We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations.
In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order uniformly elliptic equations with measurable coefficients on divergence form and show that this results in a precompactness result for such sequences.
In this paper we establish commmutator estimates for the Dirichlet-to-Neumann Map associated to a divergence form elliptic operator in the upper half-space $\mathbb{R}^{n+1}_+:=\{(x,t)\in \mathbb{R}^n \times (0,\infty)\}$, with uniformly complex elliptic, $L^{\infty}$, $t$-independent coefficients. By a standard pull-back mechanism, these results extend corresponding results of Kenig, Lin and Shen for the Laplacian in a Lipschitz domain, which have application to the theory of homogenization.
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are define…
The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an applic…
We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^α$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $α\in (-1, \infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direc…
We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_σ (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , ν} (\mathbb{R}^d)$ with $0 \leq ν< 2$.
We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q< p-1$, where $Δ_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $σ$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic …
We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case of solutions that are not Lipschitz continuous at x=0. We show that our estimate is sharp.
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=λ\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^*,q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $λ>0$ is a parameter.
In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with VMO$_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$ estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.
This paper investigates solvability of fully coupled systems of forward-backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.
In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even order larger than 2 with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. We constuct an approximation for the resolvent of $L_\varepsilon$ with the remainder term of order $\varepsilon^2$ in the operator $L^2$-norm. We impose no regularity conditions on the operator beyond ellipticity and boundedness of coefficients. We use two scale expansions with correctors regularized by the Steklov smoothing.
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong unique continuation property. Moreover, we also discuss the unique continuation property from measurable sets, which shows that nodal domains to these equations must have vanishing Lebesgue measure. Our methods rely on suitable Carleman estimates, for which we include the sublinear potential into the main part of the operator.
In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined, and in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point …
