Springer Science and Business Media LLC

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain $$D \subset \mathbb {R}^d$$ with $$d \ge 2$$ , for every (measurable) uniformly elliptic tensor field a and for almost every point $$y \in D$$ , there exists a unique Green’s function centred in y associated to the vectorial operator $$-\nabla \cdot a\nabla $$ in D. This result implies the existence of the fundamental solution for elliptic systems when $$d>2$$ , i.e. the Green function for $$-\nabla \cdot a\nabla $$ in $$\mathbb {R}^d$$ . In the second part, we introduce a shift-invariant ensemble $$\langle \cdot \rangle $$ over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for $$\langle |G(\cdot ; x,y)|\rangle $$ , $$\langle |\nabla _x G(\cdot ; x,y)|\rangle $$ and $$\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle $$ . These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.

In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the $$L^\infty $$ bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer–Fučík spectrum.

In this paper, we show the N-dimensional Rellich-Leray inequality with optimal constant for axisymmetric and divergence-free vector fields. This is a second-order differential version of the former work by Costin-Maz’ya (Calc Var Partial Differ Equ 32(4):523–532, 2008) on sharp Hardy–Leray inequality for such vector fields. In the proof of our main theorem, we show the vanishing of azimuthal components of axisymmetric vector fields for $$N\ge 4$$ , from which we also find a partial modification of the best constant derived in Costin-Maz’ya (Calc Var Partial Differ Equ 32(4):523–532, 2008).

We prove a compactness result with respect to $$\Gamma $$ -convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the $$\Gamma $$ -limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.

In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.