Stathis Filippas

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.

On the Blowup of Multidimensional Semilinear Heat Equations

Abstract This work is concerned with positive, blowing-up solutions of the semilinear heat equation ut − ∆u = up in Rn. No symmetry assumptions are made. Working with the equation in similarity variables, we first prove a result suggested by center manifold theory. We then calculate the refined asymptotics for u in a backward space-time parabola near a blowup point, and we obtain some information about the local structure of the blowup set. Our results suggest that in space dimension n, among solutions that follow the center manifold, there are exactly n different blowup patterns.

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

Improving L2 estimates to Harnack inequalities

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Refined asymptotics for the blowup of ut — δu = up

(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).

Critical Hardy-Sobolev inequalities

We consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrodinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometrical optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the semiclassical Wigner function which is a formal asymptotic approximation of the scaled Wigner function but also a regularization of the limit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner function. This representation is shown to be exact in the context of concrete examples. In these examples we compute both the semiclassical and the limit Wigner function, as well as the amplitude of the wave field near a fold or a cusp caustic, which evolve naturally from suitable initial data.

Series expansion for L p Hardy inequalities

We consider operators of the form L = −L − V , where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domainIR n with Dirichlet boundary conditions. We allow the boundary of to be made of various pieces of different codimension. We assume that L has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator L, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.