball $\mathbb B^n$ in the sense that $$ τ(\mathbb B^n)>τ(Ω)>τ(\mathbb B^n)-ε\quad \text{and} \quad P(ΩΔ\mathbb B^n)\le C(n)ε, $$ while at the same time the complement of $Ω$ has infinitely many connected components. Thus, $τ(Ω)$ can be made arbitrarily close to $τ(\mathbb B^n)$ even when $Ω$ has highly intricate geometry. We then establish, under a mild additional hypothesis, the equivalence between a condition formulated in terms of $τ$ and two classical criteria from the literature for open sets that admit trace inequalities. As a consequence, we obtain the John-type characterization of domains that support a trace inequality, assuming the ball separation property.

math.FA↗
arXiv2026

Sobolev trace inequalities on John domains and its applications

We prove that a trace inequality holds for John domains $Ω$ satisfying $$ \mathcal H^{n-1}(\partial Ω\setminus \partial_*Ω)=0,$$ where $\partial_*Ω$ denotes the measure-theoretic boundary, together with an upper density bound on $\partial Ω$. This class of domains includes $(ε,\,r)$-perimeter minimizers of Wulff perimeter $P_K$ which are close to the associated convex body $K$. Particularly, this result is established without requiring $\partial Ω$ to be Ahlfors regular. As a consequence, we give an alternative proof for a crucial step in the quantitative Wulff inequality, thereby providing a meaningful commentary on the seminal work of Figalli, Maggi, and Pratelli \cite{FMP2010}.

math.OC↗
arXiv2026

Global regularity and free boundary geometry in the planar Choné-Rochet model

In this paper, we study minimizers of the Choné--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly convex domains or, more generally, whenever the zero set of $u$ has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global $C^1$ regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a $C^1$ embedded curve, significantly strengthening previously known regularity results.

math.AP↗
arXiv2026

Serrin's overdetermined theorem within Lipschitz domains

Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $Ω$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus Ω,\quad Δu=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$ in the weak sense if and only if $Ω$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1].

math.AP↗
arXiv2026

Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For $n=2$, there exist Morse index $1$ solutions whose $L^\infty$ norm goes to infinity. - For $n \geq 3$, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in $L^\infty$. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori $L^\infty$ bound for finite Morse index solution in the sharp dimensional range $3\leq n\leq 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method. [In the previous version, the proof of Proposition A.1 was not complete. Indeed, Lemma A.2 was proved only for estimates centered at the origin, and Footnote 9 claimed that the same estimates hold at every point; however, no complete proof was provided for this extension. In the current version, Lemma A.2 is restricted to estimates at the origin, and the boundedness at arbitrary points is obtained by combining the results in [9] with the new version of Lemma A.2. This avoids the need to extend Lemma A.2 to arbitrary centers and yields a complete proof of Proposition A.1.]

math.AP↗
arXiv2025

Serrin's overdetermined problem in rough domains

The classical Serrin's overdetermined theorem states that a $C^2$ bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin's theorem to Lipschitz domains remained unresolved. This paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.

math.AP↗
arXiv2025

Carrot John domains in variational problems

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant concerning Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Techmüller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at $\infty$, appearing in the Mumford-Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev-Poincaré inequalities.

math.OC↗
arXiv2025

Sharp stability for critical points of the Sobolev inequality in the absence of bubbling

When $u$ is close to a single Talenti bubble $v$ of the $p$-Sobolev inequality, we show that \begin{equation*} \|Du-Dv\|_{L^p(\mathbb{R}^n)}^{\max\{1,p-1\}}\le C \|-{\rm div}(|Du|^{p-2}Du)-|u|^{p^*-2}u\|_{W^{-1,q}(\mathbb{R}^n)}, \end{equation*} where $C=C(n,p)>0$. This estimate provides a sharp stability estimate for the Struwe-type decomposition in the single bubble case, generalizing the result of Ciraolo, Figalli, and Maggi \cite{CFM2018} (focusing on the case $p=2$) to the arbitrary $p$. Also, in the Sobolev setting, this answers an open problem raised by Zhou and Zou in \cite[Remark 1.17]{ZZ2023}.

math.AP↗