14:00h – 14:40h

Heat semigroup, Sobolev inequalities and Regularity theorems for Elliptic PDEs

A semigroup (S(t))t≥0 acting in the spaces Lp(RN ) for 1 ≤ p ≤ ∞ is said
to be ultra-contractive if for some exponent a > 0 and a constant c > 0 one has for all t > 0 and all u0 ∈ L1(RN )

//S(t)u0//L2 ≤ c t^−a//S(t)u0//L1 .

We present a few results connecting the ultra-contractivity of the heat
semigroup S(t) := e^t∆ (where ∆ is the classical Laplacian in RN ) with theproof of regularity theorems for elliptc PDEs as well as the proof of Sobolev inequalities.
These proofs can be easily extended to the case of more general heat
semigroups S(t) := e^−tL generated by elliptic operators in the divergence form,

Lu := −div(a(x)∇u) + b(x) · ∇u + div(cu) + V u

acting in L2(Ω) with Ω ⊂ RN and D(L) := {u ∈ H10(Ω) ; Lu ∈ L2(Ω)}.

14:50h – 15:30h

Some recent results on weakly coupled elliptic systems