Achilles Tertikas

A unified approach to improved ^{} Hardy inequalities with best constants

We present a unified approach to improved L p Hardy inequalities in R N . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension 1 < k < N. In our main result, we add to the right hand side of the classical Hardy inequality a weighted L p norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted L q norms, q ¬= p.

Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics

We discuss a selection-migration model in population genetics, involving two alleles A1 and A2 such that A1 is at an advantage over A2 in certain subregions and at a disadvantage in others. It is shown that if A1 is at an overall disadvantage to A2 and the rate of gene flow is sufficiently large than A1 must die out; on the other hand, if the two alleles are in some sense equally advantaged overall, then A1 and A2 can coexist no matter how great the rate of gene flow.

Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent

On the Hardy Constant of Some Non-convex Planar Domains

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Improving L2 estimates to Harnack inequalities

(0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L2 setting to an open problem raised by Frank and Seiringer (“Sharp fractional Hardy inequalities in half-spaces,” in Around the research of Vladimir Maz’ya. International Mathematical Series, 2010).

OPTIMIZING IMPROVED HARDY INEQUALITIES FOR THE BIHARMONIC OPERATOR

This paper considers a semilinear elliptic equation which arises in a selection-migration model in population genetics, involving two alleles

Global bifurcation analysis and uniqueness for a semilinear problem

A_1

A global branch of positive solutions above the continuous spectrum for problems with indefinite nonlinearities

and