WOMASY - Geometric and Harmonic Analysis meets PDE

Abstract

Since the beginning of the 21st century there appeared a huge flow of papers written on the regularising effect of nonlinear semigroups. All authors of these papers follow the same approach: First the authors derive a Log-Sobolev inequality from Sobolev inequalities. Then they use the Log-Sobolev inequality to show that the function $t\mapsto log\parallel T_t\phi {\parallel }_{r\left(t\right)}$ satisfies a differential inequality which is strong enough to conclude a $L^p$-$L^q$-regularisation of the trajectories $t\mapsto T_t\phi$ of the given semigroup $\left\{T_t\right\}$. In this talk, we present a simplified approach to this regularity effect.

This is joint work with Prof. Thierry Coulhon (ANU/PSL)

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Abstract

Despite being linear and second order, the heat equation on ${ℝ}^n$ does not enjoy uniqueness in the class of all smooth solutions (not even in the class of all analytic solutions). This is due, intuitively, to the unboundedness of the domain. The proof of this is classical and due to Tychonoff. It is not a second-order phenomenon – in fact, this same non-uniqueness holds for all higher-order heat flows in any dimension. This begs the question: in what class of solutions is the solution unique? Classically work has focused on growth conditions, so that for example the solution to the Cauchy problem for the heat equation with subexponential growth at infinity for all time is unique (note that one can do better than this).

In our work, we have taken a different approach. Since this nonuniqueness phenomenon appears to be independent of the maximum principle, we take the biharmonic heat flow as a model case. We give a condition that is automatically satisfied at initial time for any (say $C_{loc}^2$) initial data, regardless of growth at infinity, and guarantees uniqueness. The condition is pointwise and so independent from existing energetic-type conditions in the literature, that in any case imply decay at infinity. The condition is also sharp by explicit example. This is joint work with Miles Simon (OvGU Magdeburg).

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Abstract

Singular integrals associated with Zygmund dilations Abstract: The theory of Calderón-Zygmund operators plays an important role in modern harmonic analysis. The core of this theory is that the regularity and cancellation conditions are invariant with respect to the one-parameter family of dilations on $R^n$ defined by $\delta \left(x_1,x_2,\cdots ,x_n\right)=\left(\delta x_1,\cdots ,\delta x_n\right),\delta >0,$ in the sense that the kernel ${\delta }^{n}K\left(\delta x\right)$ satisfies the same conditions with the same bound as $K\left(x\right)$. Indeed, the classical singular integrals, maximal functions, $A_p$ weights and multipliers are invariant with respect to such one-parameter dilations.

On the other hand, the multiparameter theory on ${ℝ}^n$ began with Zygmund’s study of the strong maximal function, and later has been extensively studied by Fefferman, Pipher, Ricci, Stein and others. It was point out by Fefferman and Pipher that the consideration of these operators associated with Zygmund dilations is a natural next step or the simplest case after those of the classical CalderónZygmund theory and the product space theory.

In this talk we provide our recent study on these multi-parameter singular integral operators which commute with Zygmund dilations. We introduce a class of singular integral operators associated with the Zygmund’s dilations and show the boundedness of these operators on $L^p$ for $1<p<\infty$, which cover those concrete examples of such operators studied by RicciStein, FeffermanPipher and NagelWainger. We also establish the weighted Hardy and BMO spaces associated with the Zygmund’s dilations and obtain the end point estimates of these singular integrals.

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Abstract

In the design of a reflector antenna, we are given a light source and a surface which is to be illuminated. We want to design a reflector such that the output light covers the given surface. We show that this is in fact an optimal transport problem. The general optimal transportation is to find an optimal mapping of transferring one mass density to another one such that the total cost is minimised. This problem was first introduced by Monge in 1781. Monge’s cost function is propositional to the distance the mass is transferred, namely $c\left(x,y\right)=|x-y|$, but more general costs are allowed. The optimal transportation has found a variety of applications. In 1940s Kantorovich introduced a dual functional, by which one can determine the optimal mapping through the associated potential function, for a large class of cost functions.

The potential function satisfies a complicated partial differential equation of Monge-Ampére type, subject to a second boundary condition. This is a fully nonlinear partial differential equation which also arises in a number of geometric settings, and has been extensively studied in the last century. In this talk we will first introduce the optimal transportation and review the existence of optimal mappings. We then show that the reflector problem is an optimal transportation with a special cost function. By studying the associated Monge-Ampére equation, sharp conditions on the cost function have been found by the speaker and his collaborators for the regularity of potential functions. For Monge’s cost function $|x-y|$, which does not satisfy the sharp conditions, we have also obtained the existence of optimal mappings, and established interesting regularity and singularity results for the mapping.

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Abstract

Recently, fully nonlinear curvature ow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds (J. A. McCoy et al., Annali di Matematica Pura ed Applicata , 2013).

Recently, we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the ow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (Andrews, 2010); these arguments for obtaining a curvature pinching estimate may be adapted to this setting due to axial symmetry.

As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the ow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for $n\ge 3$.

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Abstract

In this talk, I will review recent developments on isolated singularities for various classes of nonlinear elliptic equations, which may include Hardy-Sobolev type potentials. In particular, we shall look at fully classifying the behaviour of all positive solutions in different contexts that underline the interaction of the elliptic operator and the nonlinear part of the equation. We also provide sharp results on the existence of solutions with singularities, besides optimal conditions for the removability of all singularities. I will discuss results obtained with various collaborators including J. Ching, T.-Y. Chang and F. Robert.

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