International Conference on PDEs, Geometric Analysis and Functional Inequalities

Abstract

The essential spectrum of the Laplacian on functions over a noncompact Riemannian manifold has been extensively studied. It is known that on hyperbolic space a spectral gap appears, whereas is has been conjectured that on manifolds with uniformly subexponential volume growth and Ricci curvature bounded below the essential spectrum is the nonnegative real line. Much less is known for the spectrum of the Laplacian on differential forms.

In our work we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint and nonnegative operator on a Hilbert space. We use this criterion to study the spectrum of the Laplacian on k-forms over an open manifold. We first show that the spectrum of the Laplacian on 1-forms always contains the spectrum of the Laplacian on functions. We also study the spectrum of the Laplacian on k-forms under a continuous deformation of the metric. The results that we obtain allow us to compute the spectrum of the Laplacian on k-forms over asymptotically flat manifolds. This is joint work with Zhiqin Lu.

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Abstract

We deal with boundary value problems for a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation. Under minimal regularity assumptions on the boundary, we prove the membership to a Sobolev space of a nonlinear expression of the gradient of the solution, for any square-integrable right-hand side.

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Abstract

In this talk, we will discuss recent contributions on isolated singularities for various classes of nonlinear elliptic equations in the form

where $p>1$ and $B_1\left(0\right)$ denotes the open unit ball in ${ℝ}^N$. Under various assumptions on $A$, $p$ and $g$, we fully classify the behaviour of all positive solutions of (1), underlining the intricate interaction of the elliptic operator and the nonlinear part $g\left(x,u,\nabla u\right)$ of the equation. The talk will refer to joint work with collaborators such as T.-Y. Chang, J. Ching, and F. Robert.

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Abstract

We discuss the problem

in $\Omega$, $u,v\ge 0$ in $\Omega$, $u=v=0$ on $\partial \Omega$ for a large positive $k$; and the related problem

in $\Omega$, $u,v\ge 0$ in $\Omega$, $u=v=0$ on $\partial \Omega$. Here $f$ and $g$ are sublinear.

We have a limit problem

and we are interested when properties of the solutions of this equation yield information about the systems for large positive $k$. We are also interested in the parabolic analogue.

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Abstract

We would like to discuss some problems related to the Peierls-Nabarro model for atom edge dislocation in crystals. Along the sliding surface, the problem reduces to a fractional equation with a periodic potential. At a mesoscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with thenatural periodicity of the medium. These dislocation points evolve in dependenceto the external stress and an interior potential which is either attractive or repulsive, according to the orientation of the dislocations. Collision of dislocations with opposite orientations may occur in finite time. In this setting, we also consider a system of stationary equations with a perturbed potential and we construct heteroclinic, homoclinic and multibump orbits, providing an example of symbolic dynamics in a fractional setting.

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Abstract

We consider the large time behaviour of solutions to the porous medium equation with a Fisher-KPP type reaction term and bounded, nonnegative, and compactly supported initial function in $R^N$:

It is well known that the spatial support of the solution to this problem remains bounded for all time $t>0$. Moreover, in spatial dimension one there is a minimal speed $c^{*}>0$ for which the equation has a traveling wave solution, and for large time the spatial support of the solution is an interval $\left[-h_1\left(t\right),h_2\left(t\right)\right]$ with $\underset{t\to \infty }{lim}\left(h_i\left(t\right)-c^{*}t\right)=r_i$ for some constants $r_i\in ℝ$ $\left(i=1,2\right)$. Furthermore, in a suitable moving coordinate system the solution converges to the profile of the traveling wave with that speed. We show that in higher dimensions a logarithmic shifting occurs: There exists a second constant $c_{*}>0$ independent of the dimension $N$ such that, if the initial function is additionally assumed to be radially symmetric and hence the solution $u\left(.,t\right)$ remains radially symmetric for all future time $t>0$, then the radius $R\left(t\right)$ of its spatial support satisfies $\underset{t\to \infty }{lim}\left(\right.R\left(t\right)-c^{*}t+\left(N-1\right)c_{*}logt\left)\right.=r_0\in ℝ$. In a suitable moving coordinate system, that take into account the logarithmic correction, we still obtain convergence of the solution to the profile of the one-dimensional traveling wave. If the initial function is not radially symmetric, then there exists $R_i\left(t\right)$ and $r_i\in ℝ$ $\left(i=1,2\right)$ such that $R_1\left(t\right)\le R_2\left(t\right)$, $\underset{t\to \infty }{lim}\left(\right.R_i\left(t\right)-c^{*}t+\left(N-1\right)c_{*}logt\left)\right.=r_i$ and the boundary of the spatial support of the solution at time $t$ is contained in the spherical shell $R_1\left(t\right)\le r\le R_2\left(t\right)$ for all $t>0$.

This talk is based on joint work with Fernando Quiros (UAM, Spain) and Maolin Zhou (UNE)

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Abstract

Classification theory on the existence and non-existence of local in time solutions for initial value problems of nonlinear heat equations are investigated. Without assuming a concrete growth rate on a nonlinear term, we reveal the threshold integrability of initial data which classify existence and nonexistence of solutions via a quasi-scaling and its invariant integral. Typical nonlinear terms, for instance polynomial type, exponential type and its sum, product and composition, can be treated as applications. This is a joint work with Yohei Fujishima (Shizuoka University).

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Abstract

Steady states of the surface diffusion equation

will be studied. Here $V$ is the normal velocity of the evolving surfaces $\Gamma \left(t\right)$, $H$ is the mean curvature of $\Gamma \left(t\right)$, and ${\Delta }_{\Gamma }$ is the Laplace-Beltrami operator on $\Gamma \left(t\right)$. (2) was first derived by Mullins [1] to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. Also, (2) is obtained as the $H^{-1}$-gradient flow of the area functional for the evolving surfaces $\Gamma \left(t\right)$, so that (2) has a variational structure that the area of the surface decreases whereas the volume of the region enclosed by the surface is preserved. This provides the constant mean curvature surfaces (CMC surfaces) as steady states for (2). In the axisymmetric case, CMC surfaces are called Delaunay surfaces, which are cylinders, unduloids, series of spheres, and nodoids for non-zero mean curvature and catenoids for zero mean curvature. In this talk, bifurcation diagrams for steady states of (2) are shown in some cases.

References

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Abstract

The $p$-Minkowski problem and the dual Minkowski problem are two interesting topics in convex geometry. To study these problems, one is led to a class of Monge-Ampère equations on the sphere. In this talk, I will discuss some recent works on these two problems.

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Abstract

In this talk, I will discuss the asymptotic behavior of radially symmetric solutions of the nonlinear heat equation on ${ℝ}^N\left(N>2\right)$ with the Sobolev critical power nonlinearity.

In the case of time-global solutions, we show that the solution is asymptotically decomposed into a finite sum of rescaled ground states that hardly interact with each other because the ratio of the rescaling parameters goes to infinity. The total energy of the solution then converges to an integer multiple of the energy of the ground state. We call this behavior “soliton resolution” of the solution.

In the case where the solution blows up in finite time, and if the blow-up is of type II in a certain sense, we show that a similar soliton resolution occurs near the blow-up point.

In this talk, I will mainly focus on the classification of the asymptotic behavior of solutions. However, if I have time, I will also give an example of solution that exhibits multiple solitons. This is joint work with Frank Merle.

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Abstract

The Riemannian Penrose inequality is a celebrated result in mathematical general relativity, giving a lower bound on the ADM mass of an asymptotically flat manifold in terms of the area of an outermost minimal surface boundary. In this talk, a ”Penrose-like” inequality is presented: Under suitable conditions, we give a lower bound for the ADM mass in terms of the boundary geometry in the case where the boundary not necessarily a minimal surface. We give some background, and a sketch of the proof is presented. The key idea is to extend the manifold at the boundary to connect it to a minimal surface with controlled area, where the usual Riemannian Penrose inequality may be applied. This is joint work with Pengzi Miao. (More: arXiv:1701.04805)

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Abstract

We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of $S^{n+1}$. Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature. Our results complement some other recent results of Claus Gerhardt.

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Abstract

In this talk we address classical Euler’s elastica problem formulated as the minimizing problem of the total squared curvature energy defined for planar curves of fixed length and clamped endpoints. From a new viewpoint of singular perturbation, we first reveal the asymptotic shape of global minimizers of the modified total squared curvature as the effect of bending tends to be small. We then draw a similar conclusion about original Euler’s problem in a straightening process, at least in a subsequential convergence sense.

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Abstract

We present some recent results for Sobolev inequalities on ${ℝ}^n$ and on half spaces. For the sharp Sobolev inequality on ${ℝ}^n$, equality is achieved by an $\left(n+2\right)$-dimensional family of functions known as Talenti functions. In the case $p\ge 2$, we show that a function that almost attains equality is quantitatively close to a Talenti function, with closeness measured in terms of the $L^p$ norm of the gradient. On the half-space, we prove a new one-parameter family of sharp trace-Sobolev inequalities, with the classical Sobolev and Escobar inequalities as particular cases, and characterize the equality cases in this family of inequalities.

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Abstract

We prove a quantitative form of the Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. In the same spirit of well known quantitative estimates for the Dirichlet eigenvalue, we show that it is possible to bound from above the Fraenkel asymmetry in terms of the eigenvalue deficit.

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Abstract

In this talk, I would like to talk on the large time behavior of the solution to the drift-diffusion system in higher space dimensions than two. The two dimensional problem is the critical problem and the classification of the solutions in time is almost completely done and there exists a threshold number that separates the global behavior of the solutions in $L^1$ norm. However for the higher dimensional case it is not clear yet. One of the main difficulty stems from the system is super critical structure in the scaling view point if the dimension is high. We first show the semi-linear problem has an unstable large time behavior under some condition on the initial data. The result is depending on the control of the entropy functional and we employ the Shannon inequality for the Boltzmann-Shannon type entropy. For the quasi-linear case, the equation is much treatable and we show the similar unstable result. This talk is based on the joint work with H. Wakui.

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Abstract

The $L^2$-gradient flow of curves for the total squared curvature, also called elastic flow, coupled with different boundary conditions, has been widely studied in the mathematical literature. Long time existence of the evolutions is generally obtained by the smoothing effect of the energy. Regarding the asymptotic behavior as $t\to \infty$, there are general results implying that the solution subconverges to a (possibly nonunique) stationary solution. However, there are few results proving the full convergence of solutions (that is, without passing to a subsequence), and they are mostly obtained in the case of closed curve.

In this talk, first we consider the gradient flow of a general geometric functional defined on planar open curves with a natural boundary condition. The purpose of this talk is to prove the full convergence of solutions of the gradient flow under a weaker condition that there are only finitely many equilibrium states at each prescribed energy level. Moreover, we apply the result to the elastic flow under some natural boundary conditions.

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Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kaehler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth, Yanir Rubinstein and Wolfgang Ziller.

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Abstract

Polyakov’s formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple local quantities. Such a formula is well known in the case of closed surfaces (Osgood, Philips, & Sarnak 1988) and surfaces with smooth boundary (Alvarez 1983). Due to the abstract nature of the definition of the zeta-regularized determinant of the Laplacian, it is typically impossible to compute an explicit formula. Nonetheless, Kokotov (genus one Kokotov & Klochko 2007, arbitrary genus Kokotov 2013) demonstrated such a formula for polyhedral surfaces! I will discuss joint work with Clara Aldana concerning the zeta regularized determinant of the Laplacian on Euclidean domains with corners. We determine a Polyakov formula which expresses the dependence of the determinant on the opening angle at a corner. Our ultimate goal is to determine an explicit formula, in the spirit of Kokotov’s results, for the determinant on polygonal domains. The results which shall be presented here are the crucial first steps towards such a formula.

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Abstract

In this talk, we are concerned with the critical and subcritical Trudinger-Moser type inequalities for functions in a fractional Sobolev space on the whole real line.

We prove the relation between the two inequalities and discuss the attainability of the suprema.

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Abstract

Generated prescribed Jacobian PDEs are a family of Monge-Ampère type equations which unify PDE aspects of optimal transportation and geometric optics. We present some of the basic underlying convexity theory of generating functions, which are nonlinear extensions of affine functions. Then we indicate how it is used to prove a classical existence result found recently in joint work with Feida Jiang.

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Abstract

We construct a set of stationary blackhole solutions to the vacuum $4+1$ dimensional Einstein equation, which possess non-spherical horizons. In particular the horizons under considerations are diffeomorphic to $S^3$, $S^2\times S^1$ and the lens spaces $L\left(p,q\right)$. The spacetimes we constructed have the symmetry group of $U{\left(1\right)}^2*ℝ$, namely the time symmetry and two axisymmetries. The Einstein metric in this setting can be identified with a harmonic map into the symmetric space $SL\left(3\right)∕SO\left(3\right)$ with boundary conditions corresponding to the topological type of the horizon as well as the asymptotic flatness. This is a joint work with Marcus Khuri and Gilbert Weinstein.

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