Abstract: Let \(\Omega \) be a bounded, smooth, uniformly convex domain in \(\mathbb {R}^n\) . We consider the following functional $$\begin{aligned} \mathcal {E}[u]=\int _{\Omega }(-u)\det D^2 u dx,\quad \Vert u\Vert _{L^{q+1}(\Omega )}=1 \qquad \qquad (0.1) \end{aligned}$$ where \(u\in C^2(\bar{\Omega })\) is convex and \(u=0\) on \(\partial \Omega \) . In this paper, the uniqueness of least energy solution of (0.1) is investigated. For \(n=2\) , we prove the least energy solution of (0.1) is unique for \(2<q<\infty \) provided it is locally uniformly convex. In particular, for \(q=+\infty \) , we show the uniqueness of the least energy solution of (0.1) and find its relation to Santalò point. PubDate: 2019-03-22

Abstract: In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied. PubDate: 2019-03-22

Abstract: We introduce an operator \(\mathbf {S}\) on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold \(\mathscr {N}\) of the Euclidean space \(\mathbb {R}^m\) , and coincides with the distributional Jacobian in case \(\mathscr {N}\) is a sphere. More precisely, the range of \(\mathbf {S}\) is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use \(\mathbf {S}\) to characterise strong limits of smooth, \(\mathscr {N}\) -valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg–Landau type functionals, with \(\mathscr {N}\) -well potentials. PubDate: 2019-03-21

Abstract: We solve \(\square _b\) on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related \(\square _b\) operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve \(\square _b\) works whenever \({\overline{\partial }}_b\) on the compact CR manifold has closed range in \(L^2\) ; in particular, as in Beals and Greiner (Calculus on Heisenberg manifolds. Annals of mathematics studies, vol 119. Princeton University Press, Princeton, 1988), it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in \({\mathbb {C}}^2\) . Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng et al. (Adv Math 308:276–347, 2017), which they then applied to study the CR Yamabe problem in 3 dimensions. PubDate: 2019-03-21

Abstract: In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. This extends a previous result by Q. Zhang where a \(C^1\) bound on the whole Riemann tensor was assumed. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field. This latter is based on a new Toponogov type lemma for manifolds with bounded Ricci curvature, and represents the first known growth estimate for the whole norm of the soliton field in the non-gradient case. PubDate: 2019-03-20

Abstract: We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weighted graphs and compare our result to the Cheeger inequality. PubDate: 2019-03-20

Abstract: We prove that a BV map with values into the projective space \({\mathbb {RP}^{d-1}}\) has a BV lifting with values into the unit sphere \({{\mathbb {S}}^{d-1}}\) that satisfies an optimal BV-estimate. As an application to liquid crystals, this result is also stated for BV maps with values into the set of uniaxial Q-tensors. In order to quantify BV liftings, we prove an explicit formula for an intrinsic BV-energy of maps with values into any compact smooth manifold. PubDate: 2019-03-20

Abstract: We show that a complete doubling metric space \((X,d,\mu )\) supports a weak 1-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points \(s,t \in X\) . This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1-Poincaré inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining s and t is a normal 1-current T, in the sense of Ambrosio and Kirchheim, with boundary \(\partial T = \delta _{t} - \delta _{s}\) , support contained in a ball of radius \(\sim d(s,t)\) around \(\{s,t\}\) , and satisfying \(\Vert T\Vert \ll \mu \) , with $$\begin{aligned} \frac{d\Vert T\Vert }{d\mu }(y) \lesssim \frac{d(s,y)}{\mu (B(s,d(s,y)))} + \frac{d(t,y)}{\mu (B(t,d(t,y)))}. \end{aligned}$$ We show that the 1-Poincaré inequality implies the existence of GPCs joining any pair of points in X. Then, we deduce the existence of PCs from a recent decomposition result for normal 1-currents due to Paolini and Stepanov. PubDate: 2019-03-20

Abstract: Let \(\Omega ={\mathbb {R}}^2{\setminus }\overline{B(0,1)}\) be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions as well as sign-changing solutions of the following boundary value problem $$\begin{aligned} -\Delta u=a(x) f(u)\ \text{ in } \Omega ,\quad u=0\ \text{ on } \partial \Omega =\partial B(0,1), \end{aligned}$$ where the nonnegative coefficient a satisfies a certain integrability condition. We are looking for solutions in the space \(D^{1,2}_0(\Omega )\) which is the completion of \(C^\infty _c(\Omega )\) with respect to the \(\Vert \nabla \cdot \Vert _{2,\Omega }\) -norm. Unlike in the situation of \({\mathbb {R}}^N\) with \(N\ge 3\) , the behavior of solutions in the borderline case \(N=2\) considered here is qualitatively significantly different, such as for example, constant-sign solutions in the borderline case are not decaying to zero at infinity, and instead are bounded away from zero. Our main tool in studying the above problem will be the Kelvin transform. We will first show that the Kelvin transform provides an isometric isomorphism between \(D^{1,2}_0(\Omega )\) and the Sobolev space \(H^1_0(B(0,1))\) , which is order-preserving. This allows us to establish a one-to-one mapping between solutions of the problem above and solutions of an associated problem in the bounded domain B(0, 1) of the form: $$\begin{aligned} -\Delta u=b(x) f(u)\ \text{ in } B(0,1),\quad u=0\ \text{ on } \partial B(0,1), \end{aligned}$$ where b satisfies an integrability condition in terms of the coefficient a. This duality approach given via the Kelvin transform allows us to handle nonlinearities under sub, super or asymptotically linear hypotheses. PubDate: 2019-03-20

Abstract: In this paper we show existence of positive solution to the problem In order to prove the main result, we study a limit problem of (P). More precisely, we study the case when \(a=0\) . Moreover, we prove the version to \({\mathbb {R}}^{N}\) of Struwe’s Global compactness result [14] for fractional Laplace operator. PubDate: 2019-03-20

Abstract: We study the curve diffusion flow for closed curves immersed in the Minkowski plane \({\mathcal {M}}\) , which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in \({\mathcal {M}}\) depending on its length. The indiactrix \(\partial {\mathcal {U}}\) (where \({\mathcal {U}}\subset {\mathbb {R}}^{2}\) is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix \(\tilde{{\mathcal {I}}}\) . This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the flow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged \(L^{2}\) sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has enclosed area equal to the enclosed area of the initial immersion. PubDate: 2019-03-20

Abstract: We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset \(\Omega \) of . The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a \(L^1\) constraint on densities, the so-called Rellich functions maximize this functional. Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the \(L^\infty \) -norm of Rellich functions may be large, depending on the shape of \(\Omega \) , we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of bang–bang functions and Rellich densities for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations. PubDate: 2019-03-20

Abstract: Let (M, g) be a closed Riemann surface. Castéras (Pac J Math 276:321–345, 2015) introduced a mean field type flow and obtained its global existence. For \(\rho \ne 8N\pi \) , assuming in addition that \(J_\rho (v(t))\ge -C\) for \(\rho >8\pi \) , they showed the convergence of the flow. In this paper, we will prove the convergence of the flow for the critical case \(\rho =8\pi \) with some appropriate initial data \(v_0\) and some geometric hypothesis. In particular, this gives a new proof of Ding–Jost–Li–Wang’s result in (Asian J Math 1:230–248, 1997). PubDate: 2019-03-19

Abstract: A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation \(\Delta u = f(u)\) in a domain \(\Omega \) subject to constant boundary data, where the function f in general is not monotone. When the domain \(\Omega \) is a perfect ring, we incorporate a new idea of radial correction into the classical moving plane method to prove the radial symmetry of a solution. When the domain is slightly shifted from a ring, we establish the stability of the solution by showing the approximate radial symmetry of the free boundary and the solution. For this purpose, we complete the proof via an evolutionary point of view, as an elliptic comparison principle is false, nevertheless a parabolic one holds. PubDate: 2019-03-19

Abstract: We investigate a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights \(w(x)= x_1 ^a\) or \(w(x)= x ^b\) where \(a,b >0\) in a suitable range. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional Laplacian, through the Caffarelli–Silvestre extension and is a first attempt to develop a parabolic theory in this setting. PubDate: 2019-03-19

Abstract: We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories. PubDate: 2019-03-18

Abstract: We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme submitted in De Lellis and Székelyhidi Jr. (Arch Ration Mech Anal 195:225–260, 2010), Székelyhidi (Ann Sci Éc Norm Supér 45(3):491–509, 2012) applied to the subsolution in Castro et al. (Mixing solutions for the Muskat problem, arXiv:1605.04822, 2016). More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance \(\mathbf {P}\) (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations. PubDate: 2019-03-08

Abstract: We establish the first Sobolev regularity and uniqueness results for minimisers of autonomous, convex variational integrals of linear growth which depend on the symmetric rather than the full gradient. This extends the results available in the literature for the BV-setting to the case of functionals whose full gradients are a priori not known to exist as finite matrix-valued Radon measures. PubDate: 2019-02-26

Abstract: This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{ u(x)-u(y) ^{N/s}}{ x-y ^{2N}}dxdy\right) (-\Delta )^{s}_{N/s}u=f(x,u)\,\, \ &{}\quad \mathrm{in}\ \Omega ,\\ u=0\ \ \ \ &{}\quad \mathrm{in}\ {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. } \end{aligned}$$ where \((-\Delta )^{s}_{N/s}\) is the fractional N / s-Laplacian operator, \(N\ge 1\) , \(s\in (0,1)\) , \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary, \(M:{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}^+_0\) is a continuous function, and \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}} \) is a continuous function behaving like \(\exp (\alpha t^{2})\) as \(t\rightarrow \infty \) for some \(\alpha >0\) . We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity. PubDate: 2019-02-26

Abstract: In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces \(\Sigma \) with nonnegative sectional curvature in \(\mathbb {H}^n\) . As an application, we prove the hyperbolic Alexandrov–Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in \(\mathbb {H}^n\) : $$\begin{aligned} \int _{\Sigma } p_{2k}\ge \omega _{n-1}\left[ \left( \frac{ \Sigma }{\omega _{n-1}}\right) ^\frac{1}{k} +\left( \frac{ \Sigma }{\omega _{n-1}}\right) ^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right] ^k, \end{aligned}$$ where \(p_i\) is the normalized i-th mean curvature. Equality holds if and only if \(\Sigma \) is a geodesic sphere in \(\mathbb {H}^n\) . For a domain \(\Omega \subset \mathbb {H}^n\) with \(\Sigma =\partial \Omega \) having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in \(\mathbb {H}^n\) : $$\begin{aligned} W_{2k+1}(\Omega )\ge \frac{\omega _{n-1}}{n}\sum _{i=0}^{k}\frac{n-1-2k}{n-1-2i} C_k^i\left( \frac{ \Sigma }{\omega _{n-1}}\right) ^\frac{n-1-2i}{n-1}, \end{aligned}$$ where \(W_i(\Omega )\) is the i-th quermassintegral in integral geometry. Equality holds if and only if \(\Sigma \) is a geodesic sphere in \(\mathbb {H}^n\) . All these inequalities were previously proved by Ge et al. (J Differ Geom 98:237–260, 2014) under the stronger condition that \(\Sigma \) is horospherical convex. PubDate: 2019-02-23