Variational Formula for Dirichlet Forms and Estimates of Principal Eigenvalues for Symmetric α-stable Processes

In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure charging no set of zero capacity, we give a new proof of an L²()-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed -stable processes and obtain conditions for some measures being gaugeable. Mathematics Subject Classifications (2000) Primary 31C25; Secondary 34L15, 60G52.     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q ₁ ^rot , EQ ₁ ^rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.     

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces

In the previous work (Zhang and Zhu in J Differ Geom, http://arxiv.org/pdf/1012.4233v3, 2012), the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1–14, 1994), Chen and Wang (Sci Sin (A) 40:384–394, 1997) and Bakry–Qian (Adv Math 155:98–153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li–Yau’s estimate for positve solutions of heat equations on Alexandrov spaces.     

Eigenvalue gaps for the Cauchy process and a Poincaré inequality

A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004) 355-423]. From this, a variational characterization for the eigenvalues λn, n?1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ₁. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ_∗−λ₁ where λ_∗ is the eigenvalue corresponding to the “first” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ_∗−λ₁ and on a weighted Poincaré-type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0<α?2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ₂−λ₁ in bounded convex domains.     

Intertwining Relations for One-Dimensional Diffusions and Application to Functional Inequalities

Following the recent work Chafaï and Joulin (Bernoulli 19:1855–1879, 2013) fulfilled in the discrete case, we provide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang (Trans. Am. Math. Soc. 349:1239–1267, 1997) together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Higher order PDEs and symmetric stable processes

We describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations. As an application, we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set D. The variational formula is more user friendly than the classical Rayleigh--Ritz formula. We illustrate this by obtaining upper bounds on the eigenvalues in terms of Dirichlet eigenvalues of the Laplacian on D. These results generalize some work of Banuelos and Kulczycki on the Cauchy process. Along the way we prove an operator inequality for the operators associated with the transition densities of Brownian motion and the Brownian motion killed upon leaving D.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

On the Dirichlet problem for asymmetric zero-range process on increasing domains

In order to obtain hitting time estimates for the asymmetric zero-range process (AZRP) on ^d, in dimensions d3, we characterize the principal eigenvalue of the generator of the AZRP with Dirichlet boundary on special domains. We obtain a Donsker-Varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions.Mathematics Subject Classification (2000):60K35, 82C22, 60J25     

Packing,counting and covering Hamilton cycles in random directed graphs

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L ^p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L ^p (R ⁿ , (φ _iρ (x))² dx) where φ _iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.     

Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

We find necessary and sufficient conditions for a finite K–bi–invariant measure on a compact Gelfand pair (G,K) to have a square–integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G,K) is a compact Riemannian symmetric pair, we study the induced transition density for G–invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K–bi–invariant Lévy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s Lévy–Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk–Hecke theorem.     

Hölder estimates for the\bar \partial - equation in some domains of finite typein some domains of finite type

We obtain Hölder estimates for the $\bar \partial - equation$ on some domains of finite type in ?ⁿ using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h ²) or \(O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)\) in the sense of L ²-norm or L ^∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.     

Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calderón–Zygmund type estimates in the variable exponent setting. In a recent work of [12], the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above $p(x)$, see (1.3) and (1.4). Here, we bridge this gap to obtain the end point case of the estimates obtained in [12], see (1.5). In order to do this, we have to obtain significantly improved a priori estimates below $p(x)$, which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.     

On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

We consider a class of weakly coupled systems of elliptic operators \({\mathcal{A}}\) with unbounded coefficients defined in \({\mathbb{R}^N}\). We prove that a semigroup (T(t))t ≥ 0 of bounded linear operators can be associated with \({\mathcal{A}}\), in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T(t)f, when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator \({\mathcal{A}}\). Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L ^p -spaces over \({\mathbb{R}^N}\), related to the Lebesgue measure, when \({p \in [1,\infty)}\). We also provide sufficient conditions for this semigroup to be analytic when \({p \in [1,\infty)}\). Finally, we prove some L ^p ?L ^q -estimates.     

Symmetries on almost symmetric numerical semigroups

The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Fröberg in J. Algebra, 188, 418–442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H ^? (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H ₁ or H ₂ is not symmetric, then the gluing of H ₁ and H ₂ is not almost symmetric.