Luisa Moschini

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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Existence and Nonexistence of Solutions of Nonlinear Dirichlet Problems with First Order Terms

(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).

Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients

Abstract A parabolic Harnack inequality for the equation is proved; in particular, this implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence between the Schrödinger operator and the weighted Laplacian when .

Harnack inequality and heat kernel estimates for the Schroedinger operator with Hardy potential

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.