Remarks on Regularity Theorems for Solutions to Elliptic Equations via the Ultracontractivity of the Heat Semigroup

Abstract

In this paper we show how elliptic regularity results can be obtained as a consequence of the ultracontractivity of the underlying heat semigroup. For instance for f ∈ L ^p(Ω) and V ∈ L ¹_loc (Ω) with V ⁻ ∈ L ^q(Ω) and min(p, q)>N/2, if u ∈ H ¹₀ (Ω) satisfies −Δu+Vu=f then, using only the fact that the heat semigroup exp(tΔ) is ultracontractive, that is for t>0 one has $\left\| {\exp \left( {t\Delta } \right)u_0 } \right\|_\infty \leqslant t^{ - N/2} \left\| {u_0 } \right\|_{L^1 }$, one may show easily that u ∈ L ^∞(Ω). The same approach can be used in order to establish regularity results, such as the Hölderianity, or L ^p estimates, for solutions to quite general elliptic equations. Indeed such results are now classical and well known, and our main point here is to present rather elementary proofs using only the maximum principle and the ultracontractivity of the underlying heat semigroup.