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Variational theory for the total scalar curvature functional for Riemannian metrics and related topics
 in Topics in Calculus of Variations (Montecatini
, 1987
"... The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We ..."
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Cited by 177 (2 self)
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The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss variational characterizations of constant curvalure m trics, and give a proof of 0bata's uniqueness theorem. Much of the material in this section can be found in papers of Berger Ebin [3], FischerMarsden [8], N. Koiso [14], and also in the recent book by A. Besse [4] where the reader will find additional references. In §2 we give a general discussion of the Yamabe problem and its resolution. We also give a detailed analysis of the solutions of the Yamabe equation for the product conformal structure on SI(T) x S~1(1), a circle of radius T crossed with a sphere of radius one. These exhibit interesting variational fea,tures uch a.s symmetry breaking and the existence of solutions with high Morse index. Since the time of the summer school in Montecatini, the beautiful survey paper of J. Lee and T. Parker [15] has appeared. This gives a detailed discussion of the
Twodimensional KellerSegel model: Optimal critical mass and qualitative properties of the solutions
 J. DIFF. EQNS
, 2006
"... The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is k ..."
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Cited by 128 (15 self)
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The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blowup occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blowsup in finite time in the whole euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with subcritical mass, this allows us to give for large times an “intermediate asymptotics ” description of the vanishing. In selfsimilar coordinates, we actually prove a convergence result to a limiting selfsimilar solution which is not a simple reflect of the diffusion.
On some conformally invariant fully nonlinear equations, Part II: Liouville, . . .
, 2005
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Harnack Type Inequality: the Method of Moving Planes
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1999
"... A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have bee ..."
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Cited by 83 (3 self)
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A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have been studied recently by various authors.
POSITIVE SOLUTIONS OF NONLINEAR PROBLEMS INVOLVING THE SQUARE ROOT OF THE LAPLACIAN
"... Abstract. We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a pri ..."
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Cited by 82 (0 self)
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Abstract. We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of GidasSpruck type. In addition, among other results, we prove a symmetry theorem of GidasNiNirenberg type. 1.
Exact Multiplicity of Positive Solutions for a Class of Semilinear Problem, II
 JOURNAL DIFFERENTIAL EQUATIONS
, 1999
"... We consider the positive solutions to the semilinear problem: An + %f ( I,) = 0, in B”. 1, = 0, on aB”. i where B ” is the unit ball in R”.)I 2 I. and I. is a positive parameter. It is well known that if f is a smooth function, then any positive solution to the equation is radially symmetric, and ..."
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Cited by 82 (33 self)
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We consider the positive solutions to the semilinear problem: An + %f ( I,) = 0, in B”. 1, = 0, on aB”. i where B ” is the unit ball in R”.)I 2 I. and I. is a positive parameter. It is well known that if f is a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be parameterized by their maximum values. We develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions f(u) and the precise shape of the global bifurcation diagrams are rigorously proved. Our technique combines the bifurcation analysis. stability analysis, and topological methods. We show that the shape of the bifurcation curve depends on the shape of the graph of function f(u)/u as well as the growth rate of f.
Infinite time aggregation for the critical PatlakKellerSegel model in R²
 COMM. PURE APPL. MATH
"... We analyze the twodimensional parabolicelliptic PatlakKellerSegel model in the whole Euclidean space R². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of ”freeenergy solutions”, namely weak solutions w ..."
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Cited by 80 (13 self)
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We analyze the twodimensional parabolicelliptic PatlakKellerSegel model in the whole Euclidean space R². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of ”freeenergy solutions”, namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of freeenergy solutions with initial data as before for the critical mass 8pi/χ. Actually, we prove that solutions blowup as a delta dirac at the center of mass when t→ ∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2
A fully nonlinear conformal flow on locally conformally flat manifolds
 JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
, 2001
"... We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2. ..."
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Cited by 64 (14 self)
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We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2.
Remark on some conformally invariant integral equations: the method of moving spheres
, 2003
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Global existence and convergence of Yamabe flow
 Centre for Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Weimin Sheng: Department of Mathematics, Zhejiang University, Hangzhou
, 1994
"... Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)= ..."
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Cited by 50 (0 self)
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Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)=