Regularity results for degenerate elliptic equations related to Gauss measure

Abstract

Abstract. In this paper we study a Dirichlet problem relative to the equationwhere L is a linear elliptic operator with lower-order terms whose ellipticity condition is given in terms of the function ϕ(x) = (2π) − n 2 exp − |x| 2 /2 , the density of the Gaussian measure.

Parabolic equations related to Boltzmann measure

“…The following imbending theorem is a straight consequence of the Sobolev logarithmic inequalities (2.5). A proof can be obtained as in [9] using properties of rearrangements of functions.…”

“…Moreover, by Logaritmic Sobolev inequality we have that the solution u ∈ L 2 (log L) 1 2 (µ, ). Proceeding as in [9] we can study how the summability of the solution u of problem (3.1) improves by improving the summability of the data in the Lorentz-Zygmund spaces. THEOREM 3.6.…”

“…We remark that even if we improve the summability of the data, we cannot have conditions that guarantee the boundedness of the solution to problem (3.1) (see [9] for an example). REMARK 3.8.…”

“…REMARK 3.8. In the framework of this section and of papers [9] and [10] we can prove comparison and regularity results when problem (3.1) has all lower order terms and also for nonlinear problem.…”

“…Existence and regularity results for solutions to degenerate elliptic equations related to Gauss measure has been proved in [6,[9][10][11] and [12]; parabolic case has been studied in [7] and the eigenvalues problem is considered in [5].…”