In subdomains of $\mathbb{R}^{d}$ we consider uniformly elliptic equations $H\big(v( x),D v( x),D^{2}v( x), x\big)=0$ with the growth of $H$ with respect to $|Dv|$ controlled by the product of a function from $L_{d}$ times $|Dv|$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_{0}\in(d/2,d)$ such that for any $p\in(d_{0},d)$ the equation with prescribed continuous boundary data has a solution in class $W^{2}_{p,\text{loc}}$. Our results are new even if $H$ is linear.
We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of "natural constraint"
We study existence of principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term via the Krein-Rutman theorem which based on regularity up to boundary of viscosity solutions. We also show the simplicity of the eigenfunctions in viscosity sense by a nonlocal version of ABP estimate.
We prove global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order differential operators in $\mathbb{R}^d$ with unbounded diffusion, drift and potential terms.
We prove the one-dimensional symmetry of solutions to elliptic equations of the form $-\dive(e^{G(x)}a(|\nabla u|)\nabla u)=f(u) e^{G(x)}$, under suitable energy conditions. Our results hold without any restriction on the dimension of the ambient space.
In this note, we obtain a version of Aleksandrov's maximum principle when the drift coefficients are in Morrey spaces, which contains $L_d$, and when the free term is in $L_p$ for some $p<d$.
We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity.
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.
We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class containing $W^{2}_{p,loc}$. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian.
We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-Δu + b^{(α)}\cdot \nabla u= f$ in a bounded domain $Ω\subset {\mathbb R^2}$ containing the origin, where $f \in W^{-1}_q(Ω) $ with $q>2$ and $b^{(α)}:=b-α\frac{x}{|x|^2}$, $b$ is a divergence-free vector field and $α\in {\mathbb R}$ is a parameter.
We consider uniformly elliptic and parabolic second-order equations with bounded zeroth-order and bounded VMO leading coefficients and possibly growing first-order coefficients. We look for solutions which are summable to the $p$-th power with respect to the usual Lebesgue measure along with their first and second-order derivatives with respect to the spatial variable.
Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.
We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d}$, and $c\geq0$ in a Morrey class containing $L_{d/2}$. We prove the solvability in Sobolev spaces of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, and of $λu-Lu=f$ in the whole space for any $λ>0$. Weak uniqueness of the martingale problem associated with such operators is also discussed.
We discuss the local properties of weak solutions to the equation $-Δu + b\cdot\nabla u=0$. The corresponding theory is well-known in the case $b\in L_n$, where $n$ is the dimension of the space. Our main interest is focused on the case $b\in L_2$. In this case the structure assumption $\operatorname{div} b=0$ turns out to be crucial.
This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coe…
For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $ω(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}ω(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamenta…
We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } Ω, \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(Ω)^n$ has non-negative weak divergence in $Ω$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(Ω)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an …
We investigate the uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of elliptic equations with a drift posed on a complete, noncompact, Riemannian manifold $M$ of infinite volume and dimension $N\ge2$. Furthermore, in the special case of a model manifold with polynomial volume growth, we show that the conditions on the drift term are sharp.
We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of the elliptic operator $A=(1+|x|^α)Δ+b|x|^{α-1}\frac{x}{|x|}\cdot \nabla-c|x|^β$ with domain $D(A_p) =\{ u \in W^{2,p}(\mathbb{R}^N)\, |\, Au \in L^p(\mathbb{R}^N)\}$ generates a strongly continuous analytic semigroup $T(\cdot)$ provided that $α>2,\,β>α-2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A. Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in [G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show that $T(\cdot)$ is…
We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - Δ+b\cdot\nabla$ and $-Δ+b\cdot\nabla$ with a divergence-free drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BM…
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space $L^{n,\infty}(Ω; \mathbb{R}^n )$. First, assuming that the drift $\mathbf{b}$ has nonnegative weak divergence in $L^{n/2, \infty }(Ω)$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Ω)$ or $D^{1,p}(Ω)$ for any $p$ with $n' = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+\varepsilon}$ or $W^{2, …
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+λu=f+\mathrm{div } \mathbf{F}\quad \text{in } Ω\quad\text{and}\quad u=0\quad \text{on } \partialΩ, \end{equation*} in bounded Lipschitz domain $Ω$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b} : Ω\rightarrow \mathbb{R}^2$, and $λ\geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $Ω$ has small Lipschitz constant, and $\mathrm{div } A,\,\mathbf{b} \in L^{2}(Ω;…
In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order $α< 1$ in a bounded smooth domain $Ω$ . The main difficulty arises because of losses of the boundary condition for sub and supersolutions due to the lower diffusive effect of the elliptic operator compared with the drift term. We consider the notion of viscosity solution with generalized boundary conditions, concluding strong comparison principles in $\barΩ$ under rather general assumptions over the drift term. As a consequence, existence and uniqueness of …
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain $Ω$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\mathbf{b}:Ω\rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b} \in L^{n}(Ω)^n$, we first establish existence and uniqueness of solutions in $L_α^{p}(Ω)$ for the Dirichlet and Neumann problems. Here $L_α^{p}(Ω)$ denotes the Sobolev space (or Bessel potential space) with the pair $(α,p…
In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $Δu+W\cdot\nabla u=0$ in ${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$ for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty}({\mathbf R}^2)}\le C_0$ and satisfies certain a priori assumption at $0$. Then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge \exp(-C_1R^{1-2/p}\log R)\quad\text{if}\quad 2<p<\infty \] and \[ \inf_…
In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=νΔ+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;ξ,τ)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension…
In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^α)Δ+b|x|^{α-1}\frac{x}{|x|}\cdot\nabla -|x|^β$ satisfies $$ k(t,x,y) \leq c_1e^{λ_0 t+ c_2t^{-γ}}\left(\frac{1+|y|^α}{1+|x|^α}\right)^{\frac{b}{2α}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(β-α)}}{1+|y|^α} e^{-\frac{\sqrt{2}}{β-α+2}\left(|x|^{\frac{β-α+2}{2}}+ |y|^{\frac{β-α+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $λ_0$ is the largest eigenvalue of the operator $A$, and $γ=\frac{β-α+2}{β+α-2}$, in the case where $N>2,\,α>2$ and $β>α-2$. The …
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
We prove global Holder estimates for solution of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions on the domain a' la Cabre'.
We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $Q\subset \mathbb{R}^{n+1}$. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift $b$ is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is…
Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained using symmetrization techniques.
We prove Holder regularity for solutions of non divergence integro-differential equations with non necessarily even kernels. The even/odd decomposition of the kernel can be understood as a sum of a diffusion and a drift term. In our case we assume that such drift have the order smaller than or equal to the diffusion and at least one. For example we can say something about the following equation $Δ^{1/2}u + |Du| = f$. The main step relies in a localized version of the Aleksandrov-Bakelman-Pucci estimate. Our estimates are also uniform as the order of the equation goes to two.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
