Upper bounds for the eigenvalues of Hessian equations

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equationsComment: 15 pages, 1 figur

Sobolev Regularity for Monge-Ampere Type Equations

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W2,1+\u3baloc for some \u3ba>0. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp\`ere equation with right hand side bounded away from zero and infinity

Collapsing Shells and the Isoperimetric Inequality for Black Holes

Recent results of Trudinger on Isoperimetric Inequalities for non-convex bodies are applied to the gravitational collapse of a lightlike shell of matter to form a black hole. Using some integral identities for co-dimension two surfaces in Minkowski spacetime, the area $A$ of the apparent horizon is shown to be bounded above in terms of the mass $M$ by the $16 \pi G^2 M^2$, which is consistent with the Cosmic Censorship Hypothesis. The results hold in four spacetime dimensions and above.Comment: 16 pages plain TE

Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto

Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.Comment: 12 pages, Corrected typo

Coupling carbon allocation with leaf and root phenology predicts tree-grass partitioning along a savanna rainfall gradient

© Author(s) 2016. The relative complexity of the mechanisms underlying savanna ecosystem dynamics, in comparison to other biomes such as temperate and tropical forests, challenges the representation of such dynamics in ecosystem and Earth system models. A realistic representation of processes governing carbon allocation and phenology for the two defining elements of savanna vegetation (namely trees and grasses) may be a key to understanding variations in tree-grass partitioning in time and space across the savanna biome worldwide. Here we present a new approach for modelling coupled phenology and carbon allocation, applied to competing tree and grass plant functional types. The approach accounts for a temporal shift between assimilation and growth, mediated by a labile carbohydrate store. This is combined with a method to maximize long-term net primary production (NPP) by optimally partitioning plant growth between fine roots and (leaves + stem). The computational efficiency of the analytic method used here allows it to be uniquely and readily applied at regional scale, as required, for example, within the framework of a global biogeochemical model. We demonstrate the approach by encoding it in a new simple carbon-water cycle model that we call HAVANA (Hydrology and Vegetation-dynamics Algorithm for Northern Australia), coupled to the existing POP (Population Orders Physiology) model for tree demography and disturbance-mediated heterogeneity. HAVANA-POP is calibrated using monthly remotely sensed fraction of absorbed photosynthetically active radiation (fPAR) and eddy-covariance-based estimates of carbon and water fluxes at five tower sites along the North Australian Tropical Transect (NATT), which is characterized by large gradients in rainfall and wildfire disturbance. The calibrated model replicates observed gradients of fPAR, tree leaf area index, basal area, and foliage projective cover along the NATT. The model behaviour emerges from complex feedbacks between the plant physiology and vegetation dynamics, mediated by shifting above- versus below-ground resources, and not from imposed hypotheses about the controls on tree-grass co-existence. Results support the hypothesis that resource limitation is a stronger determinant of tree cover than disturbance in Australian savannas

On the Regularity of Optimal Transportation Potentials on Round Spheres

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure

Harnack inequality and regularity for degenerate quasilinear elliptic equations

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,\alpha}$ local estimates for solutions of a degenerate equation in non divergence form

Low atmospheric CO2 levels during the Little Ice Age due to cooling-induced terrestrial uptake

Low atmospheric carbon dioxide (CO2) concentration during the Little Ice Age has been used to derive the global carbon cycle sensitivity to temperature. Recent evidence confirms earlier indications that the low CO2 was caused by increased terrestrial carbon storage. It remains unknown whether the terrestrial biosphere responded to temperature variations, or there was vegetation re-growth on abandoned farmland. Here we present a global numerical simulation of atmospheric carbonyl sulfide concentrations in the pre-industrial period. Carbonyl sulfide concentration is linked to changes in gross primary production and shows a positive anomaly during the Little Ice Age. We show that a decrease in gross primary production and a larger decrease in ecosystem respiration is the most likely explanation for the decrease in atmospheric CO2 and increase in atmospheric carbonyl sulfide concentrations. Therefore, temperature change, not vegetation re-growth, was the main cause of the increased terrestrial carbon storage. We address the inconsistency between ice-core CO2 records from different sites measuring CO2 and δ13CO2 in ice from Dronning Maud Land (Antarctica). Our interpretation allows us to derive the temperature sensitivity of pre-industrial CO2 fluxes for the terrestrial biosphere (γL = -10 to -90 Pg C K-1), implying a positive climate feedback and providing a benchmark to reduce model uncertainties

Coupling carbon allocation with leaf and root phenology predicts tree-grass partitioning along a savanna rainfall gradient

Abstract. The relative complexity of the mechanisms underlying savanna ecosystem dynamics, in comparison to other biomes such as temperate and tropical forests, challenges the representation of such dynamics in ecosystem and Earth system models. A realistic representation of processes governing carbon allocation and phenology for the two defining elements of savanna vegetation (namely trees and grasses) may be a key to understanding variations in tree/grass partitioning in time and space across the savanna biome worldwide. Here we present a new approach for modelling coupled phenology and carbon allocation, applied to competing tree and grass plant functional types. The approach accounts for a temporal shift between assimilation and growth, mediated by a labile carbohydrate store. This is combined with a method to maximise long-term net primary production (NPP) by optimally partitioning plant growth between fine roots and (leaves + stem). The computational efficiency of the analytic method used here allows it to be uniquely and readily applied at regional scale, as required, for example, within the framework of a global biogeochemical model. We demonstrate the approach by encoding it in a new simple carbon/water cycle model that we call HAVANA (Hydrology and Vegetation-dynamics Algorithm for Northern Australia), coupled to the existing POP (Population Orders Physiology) model for tree demography and disturbance-mediated heterogeneity. HAVANA-POP is calibrated using monthly remotely-sensed fraction of absorbed photosynthetically active radiation (fPAR) and eddy-covariance-based estimates of carbon and water fluxes at 5 tower sites along the Northern Australian Tropical Transect (NATT), which is characterized by large gradients in rainfall and wildfire disturbance. The calibrated model replicates observed gradients of fPAR, tree leaf area index, basal area and foliage projective cover along the NATT. The model behaviour emerges from complex feed-backs between the plant physiology and vegetation dynamics, mediated by shifting above- vs. below-ground resources, and not from imposed hypotheses about the controls on tree/grass co-existence. Results support the hypothesis that resource limitation is a stronger determinant of tree cover than disturbance in Australian savannas. </jats:p