This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

/pdf/from-1970-until-present-the-keller-segel-model-in-chemotaxis-23f2x935xe.pdf

From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kahler manifolds, provided that the partial C

 0-estimate posed in [T6] is true.

/pdf/kahler-einstein-metrics-with-positive-scalar-curvature-4ob1pc2lkm.pdf

Kähler-Einstein metrics with positive scalar curvature

Let n be a positive integer and let 0 < α < n. Consider the integral equation 






We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form 






with some constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn. This solves an open problem posed by Lieb 12. The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper.



Moreover, we show that the family of well-known semilinear partial differential equations 






is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.

Classification of solutions for an integral equation

where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see [8]). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is "What is the situation in dimension 2?" Two important arguments ([25], [26], [27]) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in part

Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality

(1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.

Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions

https://projecteuclid.org/download/pdf_1/euclid.acta/1485890490

Prescribing Gaussian curvature on S2

Soit D un domaine lisse de S 2 et K une fonction lisse definie sur D. Alors pour que le probleme aux limites de Neumann: Δu+K exp(2u)=1 sur D, ∂u/∂n=0 sur ∂D, ait une solution u, on a: a) quand aire(D) max x∈∂D (K(x), 0)

/pdf/conformal-deformation-of-metrics-on-s-2-29wjpg97q4.pdf

Conformal deformation of metrics on $S^2$

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/eigenvalues-of-the-laplacian-of-compact-riemann-surfaces-and-3q3g49g9xe.pdf

Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.

The scalar curvature equation on 2- and 3-spheres

In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe problem. In dimension two the analogous problem deals with the Moser-Trudinger inequality: on a compact Riemann surface M2, there exists a constant c = c(M) so that f e47w2 < c(M) if f IVw12 < 1 and f w = 0. The connection of this inequality with geometry comes through the zeta functional determinant of the Laplacian as defined by Ray-Singer: for a Riemannian metric g, let 0 < A1 < A2 < ... be the spectrum

Paul Yang

Papers

Prescribing Gaussian curvature on S2

Conformal deformation of metrics on $S^2$

Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds

The scalar curvature equation on 2- and 3-spheres

Extremal metrics of zeta function determinants on 4-manifolds