Harnack inequalities for evolving hypersurfaces

This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship     

Entropy of formulas

A probability distribution can be given to the set of isomorphism classes of models with universe {1, ..., n} of a sentence in first-order logic. We study the entropy of this distribution and derive a result from the 0–1 law for first-order sentences.      

Remarks on differential Harnack inequalities

In this paper, we give a generalization of (global and local) differential Harnack inequalities for heat equations obtained by Li and Xu [J.F. Li, X.J. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math. 226 (5) (2011) 4456–4491] and Baudoin and Garofalo [F. Baudoin, N. Garofalo, Perelman’s entropy and doubling property on Riemannian manifolds, J. Geom. Anal. 21 (2011) 1119–1131]. From this we can derive new Harnack inequalities and new lower bounds for the associated heat kernel. Also we provide some new entropy formulas with monotonicity.     

Harnack inequalities for Yamabe type equations

We give some a priori estimates of type sup×inf on Riemannian manifolds for Yamabe and prescribed curvature type equations. An application of those results is the uniqueness result for Δu+?u=uN−1 with ? small enough.     

Harnack estimates for quasi-linear degenerate parabolic differential equations

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.     

Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains

We consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We derive an optimal elliptic-type Harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times. We show several nontrivial applications of both the exponential estimate and the derived Harnack inequality.     

Geometric flows and differential Harnack estimates for heat equations with potentials

Let $M$ be a closed Riemannian manifold with a Riemannian metric $g_{ij}(t)$ evolving by a geometric flow $\partial _{t}g_{ij} = -2{S}_{ij}$ , where $S_{ij}(t)$ is a symmetric two-tensor on $(M, g(t))$ . Suppose that $S_{ij}$ satisfies the tensor inequality $2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0$ for all vector fields $X$ on $M$ , where ${\mathcal H}(S, X)$ and ${\mathcal E}(S,X)$ are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where $S_{ij} = R_{ij}$ , the Ricci tensor of $M$ , our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.     

Superposition formulas for exterior differential systems

In this article we solve an inverse problem in the theory of quotients for differential equations. We characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original. The fact that a system can be written as a quotient can be used to find the general solution to these equations. Some examples are given to demonstrate the theory.     

Harnack inequalities for fuchsian type weighted elliptic equations

Harnack type inequalities for nonnegative (weak) solutions of degenerate elliptic equations, in divergence form, are established. The asymptotic behavior of solutions of Fuchsian type weighted elliptic operators is also investigated.     

Harnack type inequalities for conformal scalar curvature equation

We derive a Harnack type inequality for the conformal scalar curvature equation on B _3R . If the positive scalar curvature function K(x) is sub-harmonic in a neighborhood of each critical point and the maximum of u over B _R is comparable to its maximum over B _3R , then the Harnack type inequality can be obtained. Zhang is supported by NSF-DMS-0600275.     

Parabolic Harnack inequality for divergence form second order differential operators

Old and recent results concerning Harnack inequalities for divergence form operators are reviewed. In particular, the characterization of the parabolic Harnack principle by simple geometric properties -Poincaré inequality and doubling property- is discussed at length. It is shown that these two properties suffice to apply Moser's iterative technique.     

Lefschetz type formulas for dg-categories

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230–256, 2012) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129–199, 2011). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity \({\varvec{w}}\) . We apply this formula to get some restrictions on the Betti numbers of a \({\mathbb Z}/2\) -equivariant module over \(k[[x_1,\ldots ,x_n]]/({\varvec{w}})\) in the case when \({\varvec{w}}(-x)={\varvec{w}}(x)\) .     

Harnack inequality for Ornstein–Uhlenbeck type operators

Archiv der Mathematik - We prove Harnack type inequalities for non-negative weak solutions in $$(0,T]times mathbb {R}^N$$ of parabolic problems related to operators of the type $$L=hbox...     

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Let be such that each is a signed measure on R ^d belonging to the Kato class K _{d, 1}. A Brownian motion in R ^d with drift is a diffusion process in R ^d whose generator can be informally written as . When each is given by U ⁱ (x)dx for some function U ⁱ , a Brownian motion with drift is a diffusion in R ^d with generator . In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is, when is replaced by a uniformly elliptic divergence form operator with C ¹ coefficients or a uniformly elliptic non-divergence form operator with C ¹ coefficients. The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported by Research Settlement Fund for the new faculty of Seoul National University.     

An extension of Harnack type determinantal inequality

We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixties and give an extension of the inequality involving multiple positive semidefinite matrices .