We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.

In this paper, we study the asymptotic behavior of positive solutions of fractional Hardy-Henon equation with an isolated singularity at the origin. We give a classification of isolated singularities of positive solutions near x = 0. Further, we prove the asymptotic behavior of positive singular solutions. These results parallel those known for the Laplacian counterpart proved by Caffarelli, Gidas and Spruck (Caffarelli, Gidas and Spruck in Comm Pure Appl Math, 1981, 1989), but the technique is very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with a blow up (down) argument, the Kelvin transformation and uniqueness of solutions of related degenerate equations on semi spherical surface. We also investigate isolated singularities located at infinity of fractional Hardy-Henon equation.

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This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.

We establish two nonlinear compactness theorems in Lp(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) [1]. The first theorem sharpens a result by Maitre (2003) [10] and is important in the study of doubly nonlinear elliptic–parabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskii˘ (1965) [5], which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type bead–spring chains model).

We consider a L 2-contraction (a L 2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L 2-norm, then the L 2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the L-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L 2-contraction holds

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

We consider the problem of the random fluctuations in the solutions to elliptic PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not square integrable locally), the leading corrector to the averaged solution may be either deterministic or random, or both in a sense we shall explain. Our main application is the solution of an elliptic problem with random Robin boundary condition that may be used to model diffusion of signaling molecules through a layer of cells into a bulk of extracellular medium. The problem is then described by an elliptic pseudo-differential operator (a Dirichlet-to-Neumann operator) on the boundary of the domain with random potential. In the physical setting of a three dimensional extracellular medium on top of a two-dimensional surface of cells forming a layer of epithelium, we show that the approximate corrector to averaging consists of a deterministic correction plus a Gaussian field of amplitude proportional to the correlation length of the random medium. The result is obtained under some assumptions on the four-point correlation function in the medium. We provide examples of such random media based on Gaussian and Poisson statistics.

For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.