A Refined Subsolution Estimate of Weak Subsolutions to Second Order Linear Elliptic Equations with a Singular Vector Field

Abstract

We consider second order linear elliptic equations −div(A(x)∇u) + b(x) • ∇u = 0 with a singular vector field b. We prove a refined subsolution estimate, which contains a precise dependence of the quantities of b, for weak subsolutions and a weak Harnack inequality for weak supersolutions under certain assumptions on b.

Sharp solvability for singular SDEs

“…Similarly to these papers, we apply a tightness argument to construct a martingale solution once Proposition 2 is established, see Section 2.2. We also refer to Hara [5] for the proof of Hölder continuity of solutions to elliptic equations with b ∈ F δ , δ < 1 using Moser's method in L 2 . Finally, we note that passing to the L p variant of De Giorgi's method does not exclude other singular drift perturbations known to be amenable in L 2 (cf.…”

“…Finally, we note that passing to the L p variant of De Giorgi's method does not exclude other singular drift perturbations known to be amenable in L 2 (cf. [5,18]). For instance, the assertion of Theorem 1 is also valid for…”

Accretivity of the general second order linear differential operator

“…XI.7], Corollary 6.1, Theorems 6.2 and 7.2). See also [9], [22], as well as the discussion in Sec. 2.4 and Sec.…”

“…Both the accretivity and form boundedness problems have numerous applications, including mathematical quantum mechanics ( [29], [30]), elliptic and parabolic PDE with singular coefficients ( [5], [7], [14], [15], [20], [26], [9], [27]), fluid mechanics and Navier-Stokes equations ( [8], [16], [31], [33]), semigroups and Markov processes ( [17]), homogenization theory ( [34]), harmonic analysis ( [4], [6]), etc.…”

Regularity for parabolic equations with singular non-zero divergence vector fields

“…Also, considering a form-bounded b but requiring additionally the form-boundedness of the derivatives of a ij , which still allows a to have critical discontinuities, one obtains the existence of a unique weak solution to the corresponding SDE [KS5,KM]. In general, there is a quantitative dependence of the regularity properties of solutions to (1), (2) on the value of the form-bound δ, see [KS1,H]. The form-bound δ arises as a key integral characteristic of the vector field b responsible for the regularity theory of (1) and (2).…”