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Abstract: Abstract It was conjectured that if \(f\in C^1({\mathbb {R}}^n,{\mathbb {R}}^n)\) satisfies \(\mathrm{rank\,}Df\le m<n\) everywhere in \({\mathbb {R}}^n\) , then f can be uniformly approximated by \(C^\infty \) -mappings g satisfying \(\mathrm{rank\,}Dg\le m\) everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when \(m=1\) . More precisely, if \(m=1\) , our result yields an almost-uniform approximation of locally Lipschitz mappings \(f:\Omega \rightarrow {\mathbb {R}}^n\) , satisfying \(\mathrm{rank\,}Df\le 1\) a.e., by \(C^\infty \) -mappings g with \(\mathrm{rank\,}Dg\le 1\) , provided \(\Omega \subset {\mathbb {R}}^n\) is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ( \({\mathbb {R}}\) -trees). PubDate: 2023-01-20
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Abstract: Abstract Motivated by the stability result of the Rayleigh–Bénard problem in a fixed slab domain in Jiang and Liu (Nonlinearity 33:1677–1704, 2020) and the global-in-time well-posedness of an incompressible viscoelastic fluid system with an upper free boundary in Xu et al. (Arch Ration Mech Anal 208:753–803, 2013), we further investigate the Rayleigh–Bénard problem for an incompressible viscoelastic fluid in a three-dimensional horizontally periodic domain with the lower fixed boundary and with the upper free boundary. By a careful energy method, we establish an explicit stability condition, under which the viscoelastic Rayleigh–Bénard problem has a unique global-in-time solution with exponential time-decay. Our result presents that the elasticity can inhibit the thermal instability for sufficiently large elasticity coefficient \(\kappa \) . PubDate: 2023-01-20
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Abstract: Abstract In this paper we establish boundary regularity for viscosity solutions of nonlinear singular elliptic equations $$\begin{aligned} \left\{ \begin{array}{rclll} F\left( D^2u,Du,u,\frac{u_{x_n}}{x_n},\frac{u}{x_n^2},x\right) &{}=&{}f(x)&{}\textrm{in}&{}B'_1\times (0,1),\\ u(x)&{}=&{}\phi (x')&{}\textrm{on}&{}B'_1\times \{0\}. \end{array}\right. \end{aligned}$$ We prove that if there exists a smooth approximation solution \(u_*\) which satisfies \(u-u_*=O(x_n^\mu )\) , then for any \(\epsilon >0\) , \(u\in C^{\mu -\epsilon }\) up to the boundary \(\{x_n=0\}\) . PubDate: 2023-01-20
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Abstract: Abstract We prove that the hydrodynamic pressure p associated to the velocity \(u\in C^\theta (\Omega )\) , \(\theta \in (0,1)\) , of an inviscid incompressible fluid in a bounded and simply connected domain \(\Omega \subset {\mathbb {R}}^d\) with \(C^{2+}\) boundary satisfies \(p\in C^{\theta }(\Omega )\) for \(\theta \le \frac{1}{2}\) and \(p\in C^{1,2\theta -1}(\Omega )\) for \(\theta >\frac{1}{2}\) . Moreover, when \(\partial \Omega \in C^{3+}\) , we prove that an almost double Hölder regularity \(p\in C^{2\theta -}(\Omega )\) holds even for \(\theta <\frac{1}{2}\) . This extends and improves the recent result of Bardos and Titi (Philos Trans R Soc A, 2022) obtained in the planar case to every dimension \(d\ge 2\) and it also doubles the pressure regularity. Differently from Bardos and Titi (2022), we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the d-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero. PubDate: 2023-01-20
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Abstract: Abstract We study the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for a variation of Perelman’s \(\lambda \) functional adapted to the ALE situation and denoted \(\lambda _{{\text {ALE}}}\) . This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Łojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5. PubDate: 2023-01-12
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Abstract: Abstract We generalize the classical Bochner formula for the heat flow on evolving manifolds \((M,g_{t})_{t \in [0,T]}\) to an infinite-dimensional Bochner formula for martingales on parabolic path space \(P{\mathcal {M}}\) of space-time \({\mathcal {M}} = M \times [0,T]\) . Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer–Naber (J Eur Math Soc 20(5):1269–1302, 2018a). Our results are parabolic counterparts of the recent results in the elliptic setting from Haslhofer–Naber (Commun Pure Appl Math 71(6):1074–1108, 2018b). PubDate: 2023-01-12
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Abstract: Abstract We revisit the well known prescribed scalar curvature problem $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\big (1+\varepsilon K(x)\big )u^{2^*-1}, u(x)>0,~~ &{}{x\in \mathbb {R}^N},\\ u\in \mathcal {D}^{1,2}(\mathbb {R}^N),\\ \end{array}\right. } \end{aligned}$$ where \(2^*=\frac{2N}{N-2}\) , \(N\ge 5\) , \(\varepsilon >0\) and \(K(x)\in C^1(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)\) . It is known that there are a number of results related to the existence of solutions concentrating at the isolated critical points of K(x). However, if K(x) has non-isolated critical points with different degenerate rates along different directions, whether there exist solutions concentrating at these points is still an open problem. We give a certain positive answer to this problem via applying a blow-up argument based on local Pohozaev identities and modified finite dimensional reduction method when the dimension of critical point set of K(x) ranges from 1 to \(N-1\) , which generalizes some results in Cao et al. (Calc Var Partial Differ Equ 15:403–419, 2002) and Li (J Differ Equ 120:319–410, 1995; Commun Pure Appl Math 49:541–597, 1996). PubDate: 2023-01-11
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Abstract: Abstract In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice \(\mathbb {Z}^d\) as \(d \rightarrow \infty \) . In the process, we proved some Hardy-type inequalities for the operators \(\Delta ^m\) and \(\nabla (\Delta ^m)\) for non-negative integers m on a d dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as d and \(d^2\) respectively as \( d \rightarrow \infty \) . PubDate: 2023-01-11
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Abstract: Abstract We prove that positive solutions of the superlinear Lane–Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. In addition, the boundedness may fail if the exponents are not comparable. PubDate: 2023-01-11
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Abstract: Abstract The purpose of this article is to study the strict convexity of the Mabuchi functional along a \({{\mathcal {C}}}^{1,{{\bar{1}}}}\) -geodesic, with the aid of the \(\varepsilon \) -geodesics. We proved the \(L^2\) -convergence of the fiberwise volume element of the \(\varepsilon \) -geodesic. Moreover, the geodesic is proved to be uniformly fiberwise non-degenerate if the Mabuchi functional is \(\varepsilon \) -affine. PubDate: 2023-01-11
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Abstract: Abstract We consider a free boundary problem arising in the study of the equilibrium of a confined Tokamak plasma in dimensional two. By choosing a suitable flux constant on each connected component of the boundary of the domain, we construct solutions with many sharp peaks near the boundary and prove that the number of solutions of this problem goes to infinity as parameter tends to infinity. PubDate: 2023-01-11
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Abstract: Abstract Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren’s theorem about minimal Lagrangian diffeomorphism in Euclidean metric space. PubDate: 2023-01-11
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Abstract: Abstract This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary, we reprove the classical result that two Euclidean cyclic polygons (or hyperbolic cyclic polygons) are congruent if the lengths of their sides are equal. PubDate: 2023-01-11
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Abstract: Abstract We prove two Hölder regularity results for solutions of generated Jacobian equations. First, that under the A3 condition and the assumption of nonnegative \(L^p\) valued data solutions are \(C^{1,\alpha }\) for an \(\alpha \) that is sharp. Then, under the additional assumption of positive Dini continuous data, we prove a \(C^{2}\) estimate. Thus the equation is uniformly elliptic and when the data is Hölder continuous solutions are in \(C^{2,\alpha }\) . PubDate: 2023-01-11
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Abstract: Abstract For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an \(\epsilon \) -splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding n directional points by induction and stratified almost Gou–Gu Theorem. The other one is the error estimates of projections, which guarantee the n directional points we find really determine n different directions. PubDate: 2023-01-11
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Abstract: Abstract The well known phenomenon of exponential contraction for solutions to the viscous Hamilton–Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent \(\lambda (\nu )\) characterizing the exponential rate of contraction depends on the viscosity \(\nu \) . The Markov mechanism provides only a lower bound for \(\lambda (\nu )\) which vanishes in the limit \(\nu \rightarrow 0\) . At the same time, in the inviscid case \(\nu =0\) one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for \(\lambda (\nu )\) which is valid for all \(\nu \ge 0\) . The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory. PubDate: 2023-01-11
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Abstract: Abstract In this paper, we study the existence of positive solutions to the nonlinear elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+ u_1+(\mu _{11}\phi _{u_1}-\mu _{12}\phi _{u_2})u_1=\frac{1}{2\pi }\int _0^{2\pi } u_1+e^{i\theta }u_2 ^{p-1}(u_1+e^{i\theta }u_2)d\theta \ \ \text{ in } \ \ {{\mathbb {R}}}^3,\\ -\Delta u_2+ u_2+(\mu _{22}\phi _{u_2}-\mu _{12}\phi _{u_1})u_2=\frac{1}{2\pi }\int _0^{2\pi } u_2+e^{i\theta }u_1 ^{p-1}(u_2+e^{i\theta }u_1) d\theta \ \ \text{ in } \ \ {{\mathbb {R}}}^3, \end{array}\right. } \end{aligned}$$ which is derived from taking the nonrelativistic limit of the nonlinear Maxwell–Klein–Gordon equations under the decomposition of waves functions into positron and electron parts. We characterize the existence and nonexistence of positive vector solutions, depending on parameters p and \(\mu _{ij}\) . PubDate: 2023-01-07
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Abstract: Abstract In this paper, we focus on the solutions to the energy critical Hartree equations $$\begin{aligned} -\Delta u=\lambda u+\mu ( x ^{-\alpha }* u ^{2})u+( x ^{-4}* u ^{2})u,\ \ x\in \mathbb {R}^{N} \end{aligned}$$ under the normalized constraint $$\begin{aligned} \int _{{\mathbb {R}^N}} {{u}^2}=c>0, \end{aligned}$$ where \(N\ge 5\) , \(\mu \in \mathbb {R}\) , \(0<\alpha <4\) , and the frequency \(\lambda \in \mathbb {R}\) is a part of unknown and appears as Lagrange multiplier. Under different assumptions on c, \(\mu \) and \(\alpha \) , we prove some existence, non-existence, multiplicity and asymptotic results of normalized solutions to the above problem. In addition, the stability of the corresponding standing waves to the related time-dependent problem is discussed. These results are a continuation of our previous works, Luo (J Differ Equ, 195:455–467, 2019) and Cao et al. (J Differ Equ, 276:228–263, 2021), concerning normalized solutions to Hartree equations from energy subcritical to energy critical case. PubDate: 2023-01-07
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Abstract: Abstract We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth. PubDate: 2023-01-07
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Abstract: Abstract Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients. PubDate: 2023-01-07
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