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We study second order elliptic operators whose diffusion coefficients degenerate at the boundary in first order and whose drift term strongly points outward. It is shown that these operators generate analytic semigroups in L² where they are equipped with their natural domain without boundary conditions. Hence, the corresponding parabolic problem can be solved with optimal regularity. In a previous work we had treated the case of inward pointing drift terms. [ABSTRACT FROM AUTHOR]

We establish local and global higher integrability estimates for the gradient of vector-valued parabolic quasi-minimizers of integral functionals with polynomial p -growth, p > 2 n n + 2 , on a parabolic domain Ω T in R n + 1 , n ≥ 2 and under a weak regularity assumption on the parabolic boundary of Ω T . [ABSTRACT FROM AUTHOR]

Let L be a Lévy process with characteristic measure ν, which has an absolutely continuous lower bound w.r.t. the Lebesgue measure on ℝ. By using Malliavin calculus for jump processes, we investigate Bismut formula, gradient estimates and coupling property for the semigroups associated to semilinear SDEs forced by Lévy process L. [ABSTRACT FROM AUTHOR]

We present a new method of deriving estimates of the derivatives of the heat kernel on complete noncompact manifolds, based on an iteration argument. Applying these estimates we study the L p -boundedness of Littlewood–Paley–Stein operators on a class of locally symmetric spaces. [ABSTRACT FROM AUTHOR]

Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory. [ABSTRACT FROM AUTHOR]

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation: on complete noncompact manifolds with Ricci curvature bounded from below, where 0 < σ < 1 is a real constant, and b( x, t) is a function which is C in the x-variable and C in the t-variable. [ABSTRACT FROM AUTHOR]

This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued. [ABSTRACT FROM AUTHOR]

We investigate the growth of entire positive functions u(x) and their gradients Du in Sobolev spaces when a polynomial growth is assumed for their image Lu through a linear second-order uniform elliptic operator L. In particular, under suitable assumptions on the coefficients, we show that if Lu is bounded, then u(x) may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we prove the sharp global Li-Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman-type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu-Ni-Vázquez-Villani (JMPA 2009) and Kotschwar-Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The porous medium equation arises in different applications to model diffiusive phenomena. In this paper, we obtain several gradient estimates for some porous medium type equations on smooth metric measure space with N-Bakry-Emery Ricci tensor bounded from below. In particular, we improve and generalize some current gradient estimates for the porous medium equations. [ABSTRACT FROM AUTHOR]

We obtain a global weighted L estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hölder continuity of the solution. [ABSTRACT FROM AUTHOR]

Consider a general domain $$\varOmega \subseteq {\mathbb {R}}^n, n\ge 2$$ , and let $$1 < q <\infty $$ . Our first result is based on the estimate for the gradient $$\nabla p \in G^q(\varOmega )$$ in the form $$\Vert \nabla p\Vert _q \le C \,\sup |\langle \nabla p,\nabla v\rangle _{\varOmega }|/\Vert \nabla v\Vert _{q'}$$ , $$\nabla v \in G^{q'}(\varOmega ), q' = \frac{q}{q-1}$$ , with some constant $$C=C(\varOmega ,q)>0$$ . This estimate was introduced by Simader and Sohr (Mathematical Problems Relating to the Navier-Stokes Equations. Series on Advances in Mathematics for Applied Sciences, vol. 11, pp. 1-35. World Scientific, Singapore, ) for smooth bounded and exterior domains. We show for general domains that the validity of this gradient estimate in $$G^q(\varOmega )$$ and in $$G^{q'}(\varOmega )$$ is necessary and sufficient for the validity of the Helmholtz decomposition in $$L^q(\varOmega )$$ and in $$L^{q'}(\varOmega )$$ . A new aspect concerns the estimate for divergence free functions $$f_0 \in L^q_{\sigma }(\varOmega )$$ in the form $$\Vert f_0\Vert _q \le C \sup |\langle f_0,w\rangle _{\varOmega }|/ \Vert w\Vert _{q'}, w\in L^{q'}_{\sigma }(\varOmega )$$ , for the second part of the Helmholtz decomposition. We show again for general domains that the validity of this estimate in $$L^q_{\sigma }(\varOmega )$$ and in $$L^{q'}_{\sigma }(\varOmega )$$ is necessary and sufficient for the validity of the Helmholtz decomposition in $$L^q(\varOmega )$$ and in $$L^{q'}(\varOmega )$$ . [ABSTRACT FROM AUTHOR]

We consider energy solutions of the inhomogeneous parabolic p-Laplacian system . We show in the case that if the right hand side g is locally in , then u is locally in , where is the 1-Hölder–Zygmund space. This is the borderline case of the Calderón–Zygmund theory. We provide local quantitative estimates. We also show that finer properties of g are conserved by Du, e.g. Hölder continuity. Moreover, we prove a new decay for gradients of p-caloric solutions for all . [ABSTRACT FROM AUTHOR]

In this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems. [ABSTRACT FROM AUTHOR]

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented. [ABSTRACT FROM AUTHOR]

In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient almost Ricci solitons possessing certain metric properties on the base of the warped product. [ABSTRACT FROM AUTHOR]

Let u,v be two harmonic functions in {∣z∣ < 2} ⊂ C which have exactly the same set Z of zeros. We observe that ∣∇log∣u/v∣∣ is bounded in the unit disk by a constant which depends on Z only. In case Z = ∅ this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives only Hölder estimates on log ∣u/v∣. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

In this paper, we prove existence and uniqueness of Lipschitz continuous viscosity solutions for Dirichlet problems involving a class a fully non-linear operators on Lie groups. In particular, we consider the elementary symmetric functions of the eigenvalues of the Hessian built with left-invariant vector fields. [ABSTRACT FROM AUTHOR]

In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ℍ. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely a CD(0, ∞ ) inequality), see Theorem 4.5, whereas the operator itself does not satisfy any CD( r, ∞ ) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper (Qian, Bull Sci Math 135:262-278, ). [ABSTRACT FROM AUTHOR]

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated with nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover, we exhibit a class of operators which satisfy our conditions. [ABSTRACT FROM AUTHOR]

In this paper we survey some recent results concerned with nonau-tonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator [ABSTRACT FROM AUTHOR]

Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L( dx) → L( dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein-Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein-Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known. [ABSTRACT FROM AUTHOR]

Given some smooth vector fields X, X,..., X on a compact manifold M, if they satisfy Hörmander's condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations and where a, b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$. We partially generalize the results of Cao and Yau (Math. Z. 211:485-504, ). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove a derivative formula of Bismut–Elworthy–Li’s type as well as a gradient estimate for stochastic differential equations driven by -stable noises, where . As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented. [Copyright &y& Elsevier]

Abstract: We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption where , , and . Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as for , optimal decay estimates as for , or extinction in finite time for . In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton–Jacobi equation. [Copyright &y& Elsevier]

Abstract: For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal -regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level. [Copyright &y& Elsevier]

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol. [ABSTRACT FROM AUTHOR]

Abstract: Let be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation on . We also obtain a theorem of Liouville type for positive solutions of this nonlinear equation. This paper extends the result of Souplet and Zhang [P. Souplet, Qi S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006) 1045–1053]. [Copyright &y& Elsevier]

We find an optimal gradient estimate in Orlicz spaces for a nonlinear elliptic equation in divergence form with discontinuous nonlinearity in a non-smooth domain. [ABSTRACT FROM AUTHOR]

Let ( M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation on M × (0,∞), where a is a constant and q is a C function. This result can be compared with the ones of Ma ( JFA, 241, 374-382 (2006)) and Yang ( PAMS, 136, 4095-4102 (2008)). Also, we obtain Hamilton's gradient estimate for the Schödinger equation. This can be compared with the result of Ruan ( JGP, 58, 962-966 (2008)). [ABSTRACT FROM AUTHOR]

Abstract: In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group . In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a inequality), whereas the operator itself does not satisfy any inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. [Copyright &y& Elsevier]

Given manifolds with a smooth measure ( M, g, e dV), we consider gradient estimates for positive harmonic functions of the drifting Laplacian. If the ∞-Bakry-Emery Ricci tensor is bounded from below and $${|\nabla f|}$$ is bounded, we obtain a Liouville-type theorem. This extends a classical result of Cheng and Yau. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain gradient estimates in Sobolev spaces and Orlicz spaces for weak solutions of parabolic -harmonic equations under the assumptions that satisfies some proper conditions and the given function satisfies some moderate growth condition. [Copyright &y& Elsevier]

We prove space-time parabolic Besov regularity in terms of integrability of Besov norms in the space variable for solutions of the heat equation on cylindrical regions based on Lipschitz domains. [ABSTRACT FROM AUTHOR]

In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in n, n ≥ 2. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions. [ABSTRACT FROM AUTHOR]

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2 n, m), there exists a constant C > 0 such that [ABSTRACT FROM AUTHOR]

Abstract: Let be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation where is a real constant. Our method is based on the refined global gradient estimates for the corresponding evolution equation, which is due to Yau [S.T. Yau, Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett. 2 (1995) 387–399]. We partially generalize the result of Yang [Y.Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008) 4095–4102]. In the particular case of complete Riemannian manifolds with nonnegative curvature, we get a sharp upper bound for the positive solutions; this upper bound is independent of the dimension of the manifolds. [Copyright &y& Elsevier]

The author obtains sharp gradient estimates for the heat kernels in two kinds of higher dimensional Heisenberg groups — the non-isotropic Heisenberg group and the Heisenberg type group ℍ n,m. The method used here relies on the positive property of the Bakry-Émery curvature Γ2 on the radial functions and some associated semigroup technics. [ABSTRACT FROM AUTHOR]

Abstract: For the reflected diffusion generated by on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of of the Poisson equation in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger''s isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided. [Copyright &y& Elsevier]

Abstract: When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the other part blows up. Using the decomposition, we derive the best possible estimates for the blow-up of the gradient. We then consider the case when the inclusions have positive permittivities. We show quantitatively that in this case the size of the blow-up is reduced. [Copyright &y& Elsevier]

Abstract: Let be a family of elliptic differential operators with unbounded coefficients defined in . In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator generates a semigroup of positive contractions in for every , where ν is an infinitesimally invariant measure of . Here, under some additional conditions on the growth of the coefficients of , which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator of in for , and we also give a partial characterization of . Finally, we extend the results so far obtained to the case when the coefficients of the operator are T-periodic with respect to the variable s for some . [Copyright &y& Elsevier]

Abstract: Jackson’s theorem is established in a new kind of holomorphic function space related to measures in any starlike circular domain in . Particularly, the result covers many spaces including , , , and spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem. [Copyright &y& Elsevier]

Abstract: In this paper we generalize classical , , estimates of the gradient to the Orlicz space for weak solutions of quasilinear elliptic equations of p-Laplacian type. [Copyright &y& Elsevier]

Abstract: We aim to prove inequalities of the form for solutions of on a domain , where is the parabolic distance of to parabolic boundary of Ω, is the one-sided Hardy–Littlewood maximal operator in the time variable on , is a Calderón–Scott type d-dimensional elliptic maximal operator in the space variable on the domain D in , and . As a consequence, when D is a bounded Lipschitz domain, we obtain estimates for the norm of in terms of some mixed norm for the space with denotes the Besov norm in the space variable x and where . [Copyright &y& Elsevier]

Abstract: We establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem. We establish gradient estimates and use the fixed point theorem of Caristi’s, which does not require a compactness of the map, to establish the existence result. We provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region. [Copyright &y& Elsevier]