Gradient estimates and comparison principle for some nonlinear elliptic equations

Abstract

We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators

On the Problem of Maximal $$L^q$$-regularity for Viscous Hamilton–Jacobi Equations

“…There is a vast literature on such equations and more general quasilinear problems. While the existence of classical (or strong) solutions was investigated first (see for example [3,26,29,38]), the attention has since been largely focused on the existence (and uniqueness) of solutions u ∈ W 1,γ (Q) satisfying (1) in the weak or generalized sense (typically with Dirichlet boundary conditions; see, for example, [1,6,[8][9][10]14,18,22,35] and more recent works [2,7,16,17]. It has been observed that due to the superlinear nature of the problem, its (weak) solvability requires f ∈ L q , where…”

On the optimal $L^q$-regularity for viscous Hamilton-Jacobi equations with sub-quadratic growth in the gradient

“…The borderline summability threshold q = d γ ′ is in general dealt with smallness assumptions, see e.g. [32, Theorem 1.1] for the case of integrability estimates or even [3,8] for the case of data in Lorentz spaces. When instead λ > 0, which can be regarded as the closest regime to the parabolic framework, one can encompass even the borderline case q = d γ ′ , where, however, the constant of the estimate does not depend only on f L q , but remains bounded when f varies in a set which is bounded and equi-integrable in L q , see Remark 4.6.…”

“…Then, d γ ′ > 2d d+2 precisely when γ > d+2 d , which is indeed the critical threshold guaranteeing the validity of the energy formulation of the problem, see also [32]. One expects to address the case below γ = d+2 d using different techniques and notion of solutions, cf [8] and the references therein. In this section, we denote by L the Lagrangian of H, defined as its Legendre transform, i.e.…”

“…We refer to [31,21,22,47] and the references therein for further developments and earlier results through this nonlinear duality method. It is worth pointing out that some maximal regularity results have been already obtained for "small values of γ" in [8] (precisely when γ ≤ d+2 2 , which prevents from the use of energy formulations), when the source f belongs to finer Lorentz classes without zero-th order terms, see also the references therein and Remark 5.2. Still, a quite general regularity theory on spaces of maximal regularity for such equations, mostly for systems of Hamilton-Jacobi-Bellman equations with bounded data and first-order terms having at most quadratic growth, can be found in [7].…”

Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

“…Let us mention that in the elliptic framework such a kind of results have been studied in several papers using different techniques. Let us recall the papers [3], [4] [10], [11], [12], [24] (and references cited therein) where unbounded solutions for quasilinear equations have been treated. We want also to highlight the results of [28] (see also [6], [22] and [21]) that have inspired our work, where the comparison principle among unbounded sub/supersolutions has been proved, for sub/supersolutions that have a suitable power that belongs to the energy space.…”